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{{DISPLAYTITLE:Differential Logic and Dynamic Systems}}
{{DISPLAYTITLE:Differential Logic and Dynamic Systems}}
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| '''''NOTE.''' The current version of this document is '''[[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems 2.0]].'''''
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| align="left" | ''Stand and unfold yourself.''
| align="right" | Hamlet: Francsico—1.1.2
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This article develops a differential extension of [[propositional calculus]] and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.
This article develops a differential extension of [[propositional calculus]] and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.
The briefest expression for logical truth is the empty word, usually denoted by ε or λ in formal languages, where it forms the identity element for concatenation. To make it visible in this text, I denote it by the equivalent expression "(())", or, especially if operating in an algebraic context, by a simple "1". Also when working in an algebraic mode, I use the plus sign "+" for exclusive disjunction. Thus, we may express the following paraphrases of algebraic forms:
The briefest expression for logical truth is the empty word, usually denoted by ε or λ in formal languages, where it forms the identity element for concatenation. To make it visible in this text, I denote it by the equivalent expression "(())", or, especially if operating in an algebraic context, by a simple "1". Also when working in an algebraic mode, I use the plus sign "+" for exclusive disjunction. Thus, we may express the following paraphrases of algebraic forms:
:{|
:{| cellpadding="4"
| ''A'' + ''B'' || = || (''A'', ''B'')
| ''A'' + ''B''
| =
| (''A'', ''B'')
|-
|-
| ''A'' + ''B'' + ''C'' || = || ((''A'', ''B''), ''C'') || = || (''A'', (''B'', ''C''))
| ''A'' + ''B'' + ''C''
| =
| ((''A'', ''B''), ''C'')
| =
| (''A'', (''B'', ''C''))
|}
|}
One should be careful to observe that these last two expressions are not equivalent to the form (''A'', ''B'', ''C'').
One should be careful to observe that these last two expressions are not equivalent to the form (''A'', ''B'', ''C'').
<font face="courier new">
<font face="courier new">
==A Functional Conception of Propositional Calculus==
==A Functional Conception of Propositional Calculus==
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Out of the dimness opposite equals advance . . . .<br>
Always substance and increase,<br>
Always substance and increase,<br>
Always a knit of identity . . . . always distinction . . . .<br>
Always a knit of identity . . . . always distinction . . . .<br>
always a breed of life.</p>
always a breed of life.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
|}
In the general case, we start with a set of logical features {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>} that represent properties of objects or propositions about the world. In concrete examples the features {''a''<sub>''i''</sub>} commonly appear as capital letters from an ''alphabet'' like {''A'', ''B'', ''C'', …} or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>} as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.
In the general case, we start with a set of logical features {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>} that represent properties of objects or propositions about the world. In concrete examples the features {''a''<sub>''i''</sub>} commonly appear as capital letters from an ''alphabet'' like {''A'', ''B'', ''C'', …} or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>} as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.
===Qualitative Logic and Quantitative Analogy===
===Qualitative Logic and Quantitative Analogy===
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''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.
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| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56]
|}
These concepts and notations can now be explained in greater detail. In order to begin as simply as possible, I distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis, I take spaces like '''B''', '''B'''<sup>''n''</sup>, and ('''B'''<sup>''n''</sup> → '''B''') at face value and treat them as the primary objects of interest. On the second level of analysis, I use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.
These concepts and notations can now be explained in greater detail. In order to begin as simply as possible, I distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis, I take spaces like '''B''', '''B'''<sup>''n''</sup>, and ('''B'''<sup>''n''</sup> → '''B''') at face value and treat them as the primary objects of interest. On the second level of analysis, I use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.
===Philosophy of Notation : Formal Terms and Flexible Types===
===Philosophy of Notation : Formal Terms and Flexible Types===
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Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.
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| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation ''f''<sup>–1</sup> ⊆ '''B''' × '''B'''<sup>''n''</sup>, or what is the same thing, ''f''<sup>–1</sup> : '''B''' → ''Pow''('''B'''<sup>''n''</sup>), and the ''fibers'' or inverse images ''f''<sup>–1</sup>(0) and ''f''<sup>–1</sup>(1), associated with each boolean function ''f'' : '''B'''<sup>''n''</sup> → '''B''' that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets ''f''<sup>–1</sup>(''b''), for ''b'' ∈ '''B''', is part and parcel of understanding the denotative uses of each propositional function ''f''.
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation ''f''<sup>–1</sup> ⊆ '''B''' × '''B'''<sup>''n''</sup>, or what is the same thing, ''f''<sup>–1</sup> : '''B''' → ''Pow''('''B'''<sup>''n''</sup>), and the ''fibers'' or inverse images ''f''<sup>–1</sup>(0) and ''f''<sup>–1</sup>(1), associated with each boolean function ''f'' : '''B'''<sup>''n''</sup> → '''B''' that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets ''f''<sup>–1</sup>(''b''), for ''b'' ∈ '''B''', is part and parcel of understanding the denotative uses of each propositional function ''f''.
Among the <math>2^{2^n}</math> propositions or functions in ('''B'''<sup>''n''</sup> → '''B''') are several fundamental sets of 2<sup>''n''</sup> propositions each that take on special forms with respect to a given basis <font face="lucida calligraphy">A</font> = {''a''<sub>''i''</sub>}. Three of these forms are especially common, the ''linear'', the ''positive'', and the ''singular''
Among the <math>2^{2^n}</math> propositions or functions in ('''B'''<sup>''n''</sup> → '''B''') are several fundamental sets of 2<sup>''n''</sup> propositions each that take on special forms with respect to a given basis <font face="lucida calligraphy">A</font> = {''a''<sub>''i''</sub>}. Three of these forms are especially common, the ''linear'', the ''positive'', and the ''singular''
propositions. Each set is naturally parameterized by the coordinate vectors in '''B'''<sup>''n''</sup> and falls into ''n''+1 ranks, with a binomial coefficient ''C''(''n'', ''k'') giving the number of propositions that have rank or weight ''k''.
propositions. Each set is naturally parameterized by the coordinate vectors in '''B'''<sup>''n''</sup> and falls into ''n''+1 ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight ''k''.
The ''linear propositions'', {hom : '''B'''<sup>''n''</sup> → '''B'''} = ('''B'''<sup>''n''</sup> <font face=symbol>'''+>'''</font> '''B'''), may be expressed as sums of the following form:
The ''linear propositions'', {hom : '''B'''<sup>''n''</sup> → '''B'''} = ('''B'''<sup>''n''</sup> <font face=symbol>'''+>'''</font> '''B'''), may be expressed as sums of the following form:
===The Analogy Between Real and Boolean Types===
===The Analogy Between Real and Boolean Types===
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Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.
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| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}
There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.
There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.
===Theory of Control and Control of Theory===
===Theory of Control and Control of Theory===
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You will hardly know who I am or what I mean,<br>
But I shall be good health to you nevertheless,<br>
But I shall be good health to you nevertheless,<br>
And filter and fibre your blood.</p>
And filter and fibre your blood.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}
In the boolean context, a function ''f'' : ''X'' → '''B''' is tantamount to a ''proposition'' about elements of ''X'', and the elements of ''X'' constitute the ''interpretations'' of that proposition. The fiber ''f''<sup>–1</sup>(1) comprises the set of ''models'' of ''f'', or examples of elements in ''X'' satisfying the proposition ''f''. The fiber ''f''<sup>–1</sup>(0) collects the complementary set of ''anti-models'', or the exceptions to the proposition ''f'' that exist in ''X''. Of course, the space of functions (''X'' → '''B''') is isomorphic to the set of all subsets of X, called the ''power set'' of ''X'' and often denoted as <font face="lucida calligraphy">Pow</font>(''X'') or 2<sup>''X''</sup>.
In the boolean context, a function ''f'' : ''X'' → '''B''' is tantamount to a ''proposition'' about elements of ''X'', and the elements of ''X'' constitute the ''interpretations'' of that proposition. The fiber ''f''<sup>–1</sup>(1) comprises the set of ''models'' of ''f'', or examples of elements in ''X'' satisfying the proposition ''f''. The fiber ''f''<sup>–1</sup>(0) collects the complementary set of ''anti-models'', or the exceptions to the proposition ''f'' that exist in ''X''. Of course, the space of functions (''X'' → '''B''') is isomorphic to the set of all subsets of X, called the ''power set'' of ''X'' and often denoted as <font face="lucida calligraphy">Pow</font>(''X'') or 2<sup>''X''</sup>.
===Reality at the Threshold of Logic===
===Reality at the Threshold of Logic===
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But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.
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| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7]
|}
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X'' = 〈''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>〉 <math>\cong</math> '''R'''<sup>''n''</sup>. The coordinate
For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X'' = 〈''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>〉 <math>\cong</math> '''R'''<sup>''n''</sup>. The coordinate
system <font face=lucida calligraphy">X</font> = {''x''<sub>''i''</sub>} is a set of maps ''x''<sub>''i''</sub> : '''R'''<sub>''n''</sub> → '''R''', also known as the coordinate projections. Given a "dataset" of points ''x'' in '''R'''<sub>''n''</sub>, there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each ''i'' we choose an ''n''-ary relation ''L''<sub>''i''</sub> on '''R''', that is, a subset of '''R'''<sub>''n''</sub>, and then we define the ''i''<sup>th</sup> threshold map, or ''limen'' <u>''x''</u><sub>''i''</sub> as follows:
system <font face=lucida calligraphy">X</font> = {''x''<sub>''i''</sub>} is a set of maps ''x''<sub>''i''</sub> : '''R'''<sup>''n''</sup> → '''R''', also known as the coordinate projections. Given a "dataset" of points ''x'' in '''R'''<sup>''n''</sup>, there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each ''i'' we choose an ''n''-ary relation ''L''<sub>''i''</sub> on '''R''', that is, a subset of '''R'''<sup>''n''</sup>, and then we define the ''i''<sup>th</sup> threshold map, or ''limen'' <u>''x''</u><sub>''i''</sub> as follows:
: <u>''x''</u><sub>''i''</sub> : '''R'''<sub>''n''</sub> → '''B''' such that:
: <u>''x''</u><sub>''i''</sub> : '''R'''<sup>''n''</sup> → '''B''' such that:
: <u>''x''</u><sub>''i''</sub>(''x'') = 1 if ''x'' ∈ ''L''<sub>''i''</sub>,
: <u>''x''</u><sub>''i''</sub>(''x'') = 1 if ''x'' ∈ ''L''<sub>''i''</sub>,
Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''<sub>''i''</sub>} and the coordinate axes corresponding to {''x''<sub>''i''</sub>}, aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''<sub>''i''</sub> is bounded by some hyperplane that intersects the ''i''<sup>th</sup> axis at a unique threshold value ''r''<sub>''i''</sub> ∈ '''R'''. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set ''L''<sub>''i''</sub> has points on the ''i''<sup>th</sup> axis, that is, points of the form ‹0, …, 0, ''r''<sub>''i''</sub>, 0, …, 0› where only the ''x''<sub>''i''</sub> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''<sub>''i''</sub>} and the coordinate axes corresponding to {''x''<sub>''i''</sub>}, aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''<sub>''i''</sub> is bounded by some hyperplane that intersects the ''i''<sup>th</sup> axis at a unique threshold value ''r''<sub>''i''</sub> ∈ '''R'''. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set ''L''<sub>''i''</sub> has points on the ''i''<sup>th</sup> axis, that is, points of the form ‹0, …, 0, ''r''<sub>''i''</sub>, 0, …, 0› where only the ''x''<sub>''i''</sub> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''<sup>''n''</sup> can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font> = {<u>''x''</u><sub>''i''</sub>} as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''<sup>th</sup> threshold map. This can
States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''<sup>''n''</sup> can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font> = {<u>''x''</u><sub>''i''</sub>} as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''<sup>th</sup> threshold map. This can help to remind us that the ''threshold operator'' (<u> </u>)<sub>''i''</sub> acts on ''x'' by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition <u>''x''</u><sub>''i''</sub> asserts that the representative point ''x'' resides ''above'' the ''i''<sup>th</sup> threshold.
help to remind us that the ''threshold operator'' <u> </u>)<sub>''i''</sub> acts on ''x'' by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition <u>''x''</u><sub>''i''</sub> asserts that the representative point ''x'' resides ''above'' the ''i''<sup>th</sup> threshold.
Primitive assertions of the form <u>''x''</u><sub>''i''</sub>(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems. Parentheses "( )" may be used to indicate negation. Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "( , , , )", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <u>''X''</u> = 〈<font face="lucida calligraphy"><u>X</u></font>〉 <math>\cong</math> '''B'''<sup>''n''</sup>, and
Primitive assertions of the form <u>''x''</u><sub>''i''</sub>(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems. Parentheses "( )" may be used to indicate negation. Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "( , , , )", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <u>''X''</u> = 〈<font face="lucida calligraphy"><u>X</u></font>〉 <math>\cong</math> '''B'''<sup>''n''</sup>, and
===Tables of Propositional Forms===
===Tables of Propositional Forms===
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To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.
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| align="right" colspan="3" | — W.V. Quine, ''Mathematical Logic'', [Qui, 7–8]
|}
To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.
To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.
==A Differential Extension of Propositional Calculus==
==A Differential Extension of Propositional Calculus==
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Fire over water:<br>
The image of the condition before transition.<br>
The image of the condition before transition.<br>
Thus the superior man is careful<br>
Thus the superior man is careful<br>
In the differentiation of things,<br>
In the differentiation of things,<br>
So that each finds its place.</p>
So that each finds its place.
|-
|
| align="right" | — ''I Ching'', Hexagram 64, [Wil, 249]
|}
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.
===An Interlude on the Path===
===An Interlude on the Path===
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instead, speech would proceed from me,<br>
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while I stood in its path - a slender gap -<br>
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.
the point of its possible disappearance.</p>
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| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}
It may help to get a sense of the relation between '''B''' and '''D''' by considering the ''path classifier'' (or equivalence class of curves) approach to tangent vectors. As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [<font face="lucida calligraphy">X</font>]. Given the boolean value system, a path in the space ''X'' = 〈<font face="lucida calligraphy">X</font>〉 is a map ''q'' : '''B''' → ''X''. In this case, the set of paths ('''B''' → ''X'') is isomorphic to the cartesian square ''X''<sup>2</sup> = ''X'' × ''X'', or the set of ordered pairs from ''X''.
It may help to get a sense of the relation between '''B''' and '''D''' by considering the ''path classifier'' (or equivalence class of curves) approach to tangent vectors. As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [<font face="lucida calligraphy">X</font>]. Given the boolean value system, a path in the space ''X'' = 〈<font face="lucida calligraphy">X</font>〉 is a map ''q'' : '''B''' → ''X''. In this case, the set of paths ('''B''' → ''X'') is isomorphic to the cartesian square ''X''<sup>2</sup> = ''X'' × ''X'', or the set of ordered pairs from ''X''.
===The Extended Universe of Discourse===
===The Extended Universe of Discourse===
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At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.
| width="4%" |
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:
===Intentional Propositions===
===Intentional Propositions===
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| width="40%" |
| width="60%" |
Do you guess I have some intricate purpose?<br>
Well I have . . . . for the April rain has, and the mica on<br>
Well I have . . . . for the April rain has, and the mica on<br>
the side of a rock has.</p>
the side of a rock has.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 45]
|}
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need
===Life on Easy Street===
===Life on Easy Street===
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| width="40%" |
| width="60%" |
Failing to fetch me at first keep encouraged,<br>
Missing me one place search another,<br>
Missing me one place search another,<br>
I stop some where waiting for you</p>
I stop some where waiting for you
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}
The finite character of the extended universe [E<font face="lucida calligraphy">A</font>] makes the problem of solving differential propositions relatively straightforward, at least,
The finite character of the extended universe [E<font face="lucida calligraphy">A</font>] makes the problem of solving differential propositions relatively straightforward, at least,
In view of these constraints and contingencies, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.
In view of these constraints and contingencies, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.
==Back to the Beginning : Some Exemplary Universes==
==Back to the Beginning : Exemplary Universes==
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borne way beyond all possible beginnings.</p>
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I would have preferred to be enveloped in words, borne way beyond all possible beginnings.
| width="4%" |
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.
===A One-Dimensional Universe===
===A One-Dimensional Universe===
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There was never any more inception than there is now,<br>
Nor any more youth or age than there is now;<br>
Nor any more youth or age than there is now;<br>
And will never be any more perfection than there is now,<br>
And will never be any more perfection than there is now,<br>
Nor any more heaven or hell than there is now.</p>
Nor any more heaven or hell than there is now.
|-
|
| align="right" | — Walt Whitman, Leaves of Grass, [Whi, 28]
|}
Let <font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>} = {''A''} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "''A''" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element
Let <font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>} = {''A''} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "''A''" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element
It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time. A time variable is actually a reference to a ''clock'', that is, a canonical or a convenient process that is established or accepted as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.
It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time. A time variable is actually a reference to a ''clock'', that is, a canonical or a convenient process that is established or accepted as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.
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eternity indicate?</p>
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The clock indicates the moment . . . . but what does<br>
eternity indicate?
|-
|
| align="right" | — Walt Whitman, 'Leaves of Grass', [Whi, 79]
|}
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(d''A''), d''A''} are preserved or changed in the next instance. In order to know this, we would have to determine d<sup>2</sup>''A'', and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d<sup>''k''</sup>''A'' = 0 for all ''k'' greater than some fixed value ''M''. Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(d''A''), d''A''} are preserved or changed in the next instance. In order to know this, we would have to determine d<sup>2</sup>''A'', and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d<sup>''k''</sup>''A'' = 0 for all ''k'' greater than some fixed value ''M''. Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.
===Example 1. A Square Rigging===
===Example 1. A Square Rigging===
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| width="40%" |
Always the procreant urge of the world.</p>
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Urge and urge and urge,<br>
Always the procreant urge of the world.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
|}
By way of example, suppose that we are given the initial condition ''A'' = d''A'' and the law d<sup>2</sup>''A'' = (''A''). Then, since "''A'' = d''A''" ⇔ "''A'' d''A'' or (''A'')(d''A'')", we may infer two possible trajectories, as displayed in Table 11. In either of these cases, the state ''A''(d''A'')(d<sup>2</sup>''A'') is a stable attractor or a terminal condition for both starting points.
By way of example, suppose that we are given the initial condition ''A'' = d''A'' and the law d<sup>2</sup>''A'' = (''A''). Then, since "''A'' = d''A''" ⇔ "''A'' d''A'' or (''A'')(d''A'')", we may infer two possible trajectories, as displayed in Table 11. In either of these cases, the state ''A''(d''A'')(d<sup>2</sup>''A'') is a stable attractor or a terminal condition for both starting points.
Because the initial space ''X'' = 〈''A''〉 is one-dimensional, we can easily fit the second order extension E<sup>2</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A''〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.
Because the initial space ''X'' = 〈''A''〉 is one-dimensional, we can easily fit the second order extension E<sup>2</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A''〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]</p>
<p><center><font size="+1">'''Figure 12. The Anchor'''</font></center></p>
Figure 12. The Anchor
</pre>
If we eliminate from view the regions of E<sup>2</sup>''X'' that are ruled out by the dynamic law d<sup>2</sup>''A'' = (''A''), then what remains is the quotient structure that is shown in Figure 13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties ''A'' and d<sup>2</sup>''A''. As it happens, this fact might have been expressed "right off the bat" by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as (''A'', d<sup>2</sup>''A'').
If we eliminate from view the regions of E<sup>2</sup>''X'' that are ruled out by the dynamic law d<sup>2</sup>''A'' = (''A''), then what remains is the quotient structure that is shown in Figure 13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties ''A'' and d<sup>2</sup>''A''. As it happens, this fact might have been expressed "right off the bat" by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as (''A'', d<sup>2</sup>''A'').
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]</p>
<p><center><font size="+1">'''Figure 13. The Tiller'''</font></center></p>
Figure 13. The Tiller
</pre>
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an ''n''-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a ''n''-cube without necessarily being forced to actualize all of its points.
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an ''n''-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a ''n''-cube without necessarily being forced to actualize all of its points.
===Back to the Feature===
===Back to the Feature===
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green stuff woven.</p>
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I guess it must be the flag of my disposition, out of hopeful<br>
green stuff woven.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 31]
|}
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E<font face="lucida calligraphy">X</font>] = [''A'', d''A'']. Over the extended alphabet E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, d''x''<sub>1</sub>} = {''A'', d''A''}, of cardinality 2<sup>''n''</sup> = 2, we generate the set of points, E''X'', of cardinality 2<sup>2''n''</sup> = 4, that bears the following chain of equivalent descriptions:
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E<font face="lucida calligraphy">X</font>] = [''A'', d''A'']. Over the extended alphabet E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, d''x''<sub>1</sub>} = {''A'', d''A''}, of cardinality 2<sup>''n''</sup> = 2, we generate the set of points, E''X'', of cardinality 2<sup>2''n''</sup> = 4, that bears the following chain of equivalent descriptions:
===Tacit Extensions===
===Tacit Extensions===
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I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.
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|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}
Strictly speaking, however, there is a subtle distinction in type between the function ''f''<sub>''i''</sub> : ''X'' → '''B''' and the corresponding function ''g''<sub>''j''</sub> : E''X'' → '''B''', even though they share the same logical expression. Being human, we insist on preserving all the aesthetic delights afforded by the abstractly unified form of the "cake" while giving up none of the diverse contents that its substantive consummation can provide. In short, we want to maintain the logical equivalence of expressions that represent the same proposition, while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
Strictly speaking, however, there is a subtle distinction in type between the function ''f''<sub>''i''</sub> : ''X'' → '''B''' and the corresponding function ''g''<sub>''j''</sub> : E''X'' → '''B''', even though they share the same logical expression. Being human, we insist on preserving all the aesthetic delights afforded by the abstractly unified form of the "cake" while giving up none of the diverse contents that its substantive consummation can provide. In short, we want to maintain the logical equivalence of expressions that represent the same proposition, while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
===Example 2. Drives and Their Vicissitudes===
===Example 2. Drives and Their Vicissitudes===
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I open my scuttle at night and see the far-sprinkled systems,<br>
And all I see, multiplied as high as I can cipher, edge but<br>
And all I see, multiplied as high as I can cipher, edge but<br>
the rim of the farther systems.</p>
the rim of the farther systems.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 81]
|}
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].
Given this language, the particular Example that I take up here can be described as the family of 4<sup>th</sup> gear curves through E<sup>4</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A'', d<sup>3</sup>''A'', d<sup>4</sup>''A''〉. These are the trajectories generated subject to the dynamic law d<sup>4</sup>''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0. Since d<sup>4</sup>''A'' and all higher d<sup>''k''</sup>''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E<sup>3</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A'', d<sup>3</sup>''A''〉. Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E<sup>3</sup>''X''. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figures 16-a and 16-b. (NB. I leave it as an exercise for the reader to connect the dots in the second figure.)
Given this language, the particular Example that I take up here can be described as the family of 4<sup>th</sup> gear curves through E<sup>4</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A'', d<sup>3</sup>''A'', d<sup>4</sup>''A''〉. These are the trajectories generated subject to the dynamic law d<sup>4</sup>''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0. Since d<sup>4</sup>''A'' and all higher d<sup>''k''</sup>''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E<sup>3</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A'', d<sup>3</sup>''A''〉. Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E<sup>3</sup>''X''. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]</p>
<p><center><font size="+1">'''Figure 16. A Couple of Fourth Gear Orbits'''</font></center></p>
Figure 16-a. A Couple of Fourth Gear Orbits: 1
</pre>
<pre>
Figure 16-b. A Couple of Fourth Gear Orbits: 2
</pre>
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states ''q'' in E<sup>''m''</sup>''X'' with the dyadic rationals (or the binary fractions) in the half-open interval [0, 2). Formally and canonically, a state ''q''<sub>''r''</sub> is indexed by a fraction ''r'' = ''s''/''t'' whose denominator is the power of two ''t'' = 2<sup>''m''</sup> and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state ''q'' are just the values d<sup>''k''</sup>''A''(''q''), for ''k'' = 0 to ''m'', where d<sup>0</sup>''A'' is defined as being identical to ''A''. To form the binary index d<sub>0</sub>'''.'''d<sub>1</sub>…d<sub>''m''</sub> of the state ''q'' the coefficient d<sup>''k''</sup>''A''(''q'') is read off as the binary digit ''d''<sub>''k''</sub> associated with the place value 2<sup>–''k''</sup>. Expressed by way of algebraic formulas, the rational index ''r'' of the state ''q'' can be given by the following equivalent formulations:
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states ''q'' in E<sup>''m''</sup>''X'' with the dyadic rationals (or the binary fractions) in the half-open interval [0, 2). Formally and canonically, a state ''q''<sub>''r''</sub> is indexed by a fraction ''r'' = ''s''/''t'' whose denominator is the power of two ''t'' = 2<sup>''m''</sup> and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state ''q'' are just the values d<sup>''k''</sup>''A''(''q''), for ''k'' = 0 to ''m'', where d<sup>0</sup>''A'' is defined as being identical to ''A''. To form the binary index d<sub>0</sub>'''.'''d<sub>1</sub>…d<sub>''m''</sub> of the state ''q'' the coefficient d<sup>''k''</sup>''A''(''q'') is read off as the binary digit ''d''<sub>''k''</sub> associated with the place value 2<sup>–''k''</sup>. Expressed by way of algebraic formulas, the rational index ''r'' of the state ''q'' can be given by the following equivalent formulations:
==Transformations of Discourse==
==Transformations of Discourse==
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It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 39]
|}
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.
===Foreshadowing Transformations : Extensions and Projections of Discourse===
===Foreshadowing Transformations : Extensions and Projections of Discourse===
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| width="4%" |
| width="92%" |
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.
| width="4%" |
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]
|}
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type [<font face="lucida calligraphy">X</font>] → [<font face="lucida calligraphy">Y</font>] is implied any time that we make use of one alphabet <font face="lucida calligraphy">X</font> that happens to be included in another alphabet <font face="lucida calligraphy">Y</font>. When we are discussing differential issues we usually have in mind that the extended alphabet <font face="lucida calligraphy">Y</font> has a special construction or a specific lexical relation with respect to the initial alphabet <font face="lucida calligraphy">X</font>, one that is marked by characteristic types of accents, indices, or inflected forms.
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type [<font face="lucida calligraphy">X</font>] → [<font face="lucida calligraphy">Y</font>] is implied any time that we make use of one alphabet <font face="lucida calligraphy">X</font> that happens to be included in another alphabet <font face="lucida calligraphy">Y</font>. When we are discussing differential issues we usually have in mind that the extended alphabet <font face="lucida calligraphy">Y</font> has a special construction or a specific lexical relation with respect to the initial alphabet <font face="lucida calligraphy">X</font>, one that is marked by characteristic types of accents, indices, or inflected forms.
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type '''B'''<sup>1</sup> → '''B'''<sup>2</sup> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type '''B'''<sup>1</sup> → '''B'''<sup>2</sup> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 18-a. Extension from 1 to 2 Dimensions: Areal'''</font></center></p>
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
</pre>
Figure 18-b shows the differential extension from ''X''<sup> •</sup> = [''x''] to E''X''<sup> •</sup> = [''x'', d''x''] in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.
Figure 18-b shows the differential extension from ''X''<sup> •</sup> = [''x''] to E''X''<sup> •</sup> = [''x'', d''x''] in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]</p>
| | | |
<p><center><font size="+1">'''Figure 18-b. Extension from 1 to 2 Dimensions: Bundle'''</font></center></p>
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
</pre>
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 18-c. Extension from 1 to 2 Dimensions: Compact'''</font></center></p>
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
</pre>
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 18-d. Extension from 1 to 2 Dimensions: Digraph'''</font></center></p>
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
</pre>
====Extension from 2 to 4 Dimensions====
====Extension from 2 to 4 Dimensions====
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''<sup>2</sup> → '''B'''<sup>4</sup>. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''<sup>2</sup> → '''B'''<sup>4</sup>. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 19-a. Extension from 2 to 4 Dimensions: Areal'''</font></center></p>
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
</pre>
Figure 19-b shows the differential extension from ''U''<sup> •</sup> = [''u'', ''v''] to E''U''<sup> •</sup> = [''u'', ''v'', d''u'', d''v''] in the ''bundle of boxes'' form of venn diagram.
Figure 19-b shows the differential extension from ''U''<sup> •</sup> = [''u'', ''v''] to E''U''<sup> •</sup> = [''u'', ''v'', d''u'', d''v''] in the ''bundle of boxes'' form of venn diagram.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 19-b. Extension from 2 to 4 Dimensions: Bundle'''</font></center></p>
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
</pre>
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 19-c. Extension from 2 to 4 Dimensions: Compact'''</font></center></p>
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
</pre>
Figure 19-d gives the ''digraph'' form of representation for the differential extension ''U''<sup> •</sup> → E''U''<sup> •</sup>, where the 4 nodes marked "@" are the cells ''uv'', ''u''(''v''), (''u'')''v'', (''u'')(''v''), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.
Figure 19-d gives the ''digraph'' form of representation for the differential extension ''U''<sup> •</sup> → E''U''<sup> •</sup>, where the 4 nodes marked "@" are the cells ''uv'', ''u''(''v''), (''u'')''v'', (''u'')(''v''), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p>
<p><center><font size="+1">'''Figure 19-d. Extension from 2 to 4 Dimensions: Digraph'''</font></center></p>
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
</pre>
===Thematization of Functions : And a Declaration of Independence for Variables===
===Thematization of Functions : And a Declaration of Independence for Variables===
{| width="100%"
| align="left" |
The forms of things unknown, the poet's pen<br>
''And as imagination bodies forth''<br>
Turns them to shapes, and gives to airy nothing<br>
''The forms of things unknown, the poet's pen''<br>
A local habitation and a name.</p>
''Turns them to shapes, and gives to airy nothing''<br>
''A local habitation and a name.''
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18
|}
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.
====Thematization : Venn Diagrams====
====Thematization : Venn Diagrams====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
He consumes an eternal passion and is indifferent which chance happens<br>
| width="92%" |
and which possible contingency of fortune or misfortune and persuades<br>
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
daily and hourly his delicious pay.</p>
| width="4%" |
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 11–12]
|}
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition ''u''<b>·</b>''v'' in [''u'', ''v''].
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition ''u''<b>·</b>''v'' in [''u'', ''v''].
The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition ''u''<b>·</b>''v'' a distinctive functional name "''J'' ". Second, one has come to think explicitly about the target domain that contains the functional values of ''J'', as when one writes ''J'' : 〈''u'', ''v''〉 → '''B'''.
The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition ''u''<b>·</b>''v'' a distinctive functional name "''J'' ". Second, one has come to think explicitly about the target domain that contains the functional values of ''J'', as when one writes ''J'' : 〈''u'', ''v''〉 → '''B'''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]</p>
| | | |
<p><center><font size="+1">'''Figure 20-i. Thematization of Conjunction (Stage 1)'''</font></center></p>
In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]</p>
<p><center><font size="+1">'''Figure 20-ii. Thematization of Conjunction (Stage 2)'''</font></center></p>
Figure 20-i. Thematization of Conjunction (Stage 1)
</pre>
In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.
<pre>
<pre>
o-------------------------------o o-------------------------------o
o-------------------------------o o-------------------------------o
In Figure 20-iii we arrive at a stage where the functional equations, ''J'' = ''u''<b>·</b>''v'' and ''x'' = ''u''<b>·</b>''v'', are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [''u'', ''v'', ''J''] and [''u'', ''v'', ''x''], respectively. Subject to the cautions already noted, the function name "''J'' " can be reinterpreted as the name of a feature ''J''<sup> ¢</sup>, and the equation ''J'' = ''u''<b>·</b>''v'' can be read as the logical equivalence ((''J'', ''u'' ''v'')). To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition ''J''.
In Figure 20-iii we arrive at a stage where the functional equations, ''J'' = ''u''<b>·</b>''v'' and ''x'' = ''u''<b>·</b>''v'', are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [''u'', ''v'', ''J''] and [''u'', ''v'', ''x''], respectively. Subject to the cautions already noted, the function name "''J'' " can be reinterpreted as the name of a feature ''J''<sup> ¢</sup>, and the equation ''J'' = ''u''<b>·</b>''v'' can be read as the logical equivalence ((''J'', ''u'' ''v'')). To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition ''J''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]</p>
| | |```````````````````````````````|
<p><center><font size="+1">'''Figure 20-iii. Thematization of Conjunction (Stage 3)'''</font></center></p>
Figure 20-iii. Thematization of Conjunction (Stage 3)
</pre>
The first venn diagram represents the thematization of the conjunction ''J'' with shading in the appropriate regions of the universe [''u'', ''v'', ''J'']. Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.
The first venn diagram represents the thematization of the conjunction ''J'' with shading in the appropriate regions of the universe [''u'', ''v'', ''J'']. Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.
Figure 21 shows how the thematic extension operator θ acts on two further examples, the disjunction ((''u'')(''v'')) and the equality ((''u'', ''v'')). Referring to the disjunction as ''f''‹''u'', ''v''› and the equality as ''g''‹''u'', ''v''›, I write the thematic extensions as φ = θ''f'' and γ = θ''g''.
Figure 21 shows how the thematic extension operator θ acts on two further examples, the disjunction ((''u'')(''v'')) and the equality ((''u'', ''v'')). Referring to the disjunction as ''f''‹''u'', ''v''› and the equality as ''g''‹''u'', ''v''›, I write the thematic extensions as φ = θ''f'' and γ = θ''g''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]</p>
<p><center><font size="+1">'''Figure 21. Thematization of Disjunction and Equality'''</font></center></p>
| | |```````````````````````````````|
Figure 21. Thematization of Disjunction and Equality
</pre>
====Thematization : Truth Tables====
====Thematization : Truth Tables====
{| width="100%" cellpadding="0" cellspacing="0"
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beings or places or contingencies is a nuisance and a revolt.</p>
| width="92%" |
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
| width="4%" |
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 19]
|}
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.
===Propositional Transformations===
===Propositional Transformations===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
If only the word 'artificial' were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.
| width="4%" |
|-
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56–57]
|}
In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same. When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.
In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same. When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.
====Transformations of General Type====
====Transformations of General Type====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 34]
|}
Consider the situation illustrated in Figure 30, where the alphabets <font face="lucida calligraphy">U</font> = {''u'', ''v''} and <font face="lucida calligraphy">X</font> = {''x'', ''y'', ''z''} are used to label basic features in two different logical universes, ''U''<sup> •</sup> = [''u'', ''v''] and ''X''<sup> •</sup> = [''x'', ''y'', ''z''].
Consider the situation illustrated in Figure 30, where the alphabets <font face="lucida calligraphy">U</font> = {''u'', ''v''} and <font face="lucida calligraphy">X</font> = {''x'', ''y'', ''z''} are used to label basic features in two different logical universes, ''U''<sup> •</sup> = [''u'', ''v''] and ''X''<sup> •</sup> = [''x'', ''y'', ''z''].
===Analytic Expansions : Operators and Functors===
===Analytic Expansions : Operators and Functors===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
have practical bearings you ''conceive'' the<br>
| width="92%" |
objects of your ''conception'' to have. Then,<br>
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.
your ''conception'' of those effects is the<br>
| width="4%" |
whole of your ''conception'' of the object.</p>
|-
| align="right" colspan="3" | — C.S. Peirce, "The Maxim of Pragmatism", CP 5.438
|}
Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.
Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.
====Differential Analysis of Propositions and Transformations====
====Differential Analysis of Propositions and Transformations====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
he resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 43]
|}
The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators <font face=georgia>'''W'''</font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps <font face=georgia>'''W'''</font>''F''. The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.
The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators <font face=georgia>'''W'''</font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps <font face=georgia>'''W'''</font>''F''. The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.
=====The Secant Operator : <font face=georgia>'''E'''</font>=====
=====The Secant Operator : <font face=georgia>'''E'''</font>=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}
Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted "<font face=georgia>'''E'''</font>", which receives the principal investment of analytic attention, and on the constituent parts of <font face=georgia>'''E'''</font>, which derive their shares of significance as developed by the analysis. In the sequel, I refer to <font face=georgia>'''E'''</font> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›, and its active ingredient E is known as the ''enlargement operator''. (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that E''f''(''x'') = ''f''(''x''+1) for any suitable function ''f'', though of course the logical analogue that we take up here must have a rather different definition.)
Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted "<font face=georgia>'''E'''</font>", which receives the principal investment of analytic attention, and on the constituent parts of <font face=georgia>'''E'''</font>, which derive their shares of significance as developed by the analysis. In the sequel, I refer to <font face=georgia>'''E'''</font> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›, and its active ingredient E is known as the ''enlargement operator''. (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that E''f''(''x'') = ''f''(''x''+1) for any suitable function ''f'', though of course the logical analogue that we take up here must have a rather different definition.)
=====The Radius Operator : <font face=georgia>'''e'''</font>=====
=====The Radius Operator : <font face=georgia>'''e'''</font>=====
{| width="100%" cellpadding="0" cellspacing="0"
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And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
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| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}
The operator identified as <font face=georgia>'''d'''</font><sup>0</sup> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for ''F'' in the appropriately extended context. Construed in terms of its broadest components, <font face=georgia>'''d'''</font><sup>0</sup> is equivalent to the doubly tacit extension operator ‹<math>\epsilon</math>, <math>\epsilon</math>›, in recognition of which let us redub it as "<font face=georgia>'''e'''</font>". Pursuing a geometric analogy, we may refer to <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\epsilon</math>› = <font face=georgia>'''d'''</font><sup>0</sup> as the ''radius operator''. The operation that is intended by all of these forms is defined by the equation:
The operator identified as <font face=georgia>'''d'''</font><sup>0</sup> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for ''F'' in the appropriately extended context. Construed in terms of its broadest components, <font face=georgia>'''d'''</font><sup>0</sup> is equivalent to the doubly tacit extension operator ‹<math>\epsilon</math>, <math>\epsilon</math>›, in recognition of which let us redub it as "<font face=georgia>'''e'''</font>". Pursuing a geometric analogy, we may refer to <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\epsilon</math>› = <font face=georgia>'''d'''</font><sup>0</sup> as the ''radius operator''. The operation that is intended by all of these forms is defined by the equation:
=====The Phantom of the Operators : '''η'''=====
=====The Phantom of the Operators : '''η'''=====
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I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!
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| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]
|}
I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect. In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.
I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect. In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.
=====The Chord Operator : <font face=georgia>'''D'''</font>=====
=====The Chord Operator : <font face=georgia>'''D'''</font>=====
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What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.
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| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 45]
|}
Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.
Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.
=====The Tangent Operator : <font face=georgia>'''T'''</font>=====
=====The Tangent Operator : <font face=georgia>'''T'''</font>=====
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They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.
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| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 300]
|}
The operator tagged as <font face=georgia>'''d'''</font><sup>1</sup> in the analytic diagram (Figure 33) is called the ''tangent operator'', and is usually denoted in this text as <font face=georgia>'''d'''</font> or <font face=georgia>'''T'''</font>. Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <font face=georgia>'''T'''</font> = <font face=georgia>'''d'''</font> = ‹<math>\epsilon</math>, d›, where d is the operator that yields the first order differential d''F'' when applied to a transformation ''F'', and whose name is legion.
The operator tagged as <font face=georgia>'''d'''</font><sup>1</sup> in the analytic diagram (Figure 33) is called the ''tangent operator'', and is usually denoted in this text as <font face=georgia>'''d'''</font> or <font face=georgia>'''T'''</font>. Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <font face=georgia>'''T'''</font> = <font face=georgia>'''d'''</font> = ‹<math>\epsilon</math>, d›, where d is the operator that yields the first order differential d''F'' when applied to a transformation ''F'', and whose name is legion.
====Analytic Expansion of Conjunction====
====Analytic Expansion of Conjunction====
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<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul.</p>
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul.</p>
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p>
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p>
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|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 118]
|}
Figure 35 pictures the form of conjunction ''J'' : '''B'''<sup>2</sup> → '''B''' as a transformation from the 2-dimensional universe [''u'', ''v''] to the 1-dimensional universe [''x'']. This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition ''J'' : 〈''u'', ''v''〉 → '''B''' is being recast into the thematized role of a transformation ''J'' : [''u'', ''v''] → [''x''], where the new variable ''x'' takes the part of a thematic variable ¢(''J'').
Figure 35 pictures the form of conjunction ''J'' : '''B'''<sup>2</sup> → '''B''' as a transformation from the 2-dimensional universe [''u'', ''v''] to the 1-dimensional universe [''x'']. This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition ''J'' : 〈''u'', ''v''〉 → '''B''' is being recast into the thematized role of a transformation ''J'' : [''u'', ''v''] → [''x''], where the new variable ''x'' takes the part of a thematic variable ¢(''J'').
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]</p>
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<p><center><font size="+1">'''Figure 35. Conjunction as Transformation'''</font></center></p>
Figure 35. Conjunction as Transformation
</pre>
=====Tacit Extension of Conjunction=====
=====Tacit Extension of Conjunction=====
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I teach straying from me, yet who can stray from me?<br>
I follow you whoever you are from the present hour;<br>
I follow you whoever you are from the present hour;<br>
My words itch at your ears till you understand them.</p>
My words itch at your ears till you understand them.
|-
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| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 83]
|}
Earlier I defined the tacit extension operators <math>\epsilon</math> : ''X''<sup> •</sup> → ''Y''<sup> •</sup> as maps embedding each proposition of a given universe ''X''<sup> •</sup> in a more generously given universe ''Y''<sup> •</sup> containing ''X''<sup> •</sup>. Of immediate interest are the tacit extensions <math>\epsilon</math> : ''U''<sup> •</sup> → E''U''<sup> •</sup>, that locate each proposition of ''U''<sup> •</sup> in the enlarged context of E''U''<sup> •</sup>. In its application to the propositional conjunction ''J'' = ''u'' ''v'' in [''u'', ''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''<sup> •</sup> = [''u'', ''v'', d''u'', d''v'']. The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table 36, in effect, doing nothing more than conjoining a tautology of [d''u'', d''v''] to ''J'' in ''U''<sup> •</sup>.
Earlier I defined the tacit extension operators <math>\epsilon</math> : ''X''<sup> •</sup> → ''Y''<sup> •</sup> as maps embedding each proposition of a given universe ''X''<sup> •</sup> in a more generously given universe ''Y''<sup> •</sup> containing ''X''<sup> •</sup>. Of immediate interest are the tacit extensions <math>\epsilon</math> : ''U''<sup> •</sup> → E''U''<sup> •</sup>, that locate each proposition of ''U''<sup> •</sup> in the enlarged context of E''U''<sup> •</sup>. In its application to the propositional conjunction ''J'' = ''u'' ''v'' in [''u'', ''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''<sup> •</sup> = [''u'', ''v'', d''u'', d''v'']. The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table 36, in effect, doing nothing more than conjoining a tautology of [d''u'', d''v''] to ''J'' in ''U''<sup> •</sup>.
Figures 37-a through 37-d present several pictures of the proposition ''J'' and its tacit extension <math>\epsilon</math>''J''. Notice in these Figures how <math>\epsilon</math>''J'' in E''U''<sup> •</sup> visibly extends ''J'' in ''U''<sup> •</sup>, by annexing to the indicated cells of ''J'' all of the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue.
Figures 37-a through 37-d present several pictures of the proposition ''J'' and its tacit extension <math>\epsilon</math>''J''. Notice in these Figures how <math>\epsilon</math>''J'' in E''U''<sup> •</sup> visibly extends ''J'' in ''U''<sup> •</sup>, by annexing to the indicated cells of ''J'' all of the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]</p>
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<p><center><font size="+1">'''Figure 37-a. Tacit Extension of ''J'' (Areal)'''</font></center></p>
Figure 37-a. Tacit Extension of J (Areal)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 37-b. Tacit Extension of ''J'' (Bundle)'''</font></center></p>
Figure 37-b. Tacit Extension of J (Bundle)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]</p>
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<p><center><font size="+1">'''Figure 37-c. Tacit Extension of ''J'' (Compact)'''</font></center></p>
Figure 37-c. Tacit Extension of J (Compact)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 37-d. Tacit Extension of ''J'' (Digraph)'''</font></center></p>
Figure 37-d. Tacit Extension of J (Digraph)
</pre>
The computational scheme that was shown in Table 36 treated ''J'' as a proposition in ''U''<sup> •</sup> and formed <math>\epsilon</math>''J'' as a proposition in E''U''<sup> •</sup>. When ''J'' is regarded as a mapping ''J'' : ''U''<sup> •</sup> → ''X''<sup> •</sup> then <math>\epsilon</math>''J'' must be obtained as a mapping <math>\epsilon</math>''J'' : E''U''<sup> •</sup> → ''X''<sup> •</sup>. By default, the tacit extension of the map ''J'' : [''u'', ''v''] → [''x''] is naturally taken to be a particular map, of the following form:
The computational scheme that was shown in Table 36 treated ''J'' as a proposition in ''U''<sup> •</sup> and formed <math>\epsilon</math>''J'' as a proposition in E''U''<sup> •</sup>. When ''J'' is regarded as a mapping ''J'' : ''U''<sup> •</sup> → ''X''<sup> •</sup> then <math>\epsilon</math>''J'' must be obtained as a mapping <math>\epsilon</math>''J'' : E''U''<sup> •</sup> → ''X''<sup> •</sup>. By default, the tacit extension of the map ''J'' : [''u'', ''v''] → [''x''] is naturally taken to be a particular map, of the following form:
=====Enlargement Map of Conjunction=====
=====Enlargement Map of Conjunction=====
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No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.
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| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 62]
|}
The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u'' + d''u'' for ''u'' and ''v'' + d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.
The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u'' + d''u'' for ''u'' and ''v'' + d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.
Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'', ''v'']. The critical step of this procedure uses the facts that (0, ''x'') = 0 + ''x'' = ''x'' and (1, ''x'') = 1 + ''x'' = (''x'') for any boolean variable ''x''.
Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'', ''v'']. The critical step of this procedure uses the facts that (0, ''x'') = 0 + ''x'' = ''x'' and (1, ''x'') = 1 + ''x'' = (''x'') for any boolean variable ''x''.
</font><br>
</font><br>
Figures 40-a through 40-d present several views of the enlarged proposition EJ.
Figures 40-a through 40-d present several views of the enlarged proposition E''J''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]</p>
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<p><center><font size="+1">'''Figure 40-a. Enlargement of ''J'' (Areal)'''</font></center></p>
Figure 40-a. Enlargement of J (Areal)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 40-b. Enlargement of ''J'' (Bundle)'''</font></center></p>
Figure 40-b. Enlargement of J (Bundle)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]</p>
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<p><center><font size="+1">'''Figure 40-c. Enlargement of ''J'' (Compact)'''</font></center></p>
Figure 40-c. Enlargement of J (Compact)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 40-d. Enlargement of ''J'' (Digraph)'''</font></center></p>
Figure 40-d. Enlargement of J (Digraph)
</pre>
An intuitive reading of the proposition E''J'' becomes available at this point, and may be useful. Recall that propositions in the extended universe E''U''<sup> •</sup> express the ''dispositions'' of system and the constraints that are placed on them. In other words, a differential proposition in E''U''<sup> •</sup> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand E''J'' as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of ''J'', that is, the region of the universe where ''J'' is true. This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, E''J'' can be read as a proposition that ''enlarges'' on the meaning of ''J'', in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.
An intuitive reading of the proposition E''J'' becomes available at this point, and may be useful. Recall that propositions in the extended universe E''U''<sup> •</sup> express the ''dispositions'' of system and the constraints that are placed on them. In other words, a differential proposition in E''U''<sup> •</sup> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand E''J'' as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of ''J'', that is, the region of the universe where ''J'' is true. This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, E''J'' can be read as a proposition that ''enlarges'' on the meaning of ''J'', in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.
=====Digression : Reflection on Use and Mention=====
=====Digression : Reflection on Use and Mention=====
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Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.
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| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 57]
|}
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using "''J'' " to indicate the region ''J''<sup>–1</sup>(1) and using "''J'' " to indicate the function ''J''. You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "''J'' " is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J'' " by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise we have an inference like the following: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is.
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using "''J'' " to indicate the region ''J''<sup>–1</sup>(1) and using "''J'' " to indicate the function ''J''. You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "''J'' " is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J'' " by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise we have an inference like the following: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is.
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.
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The well-known capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.
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| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 130]
|}
=====Difference Map of Conjunction=====
=====Difference Map of Conjunction=====
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"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders. "In a tangle of forces like this it doesn't make a scrap of difference." He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.
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| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 8]
|}
With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'', ''v''].
With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'', ''v''].
The difference map D''J'' can also be given a ''dispositional'' interpretation. First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''. Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'', E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other. In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.
The difference map D''J'' can also be given a ''dispositional'' interpretation. First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''. Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'', E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other. In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.
<font face="courier new">
<br><font face="courier new">
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Figures 44-a through 44-d illustrate the difference proposition D''J''.
Figures 44-a through 44-d illustrate the difference proposition D''J''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 44-a. Difference Map of ''J'' (Areal)'''</font></center></p>
Figure 44-a. Difference Map of J (Areal)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 44-b. Difference Map of ''J'' (Bundle)'''</font></center></p>
Figure 44-b. Difference Map of J (Bundle)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 44-c. Difference Map of ''J'' (Compact)'''</font></center></p>
Figure 44-c. Difference Map of J (Compact)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 44-d. Difference Map of ''J'' (Digraph)'''</font></center></p>
Figure 44-d. Difference Map of J (Digraph)
</pre>
=====Differential of Conjunction=====
=====Differential of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.
| width="4%" |
|-
| align="right" colspan="3" | — Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]
|}
Finally, at long last, the differential proposition d''J'' can be gleaned from the difference proposition D''J'' by ranging over the cells of [''u'', ''v''] and picking out the linear proposition of [d''u'', d''v''] that is "closest" to the portion of D''J'' that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.
Finally, at long last, the differential proposition d''J'' can be gleaned from the difference proposition D''J'' by ranging over the cells of [''u'', ''v''] and picking out the linear proposition of [d''u'', d''v''] that is "closest" to the portion of D''J'' that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 144]
|}
Let us venture a guess about where these developments might be heading. From the present vantage point, it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form — the limitary concept of a self-corrective process and the coefficient concept of a completable product — are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.
Let us venture a guess about where these developments might be heading. From the present vantage point, it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form — the limitary concept of a self-corrective process and the coefficient concept of a completable product — are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.
Figures 46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects. This proposition of E''U''<sup> •</sup> is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.
Figures 46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects. This proposition of E''U''<sup> •</sup> is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 46-a. Differential of ''J'' (Areal)'''</font></center></p>
Figure 46-a. Differential of J (Areal)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 46-b. Differential of ''J'' (Bundle)'''</font></center></p>
Figure 46-b. Differential of J (Bundle)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 46-c. Differential of ''J'' (Compact)'''</font></center></p>
Figure 46-c. Differential of J (Compact)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 46-d. Differential of ''J'' (Digraph)'''</font></center></p>
Figure 46-d. Differential of J (Digraph)
</pre>
=====Remainder of Conjunction=====
=====Remainder of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
<p>I bequeath myself to the dirt to grow from the grass I love,<br>
<p>I bequeath myself to the dirt to grow from the grass I love,<br>
If you want me again look for me under your bootsoles.</p>
If you want me again look for me under your bootsoles.</p>
Missing me one place search another,<br>
Missing me one place search another,<br>
I stop some where waiting for you</p>
I stop some where waiting for you</p>
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}
Let us now recapitulate the story so far. In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages. First, we considered the equation E''J'' = <math>\epsilon</math>''J'' + D''J'', which was involved in the definition of D''J'' as the difference E''J'' – <math>\epsilon</math>''J''. Next, we contemplated the equation D''J'' = d''J'' + r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J'' = D''J'' – d''J''. This remaining proposition r''J'' can be computed as shown in Table 47.
Let us now recapitulate the story so far. In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages. First, we considered the equation E''J'' = <math>\epsilon</math>''J'' + D''J'', which was involved in the definition of D''J'' as the difference E''J'' – <math>\epsilon</math>''J''. Next, we contemplated the equation D''J'' = d''J'' + r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J'' = D''J'' – d''J''. This remaining proposition r''J'' can be computed as shown in Table 47.
Figures 48-a through 48-d illustrate the proposition r''J'' = d<sup>2</sup>''J'', which forms the remainder map of ''J'' and also, in this instance, the second order differential of ''J''.
Figures 48-a through 48-d illustrate the proposition r''J'' = d<sup>2</sup>''J'', which forms the remainder map of ''J'' and also, in this instance, the second order differential of ''J''.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 48-a. Remainder of ''J'' (Areal)'''</font></center></p>
Figure 48-a. Remainder of J (Areal)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 48-b. Remainder of ''J'' (Bundle)'''</font></center></p>
Figure 48-b. Remainder of J (Bundle)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]</p>
| |
<p><center><font size="+1">'''Figure 48-c. Remainder of ''J'' (Compact)'''</font></center></p>
Figure 48-c. Remainder of J (Compact)
</pre>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 48-d. Remainder of ''J'' (Digraph)'''</font></center></p>
Figure 48-d. Remainder of J (Digraph)
</pre>
=====Summary of Conjunction=====
=====Summary of Conjunction=====
====Analytic Series : Coordinate Method====
====Analytic Series : Coordinate Method====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 12]
|}
Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.
Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of E''J'' = ''J'' + D''J'' and D''J'' = d''J'' + r''J'' in two different styles of diagram.
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of E''J'' = ''J'' + D''J'' and D''J'' = d''J'' + r''J'' in two different styles of diagram.
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]</p>
<p><center><font size="+1">'''Figure 52. Decomposition of E''J'''''</font></center></p>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]</p>
<p><center><font size="+1">'''Figure 53. Decomposition of D''J'''''</font></center></p>
| | | | | |
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
</pre>
<pre>
</pre>
====Terminological Interlude====
====Terminological Interlude====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been "starred", in spite of their solidity …
| width="4%" |
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]
|}
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Section are intended to accomplish two goals. First, I call attention to important aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and I restress the most important structural elements that they indicate. Next, I prepare the way for taking on more complex examples of transformations, whose target universes have more than a single dimension.
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Section are intended to accomplish two goals. First, I call attention to important aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and I restress the most important structural elements that they indicate. Next, I prepare the way for taking on more complex examples of transformations, whose target universes have more than a single dimension.
=====Operator Maps : Areal Views=====
=====Operator Maps : Areal Views=====
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
Figure 56-a1. Radius Map of the Conjunction J = uv
<br>
</pre>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
Figure 56-a2. Secant Map of the Conjunction J = uv
<br>
</pre>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-a4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
=====Operator Maps : Box Views=====
Figure 56-a3. Chord Map of the Conjunction J = uv
<br>
</pre>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
Figure 56-a4. Tangent Map of the Conjunction J = uv
<br>
</pre>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
=====Operator Diagrams for the Conjunction J = uv=====
Figure 56-b1. Radius Map of the Conjunction J = uv
<br>
</pre>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-1. Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-2. Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
Figure 56-b2. Secant Map of the Conjunction J = uv
<br>
</pre>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-3. Chord Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
<pre>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]</p>
<p><center><font size="+1">'''Figure 57-4. Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>
</pre>
</pre>
<pre>
</pre>
<pre>
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
</pre>
<pre>
</pre>
===Taking Aim at Higher Dimensional Targets===
===Taking Aim at Higher Dimensional Targets===
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
The past and present wilt . . . . I have filled them and<br>
emptied them,<br>
emptied them,<br>
And proceed to fill my next fold of the future.</p>
And proceed to fill my next fold of the future.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 87]
|}
In the next Subdivision I consider a logical transformation ''F'' that has the concrete type ''F'' : [''u'', ''v''] → [''x'', ''y''] and the abstract type ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>]. From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:
In the next Subdivision I consider a logical transformation ''F'' that has the concrete type ''F'' : [''u'', ''v''] → [''x'', ''y''] and the abstract type ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>]. From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>].
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>].
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''
|- style="background:paleturquoise"
! Item
! Notation
! Description
! Type
|-
| valign="top" | ''U''<sup> •</sup>
| valign="top" | <font face="courier new">= </font>[''u'', ''v'']
| valign="top" | Source Universe
| valign="top" | ['''B'''<sup>''n''</sup>]
|-
| valign="top" | ''X''<sup> •</sup>
| valign="top" |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face="courier new">= </font>[''x'', ''y'']
| g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] |
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | [B^k x D^k] |
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
|-
|-
|
| <font face="courier new">= </font>[''f'', ''g'']
|}
|}
| valign="top" | Target Universe
| valign="top" | ['''B'''<sup>''k''</sup>]
|-
| valign="top" | E''U''<sup> •</sup>
| valign="top" | <font face="courier new">= </font>[''u'', ''v'', d''u'', d''v'']
| valign="top" | Extended Source Universe
| valign="top" | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
|-
| valign="top" | E''X''<sup> •</sup>
| valign="top" |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y'']
|-
| <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g'']
|}
|}
| valign="top" | Extended Target Universe
| valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|-
| ''F''
<br><font face="courier new">
| ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
| Transformation, or Mapping
|
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|
|-
| ''f''
|-
| ''g''
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''f'', ''g'' : ''U'' → '''B'''
|-
| ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
|-
| ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Proposition
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
| '''B'''<sup>''n''</sup> → '''B'''
|-
| ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
|-
| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
|}
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| W :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
|-
| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| u | v | f | g |
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
| | | | |
|-
| 0 | 0 | 0 | 1 |
| →
|-
| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,
|-
| for each W in the set:
|-
| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|-
<br><font face="courier new">
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
|-
| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
|-
| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
| width="4%" | =
|-
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| →
|-
| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
|-
|
|-
|-
|
|}
|}
|}
|-
|
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| width="8%" | E''f''
| <math>\epsilon</math>
| width="4%" | =
|-
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
| <math>\eta</math>
|-
| E
|-
| D
|-
|-
| width="8%" | E''g''
| d
|}
|}
|}
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Tacit Extension Operator || <math>\epsilon</math>
|-
|
| Trope Extension Operator || <math>\eta</math>
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
|-
| width="8%" | D''f''
| Enlargement Operator || E
| width="4%" | =
|-
| width="20%" | ((''u'')(''v''))
| Difference Operator || D
| width="4%" | +
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
|-
|-
| width="8%" | D''g''
| Differential Operator || d
| width="4%" | =
|}
|}
|-
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font>
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''W'''</font> :
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''f''
|-
|-
| E''f''
| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
|-
|-
| D''f''
| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
|-
| →
|-
| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
|-
|-
| d''f''
| for each <font face=georgia>'''W'''</font> in the set:
|-
|-
| r''f''
| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Operator
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
|
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''g''
|-
|-
| E''g''
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
|-
|-
| D''g''
| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
|-
|-
| d''g''
| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
|-
|-
| r''g''
| →
|-
|-
| D''f''
| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
|-
|-
|
|
|-
|-
|
|
|}
|}
|}
|-
|
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
| <font face=georgia>'''e'''</font>
|-
|-
| d''f''
| <font face=georgia>'''E'''</font>
| = || ''uv'' || <math>\cdot</math> || 0
|-
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| <font face=georgia>'''D'''</font>
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
|-
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| <font face=georgia>'''T'''</font>
|}
| valign="top" |
| colspan="2" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
|-
| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
|-
| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
|-
|-
| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
|}
|}<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
|- style="background:paleturquoise"
|
|
| align="center" | '''Operator<br>or<br>Operand'''
| align="center" | '''Proposition<br>or<br>Component'''
| align="center" | '''Transformation<br>or<br>Mapping'''
|-
|-
| d''g''
| Operand
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| valign="top" |
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
|-
|-
|
| ''F'' = ‹''f'', ''g''› : ''U'' → ''X''
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 → '''B'''
|-
| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> → '''B'''
|}
|}
| valign="top" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
| ''F'' : [''u'', ''v''] → [''x'', ''y'']
|-
| ''F'' : '''B'''<sup>''n''</sup> → '''B'''<sup>''k''</sup>
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tacit
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
|\ / \ /| |\ / \ / / \ / / \ /|
|}
|
| \ / \ / | | \ / O \ / O \ / O \ / |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| \ / \ / | | o /@ o @\ o /@ o |
| <math>\epsilon</math>''F''<sub>''i''</sub> :
|-
| \ / \ / | | | \ / \ / \ / | |
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Trope
|-
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
| \ / \ / | | \ / O \ / O \ / O \ / |
|-
| \ / \ / | | o /@ o @ o @\ o |
|
| \ / \ / | | |\ / / \ / \ / \ \ /| |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| \ / \ / | | | \ / \ / \ / | |
| Enlargement
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Difference
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Differential
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Remainder
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Radius
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Secant
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Chord
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tangent
|-
| Functor
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''F''<sub>''i''</sub> :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font>''F'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', ''y'', d''x'', d''y'']
|-
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
|}
|}<br>
[[Image:Tangent_Functor_Ferris_Wheel.gif|frame|<font size="3">'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]
===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>===
* '''Nota Bene.''' The original Figure 70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from ''U''<sup> •</sup> = [''u'', ''v''] to ''X''<sup> •</sup> = [''x'', ''y''] that is defined by the following system of equations:
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
| ''x''
| =
| ''f''‹''u'', ''v''›
| =
| ((''u'')(''v''))
|
|-
|
| ''y''
| =
| ''g''‹''u'', ''v''›
| =
| ((''u'', ''v''))
|
|}
|}
</font><br>
The component notation ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>› = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup> allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows:
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
| ‹''x'', ''y''›
| =
| ''F''‹''u'', ''v''›
| =
| ‹((''u'')(''v'')), ((''u'', ''v''))›
|
|}
|}
</font><br>
The information that defines the logical transformation ''F'' can be represented in the form of a truth table, as in Table 60. To cut down on subscripts in this example I continue to use plain letter equivalents for all components of spaces and maps.
<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 60. Propositional Transformation'''
|- style="background:paleturquoise"
| width="25%" | ''u''
| width="25%" | ''v''
| width="25%" | ''f''
| width="25%" | ''g''
|-
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 0
|-
| 1
|-
| 1
|}
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 0
|-
| 1
|}
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 1
|}
| width="25%" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
|-
| width="25%" |
| width="25%" |
| width="25%" | ((''u'')(''v''))
| width="25%" | ((''u'', ''v''))
|}
</font><br>
Figure 61 shows how one might paint a picture of the logical transformation ''F'' on the canvass that was earlier primed for this purpose (way back in Figure 30).
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]</p>
<p><center><font size="+1">'''Figure 61. Propositional Transformation'''</font></center></p>
Figure 62 extracts the gist of Figure 61, exemplifying a style of diagram that is adequate for most purposes.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]</p>
<p><center><font size="+1">'''Figure 62. Propositional Transformation (Short Form)'''</font></center></p>
Figure 63 give a more complete picture of the transformation ''F'', showing how the points of ''U''<sup> •</sup> are transformed into points of ''X''<sup> •</sup>. The lines that cross from one universe to the other trace the action that ''F'' induces on points, in other words, they depict the aspect of the transformation that acts as a mapping from points to points, and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]</p>
<p><center><font size="+1">'''Figure 63. Transformation of Positions'''</font></center></p>
Table 64 shows how the action of the transformation ''F'' on cells or points is computed in terms of coordinates.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 64. Transformation of Positions'''
|- style="background:paleturquoise"
| ''u'' ''v''
| ''x''
| ''y''
| ''x'' ''y''
| ''x'' (''y'')
| (''x'') ''y''
| (''x'')(''y'')
| ''X''<sup> •</sup> = [''x'', ''y'' ]
|-
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 0
|-
| 0 1
|-
| 1 0
|-
| 1 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 0
|-
| 0
|-
| 1
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 1
|-
| 1
|-
| 0
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1
|-
| 0
|-
| 0
|-
| 0
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 0
|-
| 0
|-
| 0
|}
| width="12%" |
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ↑
|-
| ''F''
|-
| ‹''f'', ''g'' ›
|-
| ↑
|}
|-
|
| ((''u'')(''v''))
| ((''u'', ''v''))
| ''u'' ''v''
| (''u'', ''v'')
| (''u'')(''v'')
| ( )
| ''U''<sup> •</sup> = [''u'', ''v'' ]
|}
<br>
Table 65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in the universe. The way that a transformation of positions affects the propositions, or any other structure that can be built on top of the positions, is normally called the ''induced action'' of the given transformation on the system of structures in question.
<br><font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 65. Induced Transformation on Propositions'''
|- style="background:paleturquoise"
| ''X''<sup> •</sup>
| colspan="3" |
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
| ←
| ''F'' = ‹''f'' , ''g''›
| ←
|}
| ''U''<sup> •</sup>
|- style="background:paleturquoise"
| rowspan="2" | ''f''<sub>''i''</sub>‹''x'', ''y''›
|
{| align="right" style="background:paleturquoise; text-align:right"
| ''u'' =
|-
| ''v'' =
|}
|
{| align="center" style="background:paleturquoise; text-align:center"
| 1 1 0 0
|-
| 1 0 1 0
|}
|
{| align="left" style="background:paleturquoise; text-align:left"
| = ''u''
|-
| = ''v''
|}
| rowspan="2" | ''f''<sub>''j''</sub>‹''u'', ''v''›
|- style="background:paleturquoise"
|
{| align="right" style="background:paleturquoise; text-align:right"
| ''x'' =
|-
| ''y'' =
|}
|
{| align="center" style="background:paleturquoise; text-align:center"
| 1 1 1 0
|-
| 1 0 0 1
|}
|
{| align="left" style="background:paleturquoise; text-align:left"
| = ''f''‹''u'', ''v''›
|-
| = ''g''‹''u'', ''v''›
|}
|-
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>0</sub>
|-
| ''f''<sub>1</sub>
|-
| ''f''<sub>2</sub>
|-
| ''f''<sub>3</sub>
|-
| ''f''<sub>4</sub>
|-
| ''f''<sub>5</sub>
|-
| ''f''<sub>6</sub>
|-
| ''f''<sub>7</sub>
|}
|
{| cellpadding="2" style="background:lightcyan"
| ()
|-
| (''x'')(''y'')
|-
| (''x'') ''y''
|-
| (''x'')
|-
| ''x'' (''y'')
|-
| (''y'')
|-
| (''x'', ''y'')
|-
| (''x'' ''y'')
|}
|
{| cellpadding="2" style="background:lightcyan"
| 0 0 0 0
|-
| 0 0 0 1
|-
| 0 0 1 0
|-
| 0 0 1 1
|-
| 0 1 0 0
|-
| 0 1 0 1
|-
| 0 1 1 0
|-
| 0 1 1 1
|}
|
{| cellpadding="2" style="background:lightcyan"
| ()
|-
| ()
|-
| (''u'')(''v'')
|-
| (''u'')(''v'')
|-
| (''u'', ''v'')
|-
| (''u'', ''v'')
|-
| (''u'' ''v'')
|-
| (''u'' ''v'')
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>0</sub>
|-
| ''f''<sub>0</sub>
|-
| ''f''<sub>1</sub>
|-
| ''f''<sub>1</sub>
|-
| ''f''<sub>6</sub>
|-
| ''f''<sub>6</sub>
|-
| ''f''<sub>7</sub>
|-
| ''f''<sub>7</sub>
|}
|-
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>8</sub>
|-
| ''f''<sub>9</sub>
|-
| ''f''<sub>10</sub>
|-
| ''f''<sub>11</sub>
|-
| ''f''<sub>12</sub>
|-
| ''f''<sub>13</sub>
|-
| ''f''<sub>14</sub>
|-
| ''f''<sub>15</sub>
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''x'' ''y''
|-
| ((''x'', ''y''))
|-
| ''y''
|-
| (''x'' (''y''))
|-
| ''x''
|-
| ((''x'') ''y'')
|-
| ((''x'')(''y''))
|-
| (())
|}
|
{| cellpadding="2" style="background:lightcyan"
| 1 0 0 0
|-
| 1 0 0 1
|-
| 1 0 1 0
|-
| 1 0 1 1
|-
| 1 1 0 0
|-
| 1 1 0 1
|-
| 1 1 1 0
|-
| 1 1 1 1
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''u'' ''v''
|-
| ''u'' ''v''
|-
| ((''u'', ''v''))
|-
| ((''u'', ''v''))
|-
| ((''u'')(''v''))
|-
| ((''u'')(''v''))
|-
| (())
|-
| (())
|}
|
{| cellpadding="2" style="background:lightcyan"
| ''f''<sub>8</sub>
|-
| ''f''<sub>8</sub>
|-
| ''f''<sub>9</sub>
|-
| ''f''<sub>9</sub>
|-
| ''f''<sub>14</sub>
|-
| ''f''<sub>14</sub>
|-
| ''f''<sub>15</sub>
|-
| ''f''<sub>15</sub>
|}
|}
</font><br>
Given the alphabets <font face="lucida calligraphy">U</font> = {''u'', ''v''} and <font face="lucida calligraphy">X</font> = {''x'', ''y''}, along with the corresponding universes of discourse ''U''<sup> •</sup> and ''X''<sup> •</sup> <math>\cong</math> ['''B'''<sup>2</sup>], how many logical transformations of the general form ''G'' = ‹''G''<sub>1</sub>, ''G''<sub>2</sub>› : ''U''<sup> •</sup> → ''X''<sup> •</sup> are there?
Since ''G''<sub>1</sub> and ''G''<sub>2</sub> can be any propositions of the type '''B'''<sup>2</sup> → '''B''', there are 2<sup>4</sup> = 16 choices for each of the maps ''G''<sub>1</sub> and ''G''<sub>2</sub>, and thus there are 2<sup>4</sup><math>\cdot</math>2<sup>4</sup> = 2<sup>8</sup> = 256 different mappings altogether of the form ''G'' : ''U''<sup> •</sup> → ''X''<sup> •</sup>. The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing (''U''<sup> •</sup> → ''X''<sup> •</sup>) = {''G'' : ''U''<sup> •</sup> → ''X''<sup> •</sup>}, and so the cardinality of this ''function space'' can be most conveniently summed up by writing |(''U''<sup> •</sup> → ''X''<sup> •</sup>)| = |('''B'''<sup>2</sup> → '''B'''<sup>2</sup>)| = 4<sup>4</sup> = 256.
Given any transformation ''G'' = ‹''G''<sub>1</sub>, ''G''<sub>2</sub>› : ''U''<sup> •</sup> → ''X''<sup> •</sup> of this type, one can define a couple of further transformations, related to ''G'', that operate between the extended universes, E''U''<sup> •</sup> and E''X''<sup> •</sup>, of its source and target domains.
First, the enlargement map (or the secant transformation) E''G'' = ‹E''G''<sub>1</sub>, E''G''<sub>2</sub>› : E''U''<sup> •</sup> → E''X''<sup> •</sup> is defined by the following set of component equations:
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | E''G''<sub>''i''</sub>
| width="4%" | =
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
|}
|}
</font><br>
Second, the difference map (or the chordal transformation) D''G'' = ‹D''G''<sub>1</sub>, D''G''<sub>2</sub>› : E''U''<sup> •</sup> → E''X''<sup> •</sup> is defined in component-wise fashion as the boolean sum of the initial proposition ''G''<sub>''i''</sub> and the enlarged proposition E''G''<sub>''i''</sub>, for ''i'' = 1, 2, according to the following set of equations:
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | D''G''<sub>''i''</sub>
| width="4%" | =
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| width="4%" | +
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
|-
| width="8%" |
| width="4%" | =
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
| width="4%" | +
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
|}
|}
</font><br>
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write D''G''<sub>''i''</sub> = E''G''<sub>''i''</sub> – ''G''<sub>''i''</sub>, but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition ''q'', then to compute the enlargement E''q'', and finally to determine the difference D''q'' = ''q'' + E''q'', so we let the variant order of terms reflect this sequence of considerations.
Viewed in this light the difference operator D is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation ''G'' and its difference map D''G'', for instance, taking the function space (''U''<sup> •</sup> → ''X''<sup> •</sup>) into (E''U''<sup> •</sup> → E''X''<sup> •</sup>). Given the interpretive flexibility of contexts in which we are allowing a proposition to appear, it should be clear that an operator of this scope is not at all a trivial matter to define properly, and may take some trouble to work out. For the moment, let's content ourselves with returning to particular cases.
In their application to the present example, namely, the logical transformation ''F'' = ‹''f'', ''g''› = ‹((''u'')(''v'')), ((''u'', ''v''))›, the operators E and D respectively produce the enlarged map E''F'' = ‹E''f'', E''g''› and the difference map D''F'' = ‹D''f'', D''g''›, whose components can be given as follows, if the reader, in lieu of a special font for the logical parentheses, can forgive a syntactically bilingual formulation:
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | E''f''
| width="4%" | =
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
|-
| width="8%" | E''g''
| width="4%" | =
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
|}
|}
</font>
<br>
<font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | D''f''
| width="4%" | =
| width="20%" | ((''u'')(''v''))
| width="4%" | +
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
|-
| width="8%" | D''g''
| width="4%" | =
| width="20%" | ((''u'', ''v''))
| width="4%" | +
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
|}
|}
</font><br>
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components ''f'' and ''g'' that we earlier used on ''J''. This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii.
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))'''
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''f''
| = || ''uv'' || <math>\cdot</math> || 1
| + || ''u''(''v'') || <math>\cdot</math> || 1
| + || (''u'')''v'' || <math>\cdot</math> || 1
| + || (''u'')(''v'') || <math>\cdot</math> || 0
|-
| E''f''
| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))
| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
| D''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
| d''f''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| r''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''
| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
|}
|}
</font><br>
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))'''
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''g''
| = || ''uv'' || <math>\cdot</math> || 1
| + || ''u''(''v'') || <math>\cdot</math> || 0
| + || (''u'')''v'' || <math>\cdot</math> || 0
| + || (''u'')(''v'') || <math>\cdot</math> || 1
|-
| E''g''
| = || ''uv'' || <math>\cdot</math> || ((d''u'', d''v''))
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
|-
| D''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| d''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| r''g''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || 0
| + || (''u'')''v'' || <math>\cdot</math> || 0
| + || (''u'')(''v'') || <math>\cdot</math> || 0
|}
|}
</font><br>
Table 67 shows how to compute the analytic series for ''F'' = ‹''f'', ''g''› = ‹((''u'')(''v'')), ((''u'', ''v''))› in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations.
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 67. Computation of an Analytic Series in Terms of Coordinates'''
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u''
| ''v''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d''u''
| d''v''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| ''u''<font face="courier new">’</font>
| ''v''<font face="courier new">’</font>
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 0 || 0
|-
| 1 || 1
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 1 || 1
|-
| 0 || 0
|-
| 0 || 1
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|-
| 1 || 0
|-
| 0 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|}
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| <math>\epsilon</math>''f''
| <math>\epsilon</math>''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| E''f''
| E''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| D''f''
| D''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d''f''
| d''g''
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d<sup>2</sup>''f''
| d<sup>2</sup>''g''
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 1
|-
| 1 || 0
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 1 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 1 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 0 || 1
|-
| 1 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 0
|-
| 1 || 1
|-
| 0 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|-
| valign="top" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 1 || 1
|-
| 1 || 0
|-
| 1 || 0
|-
| 0 || 1
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 0 || 1
|-
| 1 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 0 || 1
|-
| 0 || 0
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 0
|-
| 0 || 0
|-
| 1 || 0
|}
|}
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ '''Table 68. Computation of an Analytic Series in Symbolic Terms'''
|- style="background:paleturquoise"
| ''u'' ''v''
| ''f'' ''g''
| D''f''
| D''g''
| d''f''
| d''g''
| d<sup>2</sup>''f''
| d<sup>2</sup>''g''
|-
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 0
|-
| 0 1
|-
| 1 0
|-
| 1 1
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 1
|-
| 1 0
|-
| 1 0
|-
| 1 1
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ((d''u'')(d''v''))
|-
| (d''u'') d''v''
|-
| d''u'' (d''v'')
|-
| d''u'' d''v''
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| (d''u'', d''v'')
|-
| d''v''
|-
| d''u''
|-
| ( )
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|-
| (d''u'', d''v'')
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| d''u'' d''v''
|-
| d''u'' d''v''
|-
| d''u'' d''v''
|-
| d''u'' d''v''
|}
|
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ( )
|-
| ( )
|-
| ( )
|-
| ( )
|}
|}
<br>
Figure 69 gives a graphical picture of the difference map D''F'' = ‹D''f'', D''g''› for the transformation ''F'' = ‹''f'', ''g''› = ‹((''u'')(''v'')), ((''u'', ''v''))›. This depicts the same information about D''f'' and D''g'' that was given in the corresponding rows of the computation summary in Tables 66-i and 66-ii, excerpted here:
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
|-
| D''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
|
|-
| D''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|}
|}
</font><br>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]</p>
<p><center><font size="+1">'''Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›'''</font></center></p>
Figure 70-a shows a graphical way of picturing the tangent functor map d''F'' = ‹d''f'', d''g''› for the transformation ''F'' = ‹''f'', ''g''› = ›((u)(v)), ((u, v))›. This amounts to the same information about d''f'' and d''g'' that was given in the computation summary of Tables 66-i and 66-ii, the relevant rows of which are repeated here:
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
|
|-
| d''f''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|-
| d''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|}
|}
</font><br>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p>
<p><center><font size="+1">'''Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font></center></p>
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation ''F''‹''u'', ''v''› = ‹((''u'')(''v'')), ((''u'', ''v''))›, roughly in the style of the ''bundle of universes'' type of diagram.
<p>[[Image:Tangent_Functor_Ferris_Wheel.gif|center|frame|<font size="3">'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]</p>
* '''Nota Bene.''' The original Figure 70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.
<pre>
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= du' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | dU' | =
= | o--o o--o | =
= | o--o o--o | =
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | | o--o o--o | | o--o o--o |
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | | | | |
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= u' o-----------------------o v' =
= | dU' | =
= | U' | =
= | o--o o--o | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | /////\ /\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| o//////\XXX/\\\\\\o |
o-----------------------o
o-----------------------o
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
</pre>
==Epilogue, Enchoiry, Exodus==
= du' @ u (v) o-----------------------o dv' @ u (v) =
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
It is time to explain myself . . . . let us stand up.
| width="4%" |
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 79]
|}
==References==
==References==
Recoded: 03 Jun 2007
Recoded: 03 Jun 2007
</pre>
</pre>
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