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'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
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* The boolean domain is a set of two elements, <math>\mathbb{B} = \{ 0, 1 \},</math> represented by the two distinct shadings of the regions inside the rectangle.
* The boolean domain is a set of two elements, <math>\mathbb{B} = \{ 0, 1 \},</math> represented by the two distinct shadings of the regions inside the rectangle.
* According to the conventions observed in this context, the algebraic value 0 is interpreted as the logical value <math>\operatorname{false}</math> and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value <math>\operatorname{true}</math> and represented by the darker shading.
* According to the conventions observed in this context, the algebraic value 0 is interpreted as the logical value <math>\mathrm{false}</math> and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value <math>\mathrm{true}</math> and represented by the darker shading.
* The universe of discourse <math>X\!</math> is the domain of three functions <math>u, v, w : X \to \mathbb{B}</math> called ''basic'', ''coordinate'', or ''simple'' propositions.
* The universe of discourse <math>X\!</math> is the domain of three functions <math>u, v, w : X \to \mathbb{B}</math> called ''basic'', ''coordinate'', or ''simple'' propositions.
In order to discuss questions of this type, it is useful to define several "operators" on functions. An operator is nothing more than a function between sets that happen to have functions as members.
In order to discuss questions of this type, it is useful to define several "operators" on functions. An operator is nothing more than a function between sets that happen to have functions as members.
A typical operator <math>\operatorname{F}</math> takes us from thinking about a given function <math>f\!</math> to thinking about another function <math>g\!</math>. To express the fact that <math>g\!</math> can be obtained by applying the operator <math>\operatorname{F}</math> to <math>f\!</math>, we write <math>g = \operatorname{F}f.</math>
A typical operator <math>\mathrm{F}</math> takes us from thinking about a given function <math>f\!</math> to thinking about another function <math>g\!</math>. To express the fact that <math>g\!</math> can be obtained by applying the operator <math>\mathrm{F}</math> to <math>f\!</math>, we write <math>g = \mathrm{F}f.</math>
The first operator, <math>\operatorname{E}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{E}f</math>, where <math>\operatorname{E}f : X \times X \to Y</math> is defined by the following equation:
The first operator, <math>\mathrm{E}</math>, associates with a function <math>f : X \to Y</math> another function <math>\mathrm{E}f</math>, where <math>\mathrm{E}f : X \times X \to Y</math> is defined by the following equation:
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| <math>\operatorname{E}f(x, y) ~=~ f(x + y).</math>
| <math>\mathrm{E}f(x, y) ~=~ f(x + y).</math>
|}
|}
<math>\operatorname{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away. Thus, <math>\operatorname{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.
<math>\mathrm{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away. Thus, <math>\mathrm{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.
'''Historical Note.''' The "shift operator" <math>\operatorname{E}</math> was originally called the "enlargement operator", hence the initial "E" of the usual notation.
'''Historical Note.''' The "shift operator" <math>\mathrm{E}</math> was originally called the "enlargement operator", hence the initial "E" of the usual notation.
The next operator, <math>\operatorname{D}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{D}f</math>, where <math>\operatorname{D}f : X \times X \to Y</math> is defined by the following equation:
The next operator, <math>\mathrm{D}</math>, associates with a function <math>f : X \to Y</math> another function <math>\mathrm{D}f</math>, where <math>\mathrm{D}f : X \times X \to Y</math> is defined by the following equation:
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| <math>\operatorname{D}f(x, y) ~=~ \operatorname{E}f(x, y) - f(x),</math>
| <math>\mathrm{D}f(x, y) ~=~ \mathrm{E}f(x, y) - f(x),</math>
|}
|}
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| <math>\operatorname{D}f(x, y) ~=~ f(x + y) - f(x).</math>
| <math>\mathrm{D}f(x, y) ~=~ f(x + y) - f(x).</math>
|}
|}
<math>\operatorname{D}</math> is called a "difference operator" because it tells us about the relative change in the value of <math>f\!</math> along the shift from <math>x\!</math> to <math>x + y.\!</math>
<math>\mathrm{D}</math> is called a "difference operator" because it tells us about the relative change in the value of <math>f\!</math> along the shift from <math>x\!</math> to <math>x + y.\!</math>
In practice, one of the variables, <math>x\!</math> or <math>y\!</math>, is often considered to be "less variable" than the other one, being fixed in the context of a concrete discussion. Thus, we might find any one of the following idioms:
In practice, one of the variables, <math>x\!</math> or <math>y\!</math>, is often considered to be "less variable" than the other one, being fixed in the context of a concrete discussion. Thus, we might find any one of the following idioms:
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| <math>\operatorname{D}f : X \times X \to Y,</math>
| <math>\mathrm{D}f : X \times X \to Y,</math>
|-
|-
| <math>\operatorname{D}f(c, x) ~=~ f(c + x) - f(c).</math>
| <math>\mathrm{D}f(c, x) ~=~ f(c + x) - f(c).</math>
|}
|}
Here, <math>c\!</math> is held constant and <math>\operatorname{D}f(c, x)</math> is regarded mainly as a function of the second variable <math>x\!</math>, giving the relative change in <math>f\!</math> at various distances <math>x\!</math> from the center <math>c\!</math>.
Here, <math>c\!</math> is held constant and <math>\mathrm{D}f(c, x)</math> is regarded mainly as a function of the second variable <math>x\!</math>, giving the relative change in <math>f\!</math> at various distances <math>x\!</math> from the center <math>c\!</math>.
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| <math>\operatorname{D}f : X \times X \to Y,</math>
| <math>\mathrm{D}f : X \times X \to Y,</math>
|-
|-
| <math>\operatorname{D}f(x, h) ~=~ f(x + h) - f(x).</math>
| <math>\mathrm{D}f(x, h) ~=~ f(x + h) - f(x).</math>
|}
|}
Here, <math>h\!</math> is either a constant (usually 1), in discrete contexts, or a variably "small" amount (near to 0) over which a limit is being taken, as in continuous contexts. <math>\operatorname{D}f(x, h)</math> is regarded mainly as a function of the first variable <math>x\!</math>, in effect, giving the differences in the value of <math>f\!</math> between <math>x\!</math> and a neighbor that is a distance of <math>h\!</math> away, all the while that <math>x\!</math> itself ranges over its various possible locations.
Here, <math>h\!</math> is either a constant (usually 1), in discrete contexts, or a variably "small" amount (near to 0) over which a limit is being taken, as in continuous contexts. <math>\mathrm{D}f(x, h)</math> is regarded mainly as a function of the first variable <math>x\!</math>, in effect, giving the differences in the value of <math>f\!</math> between <math>x\!</math> and a neighbor that is a distance of <math>h\!</math> away, all the while that <math>x\!</math> itself ranges over its various possible locations.
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| <math>\operatorname{D}f : X \times X \to Y,</math>
| <math>\mathrm{D}f : X \times X \to Y,</math>
|-
|-
| <math>\operatorname{D}f(x, \operatorname{d}x) ~=~ f(x + \operatorname{d}x) - f(x).</math>
| <math>\mathrm{D}f(x, \mathrm{d}x) ~=~ f(x + \mathrm{d}x) - f(x).</math>
|}
|}
This is yet another variant of the previous form, with <math>\operatorname{d}x</math> denoting small changes contemplated in <math>x\!</math>.
This is yet another variant of the previous form, with <math>\mathrm{d}x</math> denoting small changes contemplated in <math>x\!</math>.
That's the basic idea. The next order of business is to develop the logical side of the analogy a bit more fully, and to take up the elaboration of some moderately simple applications of these ideas to a selection of relatively concrete examples.
That's the basic idea. The next order of business is to develop the logical side of the analogy a bit more fully, and to take up the elaboration of some moderately simple applications of these ideas to a selection of relatively concrete examples.
Imagine that we are sitting in one of the cells of a venn diagram, contemplating the walls. There are <math>k\!</math> of them, one for each positive feature <math>x_1, \ldots, x_k</math> in our universe of discourse. Our particular cell is described by a concatenation of <math>k\!</math> signed assertions, positive or negative, regarding each of these features, and this description of our position amounts to what is called an "interpretation" of whatever proposition may rule the space, or reign on the universe of discourse. But are we locked into this interpretation?
Imagine that we are sitting in one of the cells of a venn diagram, contemplating the walls. There are <math>k\!</math> of them, one for each positive feature <math>x_1, \ldots, x_k</math> in our universe of discourse. Our particular cell is described by a concatenation of <math>k\!</math> signed assertions, positive or negative, regarding each of these features, and this description of our position amounts to what is called an "interpretation" of whatever proposition may rule the space, or reign on the universe of discourse. But are we locked into this interpretation?
With respect to each edge <math>x\!</math> of the cell we consider a test proposition <math>\operatorname{d}x</math> that determines our decision whether or not we will make a difference in how we stand regarding <math>x\!</math>. If <math>\operatorname{d}x</math> is true then it marks our decision, intention, or plan to cross over the edge <math>x\!</math> at some point within the purview of the contemplated plan.
With respect to each edge <math>x\!</math> of the cell we consider a test proposition <math>\mathrm{d}x</math> that determines our decision whether or not we will make a difference in how we stand regarding <math>x\!</math>. If <math>\mathrm{d}x</math> is true then it marks our decision, intention, or plan to cross over the edge <math>x\!</math> at some point within the purview of the contemplated plan.
To reckon the effect of several such decisions on our current interpretation, or the value of the reigning proposition, we transform that position or that proposition by making the following array of substitutions everywhere in its expression:
To reckon the effect of several such decisions on our current interpretation, or the value of the reigning proposition, we transform that position or that proposition by making the following array of substitutions everywhere in its expression:
<blockquote>
<blockquote>
<p><math>1.\!</math> Substitute "<math>(x_1, \operatorname{d}x_1)</math>" for "<math>x_1\!</math>"</p>
<p><math>1.\!</math> Substitute "<math>(x_1, \mathrm{d}x_1)</math>" for "<math>x_1\!</math>"</p>
<p><math>2.\!</math> Substitute "<math>(x_2, \operatorname{d}x_2)</math>" for "<math>x_2\!</math>"</p>
<p><math>2.\!</math> Substitute "<math>(x_2, \mathrm{d}x_2)</math>" for "<math>x_2\!</math>"</p>
<p><math>3.\!</math> Substitute "<math>(x_3, \operatorname{d}x_3)</math>" for "<math>x_3\!</math>"</p>
<p><math>3.\!</math> Substitute "<math>(x_3, \mathrm{d}x_3)</math>" for "<math>x_3\!</math>"</p>
<p><math>\ldots</math></p>
<p><math>\ldots</math></p>
<p><math>k.\!</math> Substitute "<math>(x_k, \operatorname{d}x_k)</math>" for "<math>x_k\!</math>"</p>
<p><math>k.\!</math> Substitute "<math>(x_k, \mathrm{d}x_k)</math>" for "<math>x_k\!</math>"</p>
</blockquote>
</blockquote>
And we know the value of the interpretation by whether this last expression issues in a model.
And we know the value of the interpretation by whether this last expression issues in a model.
Applying the enlargement operator <math>\operatorname{E}</math> to the initial proposition <math>q\!</math> yields:
Applying the enlargement operator <math>\mathrm{E}</math> to the initial proposition <math>q\!</math> yields:
<pre>
<pre>
</pre>
</pre>
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<math>u\ v\ w</math>", to a true value under the target proposition <code> (( u v )( u w )( v w )) </code>.
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<math>u\ v\ w</math>", to a true value under the target proposition <code> (( u v )( u w )( v w )) </code>.
The result of applying the difference operator <math>\operatorname{D}</math> to the initial proposition <math>\operatorname{q}</math>, conjoined with a query on the center cell, yields:
The result of applying the difference operator <math>\mathrm{D}</math> to the initial proposition <math>\mathrm{q}</math>, conjoined with a query on the center cell, yields:
<pre>
<pre>
</blockquote>
</blockquote>
To round out the presentation of the Polymorphous Example 1, I will go through what has gone before and lay in the graphic forms of all of the propositional expressions. These graphs, whose official botanical designation makes them out to be a species of ''painted and rooted cacti'' (PARC's), are not too far from the actual graph-theoretic data-structures that result from parsing the cactus string expressions, the ''painted and rooted cactus expressions'' (PARCE's). Finally, I will add a couple of venn diagrams that will serve to illustrate the ''difference opus'' <math>\operatorname{D}q</math>. If you apply an operator to an operand you must arrive at either an opus or an opera, no?
To round out the presentation of the Polymorphous Example 1, I will go through what has gone before and lay in the graphic forms of all of the propositional expressions. These graphs, whose official botanical designation makes them out to be a species of ''painted and rooted cacti'' (PARC's), are not too far from the actual graph-theoretic data-structures that result from parsing the cactus string expressions, the ''painted and rooted cactus expressions'' (PARCE's). Finally, I will add a couple of venn diagrams that will serve to illustrate the ''difference opus'' <math>\mathrm{D}q</math>. If you apply an operator to an operand you must arrive at either an opus or an opera, no?
Consider the polymorphous set <math>Q\!</math> of Example 1 and focus on the central cell, described by the conjunction of logical features in the expression "<math>u\ v\ w\!</math>".
Consider the polymorphous set <math>Q\!</math> of Example 1 and focus on the central cell, described by the conjunction of logical features in the expression "<math>u\ v\ w\!</math>".
And we know the value of the interpretation by whether this last expression issues in a model.
And we know the value of the interpretation by whether this last expression issues in a model.
Applying the enlargement operator <math>\operatorname{E}</math> to the initial proposition <math>q\!</math> yields:
Applying the enlargement operator <math>\mathrm{E}</math> to the initial proposition <math>q\!</math> yields:
<pre>
<pre>
</pre>
</pre>
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<math>u\ v\ w</math>", to a true value under the target proposition <code> (( u v )( u w )( v w )) </code>.
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<math>u\ v\ w</math>", to a true value under the target proposition <code> (( u v )( u w )( v w )) </code>.
The result of applying the difference operator <math>\operatorname{D}</math> to the initial proposition <math>q\!</math>, conjoined with a query on the center cell, yields:
The result of applying the difference operator <math>\mathrm{D}</math> to the initial proposition <math>q\!</math>, conjoined with a query on the center cell, yields:
<pre>
<pre>
</pre>
</pre>
Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant ''difference proposition'' <math>\operatorname{D}q</math> are marked with "<code>@</code>" signs, and the boundary crossings along each path are marked with the corresponding ''differential features'' among the collection <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. In sum, starting from the cell <math>uvw\!</math>, we have the following four paths:
Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant ''difference proposition'' <math>\mathrm{D}q</math> are marked with "<code>@</code>" signs, and the boundary crossings along each path are marked with the corresponding ''differential features'' among the collection <math>\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math>. In sum, starting from the cell <math>uvw\!</math>, we have the following four paths:
<pre>
<pre>
The next thing that one typically does is to consider the effects of various ''operators'' on the proposition of interest, which may be called the ''operand'' or the ''source'' proposition, leaving the corresponding terms ''opus'' or ''target'' as names for the result.
The next thing that one typically does is to consider the effects of various ''operators'' on the proposition of interest, which may be called the ''operand'' or the ''source'' proposition, leaving the corresponding terms ''opus'' or ''target'' as names for the result.
In our initial consideration of the proposition <math>q\!</math>, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis <math>\{ u, v, w \}</math>. As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as ''tacitly embedded'' in any number of higher dimensional spaces. Just by way of starting out, our immediate interest is with the ''first order differential analysis'' (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. Now this does not change the expression of any proposition, like <math>q\!</math>, that does not mention the extra variables, only changing how it gets interpreted as a function. A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics. In this discussion, I will invoke its application under the name of the ''[[tacit extension]]'' of a proposition to any universe of discourse based on a superset of its original basis.
In our initial consideration of the proposition <math>q\!</math>, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis <math>\{ u, v, w \}</math>. As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as ''tacitly embedded'' in any number of higher dimensional spaces. Just by way of starting out, our immediate interest is with the ''first order differential analysis'' (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set <math>\{ u, v, w, \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math>. Now this does not change the expression of any proposition, like <math>q\!</math>, that does not mention the extra variables, only changing how it gets interpreted as a function. A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics. In this discussion, I will invoke its application under the name of the ''[[tacit extension]]'' of a proposition to any universe of discourse based on a superset of its original basis.
I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function <math>q(u, v, w)\!</math> that is given by the following expression:
I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function <math>q(u, v, w)\!</math> that is given by the following expression:
</code></blockquote>
</code></blockquote>
When <math>X\!</math> is the type of space that is generated by <math>\{ u, v, w \}\!</math>, let <math>\operatorname{d}X</math> be the type of space that is generated by <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>, and let <math>X \times \operatorname{d}X</math> be the type of space that is generated by the extended set of boolean basis elements <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. For convenience, define a notation "<math>\operatorname{E}X</math>" so that <math>\operatorname{E}X = X \times \operatorname{d}X</math>. Even though the differential variables are in some abstract sense no different than other boolean variables, it usually helps to mark their distinctive roles and their differential interpretation by means of the distinguishing domain name "<math>\operatorname{d}\mathbb{B}</math>". Using these designations of logical spaces, the propositions over them can be assigned both abstract and concrete types.
When <math>X\!</math> is the type of space that is generated by <math>\{ u, v, w \}\!</math>, let <math>\mathrm{d}X</math> be the type of space that is generated by <math>\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math>, and let <math>X \times \mathrm{d}X</math> be the type of space that is generated by the extended set of boolean basis elements <math>\{ u, v, w, \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math>. For convenience, define a notation "<math>\mathrm{E}X</math>" so that <math>\mathrm{E}X = X \times \mathrm{d}X</math>. Even though the differential variables are in some abstract sense no different than other boolean variables, it usually helps to mark their distinctive roles and their differential interpretation by means of the distinguishing domain name "<math>\mathrm{d}\mathbb{B}</math>". Using these designations of logical spaces, the propositions over them can be assigned both abstract and concrete types.
For instance, consider the proposition <math>q(u, v, w)\!</math>, as before, and then consider its tacit extension <math>q(u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w)\!</math>, the latter of which may be indicated more explicitly as "<math>\operatorname{e}q\!</math>".
For instance, consider the proposition <math>q(u, v, w)\!</math>, as before, and then consider its tacit extension <math>q(u, v, w, \mathrm{d}u, \mathrm{d}v, \mathrm{d}w)\!</math>, the latter of which may be indicated more explicitly as "<math>\mathrm{e}q\!</math>".
#<p>Proposition <math>q\!</math> is abstractly typed as <math>q : \mathbb{B}^3 \to \mathbb{B}.</math></p><p>Proposition <math>q\!</math> is concretely typed as <math>q : X \to \mathbb{B}.</math></p>
#<p>Proposition <math>q\!</math> is abstractly typed as <math>q : \mathbb{B}^3 \to \mathbb{B}.</math></p><p>Proposition <math>q\!</math> is concretely typed as <math>q : X \to \mathbb{B}.</math></p>
#<p>Proposition <math>\operatorname{e}q\!</math> is abstractly typed as <math>\operatorname{e}q : \mathbb{B}^3 \times \operatorname{d}\mathbb{B}^3 \to \mathbb{B}.</math></p><p>Proposition <math>\operatorname{e}q\!</math> is concretely typed as <math>\operatorname{e}q : X \times \operatorname{d}X \to \mathbb{B}.</math></p><p>Succinctly, <math>\operatorname{e}q : \operatorname{E}X \to \mathbb{B}.</math></p>
#<p>Proposition <math>\mathrm{e}q\!</math> is abstractly typed as <math>\mathrm{e}q : \mathbb{B}^3 \times \mathrm{d}\mathbb{B}^3 \to \mathbb{B}.</math></p><p>Proposition <math>\mathrm{e}q\!</math> is concretely typed as <math>\mathrm{e}q : X \times \mathrm{d}X \to \mathbb{B}.</math></p><p>Succinctly, <math>\mathrm{e}q : \mathrm{E}X \to \mathbb{B}.</math></p>
We now return to our consideration of the effects of various differential operators on propositions. This time around we have enough exact terminology that we shall be able to explain what is actually going on here in a rather more articulate fashion.
We now return to our consideration of the effects of various differential operators on propositions. This time around we have enough exact terminology that we shall be able to explain what is actually going on here in a rather more articulate fashion.
The first transformation of the source proposition <math>q\!</math> that we may wish to stop and examine, though it is not unusual to skip right over this stage of analysis, frequently regarding it as a purely intermediary stage, holding scarcely even so much as the passing interest, is the work of the ''enlargement'' or ''shift'' operator <math>\operatorname{E}.</math>
The first transformation of the source proposition <math>q\!</math> that we may wish to stop and examine, though it is not unusual to skip right over this stage of analysis, frequently regarding it as a purely intermediary stage, holding scarcely even so much as the passing interest, is the work of the ''enlargement'' or ''shift'' operator <math>\mathrm{E}.</math>
Applying the operator <math>\operatorname{E}</math> to the operand proposition <math>q\!</math> yields:
Applying the operator <math>\mathrm{E}</math> to the operand proposition <math>q\!</math> yields:
<pre>
<pre>
</pre>
</pre>
The enlarged proposition <math>\operatorname{E}q</math> is minimally interpretable as a function on the six variables of <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}.</math> .In other words, <math>\operatorname{E}q : \operatorname{E}X \to \mathbb{B},</math> or <math>\operatorname{E}q : X \times \operatorname{d}X \to \mathbb{B}.</math>
The enlarged proposition <math>\mathrm{E}q</math> is minimally interpretable as a function on the six variables of <math>\{ u, v, w, \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}.</math> .In other words, <math>\mathrm{E}q : \mathrm{E}X \to \mathbb{B},</math> or <math>\mathrm{E}q : X \times \mathrm{d}X \to \mathbb{B}.</math>
Conjoining a query on the center cell, <math>c = u\ v\ w\!</math>, yields:
Conjoining a query on the center cell, <math>c = u\ v\ w\!</math>, yields:
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The models of this last expression tell us which combinations of feature changes among the set <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<code>u v w</code>", to a true value under the given proposition <code>(( u v )( u w )( v w ))</code>.
The models of this last expression tell us which combinations of feature changes among the set <math>\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math> will take us from our present interpretation, the center cell expressed by "<code>u v w</code>", to a true value under the given proposition <code>(( u v )( u w )( v w ))</code>.
The models of <math>\operatorname{E}q \cdot c</math> can be described in the usual ways as follows:
The models of <math>\mathrm{E}q \cdot c</math> can be described in the usual ways as follows:
* The points of the space <math>\operatorname{E}X</math> that have the following coordinate descriptions:
* The points of the space <math>\mathrm{E}X</math> that have the following coordinate descriptions:
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* The points of the space <math>\operatorname{E}X</math> that have the following conjunctive expressions:
* The points of the space <math>\mathrm{E}X</math> that have the following conjunctive expressions:
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In summary, <math>\operatorname{E}q \cdot c</math> informs us that we can get from <math>c\!</math> to a model of <math>q\!</math> by changing our position with respect to <math>u, v, w\!</math> according to the following description:
In summary, <math>\mathrm{E}q \cdot c</math> informs us that we can get from <math>c\!</math> to a model of <math>q\!</math> by changing our position with respect to <math>u, v, w\!</math> according to the following description:
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I think that it would be worth our time to diagram the models of the ''enlarged'' or ''shifted'' proposition, <math>\operatorname{E}q,</math> at least, the selection of them that we find issuing from the center cell <math>c.\!</math>
I think that it would be worth our time to diagram the models of the ''enlarged'' or ''shifted'' proposition, <math>\mathrm{E}q,</math> at least, the selection of them that we find issuing from the center cell <math>c.\!</math>
Figure 4 is an extended venn diagram for the proposition <math>\operatorname{E}q \cdot c,</math> where the shaded area gives the models of <math>q\!</math> and the "<code>@</code>" signs mark the terminal points of the requisite feature alterations.
Figure 4 is an extended venn diagram for the proposition <math>\mathrm{E}q \cdot c,</math> where the shaded area gives the models of <math>q\!</math> and the "<code>@</code>" signs mark the terminal points of the requisite feature alterations.
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Let <math>\mathcal{X} = \{ x_1, \ldots, x_k \}</math> be a finite set of variables, regarded as a formal alphabet of formal symbols but listed here without quotation marks. Starting from this initial alphabet, the following items may then be defined:
Let <math>\mathcal{X} = \{ x_1, \ldots, x_k \}</math> be a finite set of variables, regarded as a formal alphabet of formal symbols but listed here without quotation marks. Starting from this initial alphabet, the following items may then be defined:
#<p>The "(first order) differential alphabet",</p><p><math>\operatorname{d}\mathcal{X} = \{ \operatorname{d}x_1, \ldots, \operatorname{d}x_k \}.</math></p>
#<p>The "(first order) differential alphabet",</p><p><math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_k \}.</math></p>
#<p>The "(first order) extended alphabet",</p><p><math>\operatorname{E}\mathcal{X} = \mathcal{X} \cup \operatorname{d}\mathcal{X},</math></p><p><math>\operatorname{E}\mathcal{X} = \{ x_1, \dots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k \}.</math></p>
#<p>The "(first order) extended alphabet",</p><p><math>\mathrm{E}\mathcal{X} = \mathcal{X} \cup \mathrm{d}\mathcal{X},</math></p><p><math>\mathrm{E}\mathcal{X} = \{ x_1, \dots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k \}.</math></p>
Before we continue with the differential analysis of the source proposition <math>q\!</math>, we need to pause and take another look at just how it shapes up in the light of the extended universe <math>\operatorname{E}X,</math> in other words, to examine in detail its tacit extension <math>\operatorname{e}q.\!</math>
Before we continue with the differential analysis of the source proposition <math>q\!</math>, we need to pause and take another look at just how it shapes up in the light of the extended universe <math>\mathrm{E}X,</math> in other words, to examine in detail its tacit extension <math>\mathrm{e}q.\!</math>
The models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> can be comprehended as follows:
The models of <math>\mathrm{e}q\!</math> in <math>\mathrm{E}X\!</math> can be comprehended as follows:
*<p>Working in the ''summary coefficient'' form of representation, if the coordinate list <math>\mathbf{x}\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a coordinate list <math>\operatorname{e}\mathbf{x}\!</math> as a model for <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> just by appending any combination of values for the differential variables in <math>\operatorname{d}\mathcal{X}.</math></p><p>For example, to focus once again on the center cell <math>c,\!</math> which happens to be a model of the proposition <math>q\!</math> in <math>X,\!</math> one can extend <math>c\!</math> in eight different ways into <math>\operatorname{E}X,\!</math> and thus get eight models of the tacit extension <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X.\!</math></p><p>It is a trivial exercise to write these out, but it is useful to do so at least once in order to see the patterns of data involved.</p><p>The tacit extensions of <math>c\!</math> that are models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> are as follows:</p>
*<p>Working in the ''summary coefficient'' form of representation, if the coordinate list <math>\mathbf{x}\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a coordinate list <math>\mathrm{e}\mathbf{x}\!</math> as a model for <math>\mathrm{e}q\!</math> in <math>\mathrm{E}X\!</math> just by appending any combination of values for the differential variables in <math>\mathrm{d}\mathcal{X}.</math></p><p>For example, to focus once again on the center cell <math>c,\!</math> which happens to be a model of the proposition <math>q\!</math> in <math>X,\!</math> one can extend <math>c\!</math> in eight different ways into <math>\mathrm{E}X,\!</math> and thus get eight models of the tacit extension <math>\mathrm{e}q\!</math> in <math>\mathrm{E}X.\!</math></p><p>It is a trivial exercise to write these out, but it is useful to do so at least once in order to see the patterns of data involved.</p><p>The tacit extensions of <math>c\!</math> that are models of <math>\mathrm{e}q\!</math> in <math>\mathrm{E}X\!</math> are as follows:</p>
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* <p>Working in the ''conjunctive product'' form of representation, if the conjunctive proposition <math>x\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a conjunctive proposition <math>\operatorname{e}x\!</math> as a model for <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> just by appending any combination of values for the differential variables in <math>\operatorname{d}\mathcal{X}.</math></p><p>The tacit extensions of <math>c\!</math> that are models of <math>\operatorname{e}q\!</math> in <math>\operatorname{E}X\!</math> are as follows:</p>
* <p>Working in the ''conjunctive product'' form of representation, if the conjunctive proposition <math>x\!</math> is a model of <math>q\!</math> in <math>X,\!</math> then one can construct a conjunctive proposition <math>\mathrm{e}x\!</math> as a model for <math>\mathrm{e}q\!</math> in <math>\mathrm{E}X\!</math> just by appending any combination of values for the differential variables in <math>\mathrm{d}\mathcal{X}.</math></p><p>The tacit extensions of <math>c\!</math> that are models of <math>\mathrm{e}q\!</math> in <math>\mathrm{E}X\!</math> are as follows:</p>
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In short, <math>\operatorname{e}q \cdot c</math> just enumerates all of the possible changes in <math>\operatorname{E}X\!</math> that ''derive from'', ''issue from'', or ''stem from'' the cell <math>c\!</math> in <math>X.\!</math>
In short, <math>\mathrm{e}q \cdot c</math> just enumerates all of the possible changes in <math>\mathrm{E}X\!</math> that ''derive from'', ''issue from'', or ''stem from'' the cell <math>c\!</math> in <math>X.\!</math>
That was pretty tedious, and I know that it all appears to be totally trivial, which is precisely why we usually just leave it "tacit" in the first place, but hard experience, and a real acquaintance with the confusion that can beset us when we do not render these implicit grounds explicit, have taught me that it will ultimately be necessary to get clear about it, and by this ''clear'' to say ''marked'', not merely ''transparent''.
That was pretty tedious, and I know that it all appears to be totally trivial, which is precisely why we usually just leave it "tacit" in the first place, but hard experience, and a real acquaintance with the confusion that can beset us when we do not render these implicit grounds explicit, have taught me that it will ultimately be necessary to get clear about it, and by this ''clear'' to say ''marked'', not merely ''transparent''.
With that intermezzo, with all of its echoes of the opening overture, over and done, let us now return to that droller drama, already fast in progress, the differential disentanglements, hopefully toward the end of a grandly enlightening denouement, of the ever-polymorphous <math>Q.\!</math>
With that intermezzo, with all of its echoes of the opening overture, over and done, let us now return to that droller drama, already fast in progress, the differential disentanglements, hopefully toward the end of a grandly enlightening denouement, of the ever-polymorphous <math>Q.\!</math>
The next transformation of the source proposition <math>q,\!</math> that we are typically aiming to contemplate in the process of carrying out a ''differential analysis'' of its ''dynamic'' effects or implications, is the yield of the so-called ''difference'' or ''delta'' operator <math>\operatorname{D}.</math> The resultant ''difference proposition'' <math>\operatorname{D}q</math> is defined in terms of the source proposition <math>q\!</math> and the ''shifted proposition'' <math>\operatorname{E}q</math> thusly:
The next transformation of the source proposition <math>q,\!</math> that we are typically aiming to contemplate in the process of carrying out a ''differential analysis'' of its ''dynamic'' effects or implications, is the yield of the so-called ''difference'' or ''delta'' operator <math>\mathrm{D}.</math> The resultant ''difference proposition'' <math>\mathrm{D}q</math> is defined in terms of the source proposition <math>q\!</math> and the ''shifted proposition'' <math>\mathrm{E}q</math> thusly:
: <math>\operatorname{D}q = \operatorname{E}q - q = \operatorname{E}q - \operatorname{e}q.</math>
: <math>\mathrm{D}q = \mathrm{E}q - q = \mathrm{E}q - \mathrm{e}q.</math>
: Since "+" and "-" signify the same operation over <math>\mathbb{B},</math> we have:
: Since "+" and "-" signify the same operation over <math>\mathbb{B},</math> we have:
: <math>\operatorname{D}q = \operatorname{E}q + q = \operatorname{E}q + \operatorname{e}q.</math>
: <math>\mathrm{D}q = \mathrm{E}q + q = \mathrm{E}q + \mathrm{e}q.</math>
: Since "+" = "exclusive-or", cactus syntax expresses this as:
: Since "+" = "exclusive-or", cactus syntax expresses this as:
Recall that a <math>k</math>-place bracket "<math>(x_1, x_2, \ldots, x_k)\!</math>" is interpreted (in the ''existential interpretation'') to mean "Exactly one of the <math>x_j\!</math> is false", thus the two-place bracket is equivalent to the exclusive-or.
Recall that a <math>k</math>-place bracket "<math>(x_1, x_2, \ldots, x_k)\!</math>" is interpreted (in the ''existential interpretation'') to mean "Exactly one of the <math>x_j\!</math> is false", thus the two-place bracket is equivalent to the exclusive-or.
The result of applying the difference operator <math>\operatorname{D}</math> to the source proposition <math>q,\!</math> conjoined with a query on the center cell <math>c,\!</math> is:
The result of applying the difference operator <math>\mathrm{D}</math> to the source proposition <math>q,\!</math> conjoined with a query on the center cell <math>c,\!</math> is:
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The models of the difference proposition <math>\operatorname{D}q \cdot uvw\!</math> are:
The models of the difference proposition <math>\mathrm{D}q \cdot uvw\!</math> are:
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Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant ''difference proposition'' <math>\operatorname{D}q</math> are marked with "<code>@</code>" signs, and the boundary crossings along each path are marked with the corresponding ''differential features'' among the collection <math>\{ \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>. In sum, starting from the cell <math>u\ v\ w,</math> we have the following four paths:
Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant ''difference proposition'' <math>\mathrm{D}q</math> are marked with "<code>@</code>" signs, and the boundary crossings along each path are marked with the corresponding ''differential features'' among the collection <math>\{ \mathrm{d}u, \mathrm{d}v, \mathrm{d}w \}</math>. In sum, starting from the cell <math>u\ v\ w,</math> we have the following four paths:
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That sums up, but rather more carefully, the material that I ran through just a bit too quickly the first time around. Next time, I will begin to develop an alternative style of diagram for depicting these types of differential settings.
That sums up, but rather more carefully, the material that I ran through just a bit too quickly the first time around. Next time, I will begin to develop an alternative style of diagram for depicting these types of differential settings.
Another way of looking at this situation is by letting the (first order) differential features <math>\operatorname{d}u, \operatorname{d}v, \operatorname{d}w</math> be viewed as the features of another universe of discourse, called the ''tangent universe'' to <math>X\!</math> with respect to the interpretation <math>c\!</math> and represented as <math>\operatorname{d}X \cdot c</math> . In this setting, <math>\operatorname{D}q \cdot c</math> , the ''difference proposition'' of <math>q\!</math> at the interpretation <math>c\!</math> , where <math>c = u\ v\ w</math> , is marked by the shaded region in Figure 4.
Another way of looking at this situation is by letting the (first order) differential features <math>\mathrm{d}u, \mathrm{d}v, \mathrm{d}w</math> be viewed as the features of another universe of discourse, called the ''tangent universe'' to <math>X\!</math> with respect to the interpretation <math>c\!</math> and represented as <math>\mathrm{d}X \cdot c</math> . In this setting, <math>\mathrm{D}q \cdot c</math> , the ''difference proposition'' of <math>q\!</math> at the interpretation <math>c\!</math> , where <math>c = u\ v\ w</math> , is marked by the shaded region in Figure 4.
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Taken in the context of the tangent universe to <math>X\!</math> at <math>c = u\ v\ w</math> , written <math>\operatorname{d}X \cdot c</math> or <math>\operatorname{d}X \cdot u\ v\ w</math> , the shaded area of Figure 4 indicates the models of the difference proposition <math>\operatorname{d}q \cdot u\ v\ w</math> , specifically:
Taken in the context of the tangent universe to <math>X\!</math> at <math>c = u\ v\ w</math> , written <math>\mathrm{d}X \cdot c</math> or <math>\mathrm{d}X \cdot u\ v\ w</math> , the shaded area of Figure 4 indicates the models of the difference proposition <math>\mathrm{d}q \cdot u\ v\ w</math> , specifically:
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I'm going to let that settle a while.
I'm going to let that settle a while.
Table 5 sums up the facts of the physical situation at equilibrium. If we let <math>\mathbf{B} = \{ \mathrm{charged}, \mathrm{resting} \} = \{ \mathrm{moving}, \mathrm{steady} \} = \{ \mathrm{note}, \mathrm{rest} \},</math> or whatever candidates you pick for the 2-membered set in question, the Table shows a function <math>f : \mathbf{B} \times \mathbf{B} \to \mathbf{B},</math> where <math>f(x, y) = (x, y) = \operatorname{XOR}(x, y).\!</math>
Table 5 sums up the facts of the physical situation at equilibrium. If we let <math>\mathbf{B} = \{ \mathrm{charged}, \mathrm{resting} \} = \{ \mathrm{moving}, \mathrm{steady} \} = \{ \mathrm{note}, \mathrm{rest} \},</math> or whatever candidates you pick for the 2-membered set in question, the Table shows a function <math>f : \mathbf{B} \times \mathbf{B} \to \mathbf{B},</math> where <math>f(x, y) = (x, y) = \mathrm{XOR}(x, y).\!</math>
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Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.
Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.
Start with a proposition of the form "<math>x\ \operatorname{and}\ y</math>". This is graphed as two labels attached to a root node:
Start with a proposition of the form "<math>x\ \mathrm{and}\ y</math>". This is graphed as two labels attached to a root node:
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Now ask yourself: What is the value of the proposition <math>xy\!</math> at a distance of <math>\operatorname{d}x</math> and <math>\operatorname{d}y</math> from the cell <math>xy\!</math> where you are standing?
Now ask yourself: What is the value of the proposition <math>xy\!</math> at a distance of <math>\mathrm{d}x</math> and <math>\mathrm{d}y</math> from the cell <math>xy\!</math> where you are standing?
Don't think about it — just compute:
Don't think about it — just compute:
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However you draw it, these expressions follow because the expression <math>x + \operatorname{d}x,</math> where the plus sign indicates (mod 2) addition in <math>\mathbb{B},</math> and thus corresponds to an exclusive-or in logic, parses to a graph of the following form:
However you draw it, these expressions follow because the expression <math>x + \mathrm{d}x,</math> where the plus sign indicates (mod 2) addition in <math>\mathbb{B},</math> and thus corresponds to an exclusive-or in logic, parses to a graph of the following form:
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We have just met with the fact that the differential of the <math>\operatorname{and}</math> is the <math>\operatorname{or}</math> of the differentials. Briefly summarized:
We have just met with the fact that the differential of the <math>\mathrm{and}</math> is the <math>\mathrm{or}</math> of the differentials. Briefly summarized:
: <p><math>x\ \operatorname{and}\ y\ \xrightarrow{\operatorname{Diff}}\ \operatorname{d}x \ \operatorname{or}\ \operatorname{d}y</math></p>
: <p><math>x\ \mathrm{and}\ y\ \xrightarrow{\mathrm{Diff}}\ \mathrm{d}x \ \mathrm{or}\ \mathrm{d}y</math></p>
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A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
: Let <math>X\!</math> be the set of values <math>\{ (\!|x|\!), x \} = \{ \operatorname{not}\ x, x \}.</math>
: Let <math>X\!</math> be the set of values <math>\{ (\!|x|\!), x \} = \{ \mathrm{not}\ x, x \}.</math>
: Let <math>Y\!</math> be the set of values <math>\{ (\!|y|\!), y \} = \{ \operatorname{not}\ y, y \}.</math>
: Let <math>Y\!</math> be the set of values <math>\{ (\!|y|\!), y \} = \{ \mathrm{not}\ y, y \}.</math>
Then interpret the usual propositions about <math>x, y\!</math> as functions of the concrete type <math>f : X \times Y \to \mathbb{B}.</math>
Then interpret the usual propositions about <math>x, y\!</math> as functions of the concrete type <math>f : X \times Y \to \mathbb{B}.</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\mathrm{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\mathrm{F}f.</math>
The first couple of operators that we need to consider are logical analogues of those that occur in the classical ''finite difference calculus'', namely:
The first couple of operators that we need to consider are logical analogues of those that occur in the classical ''finite difference calculus'', namely:
: The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
: The ''difference operator'' <math>\Delta,\!</math> written here as <math>\mathrm{D}.</math>
: The ''enlargement operator'' <math>\Epsilon,\!</math> written here as <math>\operatorname{E}.</math>
: The ''enlargement operator'' <math>\Epsilon,\!</math> written here as <math>\mathrm{E}.</math>
These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.
These days, <math>\mathrm{E}</math> is more often called the ''shift operator''.
In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse. We mount up from the space <math>U = X \times Y</math> to its ''differential extension'':
In order to describe the universe in which these operators operate, it will be necessary to enlarge our original universe of discourse. We mount up from the space <math>U = X \times Y</math> to its ''differential extension'':
: <p><math>\operatorname{E}U = U \times \operatorname{d}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y</math></p>
: <p><math>\mathrm{E}U = U \times \mathrm{d}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y</math></p>
: <p><math>\operatorname{d}X = \{ (\!|\operatorname{d}x|\!), \operatorname{d}x \}</math></p>
: <p><math>\mathrm{d}X = \{ (\!|\mathrm{d}x|\!), \mathrm{d}x \}</math></p>
: <p><math>\operatorname{d}Y = \{ (\!|\operatorname{d}y|\!), \operatorname{d}y \}</math></p>
: <p><math>\mathrm{d}Y = \{ (\!|\mathrm{d}y|\!), \mathrm{d}y \}</math></p>
The interpretations of these new symbols can be diverse, but the easiest for now is just to say that <math>\operatorname{d}x</math> means "change <math>x\!</math> " and <math>\operatorname{d}y</math> means "change <math>y\!</math> ".
The interpretations of these new symbols can be diverse, but the easiest for now is just to say that <math>\mathrm{d}x</math> means "change <math>x\!</math> " and <math>\mathrm{d}y</math> means "change <math>y\!</math> ".
To draw the differential extension <math>\operatorname{E}U</math> of our present universe <math>U = X \times Y</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \operatorname{d}X, \operatorname{d}Y,</math> but we can project a suggestion of what it's about on the universe <math>X \times Y</math> by drawing arrows that cross designated borders, labeling the arrows as <math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>(\!|x|\!)</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>(\!|y|\!),</math> in either direction, in either case.
To draw the differential extension <math>\mathrm{E}U</math> of our present universe <math>U = X \times Y</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \mathrm{d}X, \mathrm{d}Y,</math> but we can project a suggestion of what it's about on the universe <math>X \times Y</math> by drawing arrows that cross designated borders, labeling the arrows as <math>\mathrm{d}x</math> when crossing the border between <math>x\!</math> and <math>(\!|x|\!)</math> and as <math>\mathrm{d}y</math> when crossing the border between <math>y\!</math> and <math>(\!|y|\!),</math> in either direction, in either case.
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We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition <math>(\!| \operatorname{d}x\ (\!| \operatorname{d}y |\!) |\!)</math> to say "<math>\operatorname{d}x \Rightarrow \operatorname{d}y</math> ", in other words, however you wish to take it, whether indicatively or injunctively, as saying something to the effect that there is "no change in <math>x\!</math> without a change in <math>y\!</math> ".
We can form propositions from these differential variables in the same way that we would any other logical variables, for instance, interpreting the proposition <math>(\!| \mathrm{d}x\ (\!| \mathrm{d}y |\!) |\!)</math> to say "<math>\mathrm{d}x \Rightarrow \mathrm{d}y</math> ", in other words, however you wish to take it, whether indicatively or injunctively, as saying something to the effect that there is "no change in <math>x\!</math> without a change in <math>y\!</math> ".
Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y\!, </math> the (''first order'') ''enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> in <math>\operatorname{E}U</math> that is defined by the formula <math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = f(x + \operatorname{d}x, y + \operatorname{d}y).</math>
Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y\!, </math> the (''first order'') ''enlargement'' of <math>f\!</math> is the proposition <math>\mathrm{E}f</math> in <math>\mathrm{E}U</math> that is defined by the formula <math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = f(x + \mathrm{d}x, y + \mathrm{d}y).</math>
In the example <math>f(x, y) = xy,\!</math> we obtain:
In the example <math>f(x, y) = xy,\!</math> we obtain:
: <p><math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = (x + \operatorname{d}x)(y + \operatorname{d}y).</math></p>
: <p><math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = (x + \mathrm{d}x)(y + \mathrm{d}y).</math></p>
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Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y,</math> the (''first order'') ''difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> in <math>\operatorname{E}U</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full, <math>\operatorname{D}f(x, y, \operatorname{d}x, \operatorname{d}y) = f(x + \operatorname{d}x, y + \operatorname{d}y) - f(x, y).</math>
Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y,</math> the (''first order'') ''difference'' of <math>f\!</math> is the proposition <math>\mathrm{D}f</math> in <math>\mathrm{E}U</math> that is defined by the formula <math>\mathrm{D}f = \mathrm{E}f - f,</math> or, written out in full, <math>\mathrm{D}f(x, y, \mathrm{d}x, \mathrm{d}y) = f(x + \mathrm{d}x, y + \mathrm{d}y) - f(x, y).</math>
In the example <math>f(x, y) = xy,\!</math> the result is:
In the example <math>f(x, y) = xy,\!</math> the result is:
: <p><math>\operatorname{D}f(x, y, \operatorname{d}x, \operatorname{d}y) = (x + \operatorname{d}x)(y + \operatorname{d}y) - xy.</math></p>
: <p><math>\mathrm{D}f(x, y, \mathrm{d}x, \mathrm{d}y) = (x + \mathrm{d}x)(y + \mathrm{d}y) - xy.</math></p>
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We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>xy,\!</math> in as much as if to say, at the place where <math>x = 1\!</math> and <math>y = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{xy}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that states that:
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>xy,\!</math> in as much as if to say, at the place where <math>x = 1\!</math> and <math>y = 1.\!</math> This evaluation is written in the form <math>\mathrm{D}f|_{xy}</math> or <math>\mathrm{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that states that:
: <p><math>f = xy = x\ \operatorname{and}\ y \Rightarrow \operatorname{D}f|_{xy} = (\!|(\!| \operatorname{d}x |\!)(\!| \operatorname{d}y |\!)|\!) = \operatorname{d}x\ \operatorname{or}\ \operatorname{d}y.</math></p>
: <p><math>f = xy = x\ \mathrm{and}\ y \Rightarrow \mathrm{D}f|_{xy} = (\!|(\!| \mathrm{d}x |\!)(\!| \mathrm{d}y |\!)|\!) = \mathrm{d}x\ \mathrm{or}\ \mathrm{d}y.</math></p>
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The picture illustrates the analysis of the inclusive disjunction <math>(\!|(\!| \operatorname{d}x |\!)(\!| \operatorname{d}y |\!)|\!)</math> into the following exclusive disjunction:
The picture illustrates the analysis of the inclusive disjunction <math>(\!|(\!| \mathrm{d}x |\!)(\!| \mathrm{d}y |\!)|\!)</math> into the following exclusive disjunction:
: <p><math>\operatorname{d}x\ (\!| \operatorname{d}y |\!) + \operatorname{d}y\ (\!| \operatorname{d}x |\!) + \operatorname{d}x\ \operatorname{d}y.</math></p>
: <p><math>\mathrm{d}x\ (\!| \mathrm{d}y |\!) + \mathrm{d}y\ (\!| \mathrm{d}x |\!) + \mathrm{d}x\ \mathrm{d}y.</math></p>
The latter proposition may be interpreted as saying "change <math>x\!</math> or change <math>y\!</math> or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.
The latter proposition may be interpreted as saying "change <math>x\!</math> or change <math>y\!</math> or both". And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.
We have just computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_p</math> for the proposition <math>f(x, y) = xy\!</math> at the point <math>p\!</math> where <math>x = 1\!</math> and <math>y = 1.\!</math>
We have just computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\mathrm{D}f_p</math> for the proposition <math>f(x, y) = xy\!</math> at the point <math>p\!</math> where <math>x = 1\!</math> and <math>y = 1.\!</math>
In the universe <math>U = X \times Y</math> the four propositions <math>xy,\ x (\!| y |\!),\ (\!| x |\!) y,\ (\!| x |\!)(\!| y |\!)</math> that indicate the ''cells'', or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),\ (1, 0),\ (0, 1),\ (0, 0),</math> respectively.
In the universe <math>U = X \times Y</math> the four propositions <math>xy,\ x (\!| y |\!),\ (\!| x |\!) y,\ (\!| x |\!)(\!| y |\!)</math> that indicate the ''cells'', or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),\ (1, 0),\ (0, 1),\ (0, 0),</math> respectively.
Thus, we can write <math>\operatorname{D}f_p = \operatorname{D}f|_p = \operatorname{D}f|_{(1, 1)} = \operatorname{D}f|_{xy},</math> so long as we know the frame of reference in force.
Thus, we can write <math>\mathrm{D}f_p = \mathrm{D}f|_p = \mathrm{D}f|_{(1, 1)} = \mathrm{D}f|_{xy},</math> so long as we know the frame of reference in force.
Sticking with the example <math>f(x, y) = xy,\!</math> let us compute the value of the difference proposition <math>\operatorname{D}f</math> at all of the points.
Sticking with the example <math>f(x, y) = xy,\!</math> let us compute the value of the difference proposition <math>\mathrm{D}f</math> at all of the points.
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This really just constitutes a depiction of the interpretations in <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y</math> that satisfy the difference proposition <math>\operatorname{D}f,</math> namely, these:
This really just constitutes a depiction of the interpretations in <math>\mathrm{E}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y</math> that satisfy the difference proposition <math>\mathrm{D}f,</math> namely, these:
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By inspection, it is fairly easy to understand <math>\operatorname{D}f</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by <math>f(x, y).\!</math>
By inspection, it is fairly easy to understand <math>\mathrm{D}f</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by <math>f(x, y).\!</math>
We have been studying the action of the difference operator <math>\operatorname{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>xy.\!</math> We described <math>\operatorname{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\operatorname{D}f : X \times Y \times \operatorname{d}X \times \operatorname{d}Y \to \mathbb{B}.</math> Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of <math>\operatorname{D}f</math> distribute within the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> we can depict <math>\operatorname{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U = X \times Y</math> and whose arrows are labeled with the elements of <math>\operatorname{d}U = \operatorname{d}X \times \operatorname{d}Y.</math>
We have been studying the action of the difference operator <math>\mathrm{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>xy.\!</math> We described <math>\mathrm{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\mathrm{D}f : X \times Y \times \mathrm{d}X \times \mathrm{d}Y \to \mathbb{B}.</math> Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of <math>\mathrm{D}f</math> distribute within the extended universe <math>\mathrm{E}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y,</math> we can depict <math>\mathrm{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U = X \times Y</math> and whose arrows are labeled with the elements of <math>\mathrm{d}U = \mathrm{d}X \times \mathrm{d}Y.</math>
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Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning. We will encounter more and more of these variant readings as we go.
Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning. We will encounter more and more of these variant readings as we go.
The enlargement operator <math>\operatorname{E},</math> also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, <math>f(x, y) = xy.\!</math>
The enlargement operator <math>\mathrm{E},</math> also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, <math>f(x, y) = xy.\!</math>
Introduce a suitably generic definition of the extended universe of discourse:
Introduce a suitably generic definition of the extended universe of discourse:
: For <math>U = X_1 \times \ldots \times X_k</math>,
: For <math>U = X_1 \times \ldots \times X_k</math>,
: let <math>\operatorname{E}U = U \times \operatorname{d}U = X_1 \times \ldots \times X_k \times \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.</math>
: let <math>\mathrm{E}U = U \times \mathrm{d}U = X_1 \times \ldots \times X_k \times \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k.</math>
For a proposition <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the (first order) enlargement of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}U \to \mathbb{B}</math> that is defined by:
For a proposition <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the (first order) enlargement of <math>f\!</math> is the proposition <math>\mathrm{E}f : \mathrm{E}U \to \mathbb{B}</math> that is defined by:
: <p><math>\operatorname{E}f(x_1, \ldots x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) = f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k).</math></p>
: <p><math>\mathrm{E}f(x_1, \ldots x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) = f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k).</math></p>
It should be noted that the so-called ''differential variables'' <math>\operatorname{d}x_j</math> are really just the same kind of boolean variables as the other <math>x_j.\!</math> It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.
It should be noted that the so-called ''differential variables'' <math>\mathrm{d}x_j</math> are really just the same kind of boolean variables as the other <math>x_j.\!</math> It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.
For the example <math>f(x, y) = xy,\!</math> we obtain:
For the example <math>f(x, y) = xy,\!</math> we obtain:
: <p><math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = (x + \operatorname{d}x)(y + \operatorname{d}y).</math></p>
: <p><math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = (x + \mathrm{d}x)(y + \mathrm{d}y).</math></p>
Given that this expression uses nothing more than the boolean ring operations of addition <math>(+)\!</math> and multiplication <math>(\cdot),</math> it is permissible to multiply things out in the usual manner to arrive at the result:
Given that this expression uses nothing more than the boolean ring operations of addition <math>(+)\!</math> and multiplication <math>(\cdot),</math> it is permissible to multiply things out in the usual manner to arrive at the result:
: <p><math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = xy + x\ \operatorname{d}y + y\ \operatorname{d}x + \operatorname{d}x\ \operatorname{d}y.</math>
: <p><math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = xy + x\ \mathrm{d}y + y\ \mathrm{d}x + \mathrm{d}x\ \mathrm{d}y.</math>
To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a ''disjunctive normal form'' (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for <math>\operatorname{D}f.</math> Thus, let us compute the value of the enlarged proposition <math>\operatorname{E}f</math> at each of the points in the universe of discourse <math>U = X \times Y.</math>
To understand what this means in logical terms, for instance, as expressed in a boolean expansion or a ''disjunctive normal form'' (DNF), it is perhaps a little better to go back and analyze the expression the same way that we did for <math>\mathrm{D}f.</math> Thus, let us compute the value of the enlarged proposition <math>\mathrm{E}f</math> at each of the points in the universe of discourse <math>U = X \times Y.</math>
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Given the sort of data that arises from this form of analysis, we can now fold the disjoined ingredients back into a boolean expansion or a DNF that is equivalent to the proposition <math>\operatorname{E}f.</math>
Given the sort of data that arises from this form of analysis, we can now fold the disjoined ingredients back into a boolean expansion or a DNF that is equivalent to the proposition <math>\mathrm{E}f.</math>
: <p><math>\operatorname{E}f = xy\ \operatorname{E}f_{x y} + x (\!| y |\!)\ \operatorname{E}f_{x (\!| y |\!)} + (\!| x |\!) y\ \operatorname{E}f_{(\!| x |\!) y} + (\!| x |\!)(\!| y |\!)\ \operatorname{E}f_{(\!| x |\!)(\!| y |\!)}.</math></p>
: <p><math>\mathrm{E}f = xy\ \mathrm{E}f_{x y} + x (\!| y |\!)\ \mathrm{E}f_{x (\!| y |\!)} + (\!| x |\!) y\ \mathrm{E}f_{(\!| x |\!) y} + (\!| x |\!)(\!| y |\!)\ \mathrm{E}f_{(\!| x |\!)(\!| y |\!)}.</math></p>
Here is a summary of the result, illustrated by means of a digraph picture, where the ''no change'' element <math>(\!| \operatorname{d}x |\!)(\!| \operatorname{d}y |\!)</math> is drawn as a loop at the point <math>xy.\!</math>
Here is a summary of the result, illustrated by means of a digraph picture, where the ''no change'' element <math>(\!| \mathrm{d}x |\!)(\!| \mathrm{d}y |\!)</math> is drawn as a loop at the point <math>xy.\!</math>
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We may understand the enlarged proposition <math>\operatorname{E}f</math> as telling us all the different ways to reach a model of <math>f\!</math> from any point of the universe <math>U.\!</math>
We may understand the enlarged proposition <math>\mathrm{E}f</math> as telling us all the different ways to reach a model of <math>f\!</math> from any point of the universe <math>U.\!</math>
===Propositional Forms on Two Variables===
===Propositional Forms on Two Variables===
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> For future reference, I will set here a few Tables that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> For future reference, I will set here a few Tables that detail the actions of <math>\mathrm{E}</math> and <math>\mathrm{D}</math> on each of these functions, allowing us to view the results in several different ways.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{false}</math></p>
| height="36px" | <p><math>\mathrm{false}</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
| height="36px" | <p><math>\mathrm{neither}\ x\ \mathrm{nor}\ y</math></p>
|-
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>y\ \mathrm{without}\ x</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
| height="36px" | <p><math>\mathrm{not}\ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{without}\ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
| height="36px" | <p><math>\mathrm{not}\ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{not~equal~to}\ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>\mathrm{not~both}\ x\ \mathrm{and}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{and}\ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{equal~to}\ y</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>y\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>\mathrm{not}\ x\ \mathrm{without}\ y</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>x\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>\mathrm{not}\ y\ \mathrm{without}\ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{or}\ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{true}</math></p>
| height="36px" | <p><math>\mathrm{true}</math></p>
|}
|}
|
|
| <p>0 0 0 0</p>
| <p>0 0 0 0</p>
| <p><math>(~)\!</math></p>
| <p><math>(~)\!</math></p>
| <p><math>\operatorname{false}</math></p>
| <p><math>\mathrm{false}</math></p>
| <p><math>0\!</math></p>
| <p><math>0\!</math></p>
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
| height="36px" | <p><math>\mathrm{neither}\ x\ \mathrm{nor}\ y</math></p>
|-
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>y\ \mathrm{without}\ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{without}\ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{and}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
| height="36px" | <p><math>\mathrm{not}\ x</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>x\!</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{not~equal~to}\ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{equal~to}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
| height="36px" | <p><math>\mathrm{not}\ y</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>y\!</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>\mathrm{not~both}\ x\ \mathrm{and}\ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>\mathrm{not}\ x\ \mathrm{without}\ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>\mathrm{not}\ y\ \mathrm{without}\ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
| height="36px" | <p><math>x\ \mathrm{or}\ y</math></p>
|}
|}
|
|
| <p>1 1 1 1</p>
| <p>1 1 1 1</p>
| <p><math>((~))\!</math></p>
| <p><math>((~))\!</math></p>
| <p><math>\operatorname{true}</math></p>
| <p><math>\mathrm{true}</math></p>
| <p><math>1\!</math></p>
| <p><math>1\!</math></p>
|}
|}
<br>
<br>
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
The next four Tables expand the expressions of <math>\mathrm{E}f</math> and <math>\mathrm{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
<br>
<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
|+ '''Table 3. <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
|+ '''Table 3. <math>\mathrm{E}f</math> Expanded Over Differential Features <math>\{ \mathrm{d}x, \mathrm{d}y \}</math>'''
|- style="background:#e6e6ff; height:48px"
|- style="background:#e6e6ff; height:48px"
|
|
{| align="center"
{| align="center"
|-
|-
| <math>\operatorname{T}_{11}f</math>
| <math>\mathrm{T}_{11}f</math>
|-
|-
| <math>\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
| <math>\mathrm{E}f|_{\mathrm{d}x\ \mathrm{d}y}</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| <math>\operatorname{T}_{10}f</math>
| <math>\mathrm{T}_{10}f</math>
|-
|-
| <math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
| <math>\mathrm{E}f|_{\mathrm{d}x(\mathrm{d}y)}</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| <math>\operatorname{T}_{01}f</math>
| <math>\mathrm{T}_{01}f</math>
|-
|-
| <math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
| <math>\mathrm{E}f|_{(\mathrm{d}x)\mathrm{d}y}</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| <math>\operatorname{T}_{00}f</math>
| <math>\mathrm{T}_{00}f</math>
|-
|-
| <math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
| <math>\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}</math>
|}
|}
|- style="height:36px"
|- style="height:36px"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
|+ '''Table 4. <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
|+ '''Table 4. <math>\mathrm{D}f</math> Expanded Over Differential Features <math>\{ \mathrm{d}x, \mathrm{d}y \}</math>'''
|- style="background:#e6e6ff; height:36px"
|- style="background:#e6e6ff; height:36px"
|
|
| <math>f\!</math>
| <math>f\!</math>
| <math>\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
| <math>\mathrm{D}f|_{\mathrm{d}x\ \mathrm{d}y}</math>
| <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
| <math>\mathrm{D}f|_{\mathrm{d}x(\mathrm{d}y)}</math>
| <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
| <math>\mathrm{D}f|_{(\mathrm{d}x)\mathrm{d}y}</math>
| <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
| <math>\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}</math>
|- style="height:36px"
|- style="height:36px"
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
|+ '''Table 5. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|+ '''Table 5. <math>\mathrm{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|- style="background:#e6e6ff; height:36px"
|- style="background:#e6e6ff; height:36px"
|
|
| <math>f\!</math>
| <math>f\!</math>
| <math>\operatorname{E}f|_{xy}</math>
| <math>\mathrm{E}f|_{xy}</math>
| <math>\operatorname{E}f|_{x(y)}</math>
| <math>\mathrm{E}f|_{x(y)}</math>
| <math>\operatorname{E}f|_{(x)y}</math>
| <math>\mathrm{E}f|_{(x)y}</math>
| <math>\operatorname{E}f|_{(x)(y)}</math>
| <math>\mathrm{E}f|_{(x)(y)}</math>
|- style="height:36px"
|- style="height:36px"
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x) (\mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x) (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x) (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x) (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|}
|}
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)</math>
| height="36px" | <math>(\mathrm{d}x)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)</math>
| height="36px" | <math>(\mathrm{d}x)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x)</math>
| height="36px" | <math>(\mathrm{d}x)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x)</math>
| height="36px" | <math>(\mathrm{d}x)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|}
|}
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x,\ \mathrm{d}y))</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x,\ \mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x,\ \mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x,\ \mathrm{d}y))</math>
|}
|}
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
|-
|-
| height="36px" | <math>(\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
|-
|-
| height="36px" | <math>(\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
|}
|}
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
| height="36px" | <math>((\mathrm{d}x)\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
| height="36px" | <math>(\mathrm{d}x\ (\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x\ \mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
| height="36px" | <math>((\mathrm{d}x)\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
| height="36px" | <math>(\mathrm{d}x\ (\mathrm{d}y))</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
| height="36px" | <math>(\mathrm{d}x\ (\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
| height="36px" | <math>((\mathrm{d}x)\ \mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
| height="36px" | <math>(\mathrm{d}x\ (\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
| height="36px" | <math>((\mathrm{d}x)\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|}
|}
|- style="height:36px"
|- style="height:36px"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
|+ '''Table 6. <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|+ '''Table 6. <math>\mathrm{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|- style="background:#e6e6ff; height:36px"
|- style="background:#e6e6ff; height:36px"
|
|
| <math>f\!</math>
| <math>f\!</math>
| <math>\operatorname{D}f|_{xy}</math>
| <math>\mathrm{D}f|_{xy}</math>
| <math>\operatorname{D}f|_{x(y)}</math>
| <math>\mathrm{D}f|_{x(y)}</math>
| <math>\operatorname{D}f|_{(x)y}</math>
| <math>\mathrm{D}f|_{(x)y}</math>
| <math>\operatorname{D}f|_{(x)(y)}</math>
| <math>\mathrm{D}f|_{(x)(y)}</math>
|- style="height:36px"
|- style="height:36px"
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
|-
|-
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
|}
|}
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|-
|-
| height="36px" | <math>\operatorname{d}x</math>
| height="36px" | <math>\mathrm{d}x</math>
|}
|}
|-
|-
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
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| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
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{| align="center"
{| align="center"
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| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
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| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
| height="36px" | <math>(\mathrm{d}x,\ \mathrm{d}y)</math>
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{| align="center"
{| align="center"
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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{| align="center"
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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{| align="center"
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}y</math>
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{| align="center"
{| align="center"
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| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
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| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
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| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
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{| align="center"
{| align="center"
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| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
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| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
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| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
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{| align="center"
{| align="center"
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| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
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| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
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| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
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| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
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{| align="center"
{| align="center"
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| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
| height="36px" | <math>\mathrm{d}x\ \mathrm{d}y</math>
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| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
| height="36px" | <math>\mathrm{d}x\ (\mathrm{d}y)</math>
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| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
| height="36px" | <math>(\mathrm{d}x)\ \mathrm{d}y</math>
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| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
| height="36px" | <math>((\mathrm{d}x)(\mathrm{d}y))</math>
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|- style="height:36px"
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It would be good to summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far.
It would be good to summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far.
We have been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, <math>X,\!</math> to considering a larger universe of discourse, <math>\operatorname{E}X.\!</math>
We have been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, <math>X,\!</math> to considering a larger universe of discourse, <math>\mathrm{E}X.\!</math>
Each of these operators, in general terms having the form <math>\operatorname{F} : X \to \operatorname{E}X,\!</math> acts on each proposition <math>p : X \to \mathbb{B}\!</math> of the source universe <math>X\!</math> to produce a proposition <math>\operatorname{F}p : \operatorname{E}X \to \mathbb{B}\!</math> of the target universe <math>\operatorname{E}X.\!</math>
Each of these operators, in general terms having the form <math>\mathrm{F} : X \to \mathrm{E}X,\!</math> acts on each proposition <math>p : X \to \mathbb{B}\!</math> of the source universe <math>X\!</math> to produce a proposition <math>\mathrm{F}p : \mathrm{E}X \to \mathbb{B}\!</math> of the target universe <math>\mathrm{E}X.\!</math>
The two main operators that we have worked with up to this point are the ''enlargement operator'' <math>\operatorname{E} : X \to \operatorname{E}X\!</math> and the ''difference operator'' <math>\operatorname{D} : X \to \operatorname{E}X.\!</math>
The two main operators that we have worked with up to this point are the ''enlargement operator'' <math>\mathrm{E} : X \to \mathrm{E}X\!</math> and the ''difference operator'' <math>\mathrm{D} : X \to \mathrm{E}X.\!</math>
<math>\operatorname{E}\!</math> and <math>\operatorname{D}\!</math> take a proposition in <math>X,\!</math> that is, a proposition <math>p : X \mathbb{B}\!</math> that is said to be ''about'' the subject matter of <math>X,\!</math> and produce the extended propositions <math>\operatorname{E}p, \operatorname{D}p : \operatorname{E}X \to \mathbb{B},\!</math> which may be interpreted as being about specified collections of changes that might occur in <math>X.\!</math>
<math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> take a proposition in <math>X,\!</math> that is, a proposition <math>p : X \mathbb{B}\!</math> that is said to be ''about'' the subject matter of <math>X,\!</math> and produce the extended propositions <math>\mathrm{E}p, \mathrm{D}p : \mathrm{E}X \to \mathbb{B},\!</math> which may be interpreted as being about specified collections of changes that might occur in <math>X.\!</math>
Here we have need of visual representations, some array of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us before we try to climb any higher into the ever more rarefied air of abstractions.
Here we have need of visual representations, some array of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us before we try to climb any higher into the ever more rarefied air of abstractions.
</pre>
</pre>
Each of the operators <math>\operatorname{E}, \operatorname{D} : X \to \operatorname{E}X\!</math> takes us from considering propositions <math>p : X \to \mathbb{B},\!</math> here viewed as ''scalar fields'' over <math>X,\!</math> to considering the corresponding ''differential fields'' over <math>X,\!</math> analogous to what are usually called ''vector fields'' <math>X.\!</math>
Each of the operators <math>\mathrm{E}, \mathrm{D} : X \to \mathrm{E}X\!</math> takes us from considering propositions <math>p : X \to \mathbb{B},\!</math> here viewed as ''scalar fields'' over <math>X,\!</math> to considering the corresponding ''differential fields'' over <math>X,\!</math> analogous to what are usually called ''vector fields'' <math>X.\!</math>
The structure of these differential fields can be described this way. To each point of <math>X\!</math> there is attached an object of the following type, a proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \operatorname{d}X \to \mathbb{B}.\!</math> In this setting, if <math>X\!</math> is the universe that is generated by the set of coordinate propositions <math>\{ u, v \},\!</math> then <math>\operatorname{d}X\!</math> is the differential universe that is generated by the set of differential propositions <math>\{ \operatorname{d}u, \operatorname{d}v \}.\!</math> These differential propositions may be interpreted as indicating "change in <math>u\!</math>" and "change in <math>v\!</math>", respectively.
The structure of these differential fields can be described this way. To each point of <math>X\!</math> there is attached an object of the following type, a proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \mathrm{d}X \to \mathbb{B}.\!</math> In this setting, if <math>X\!</math> is the universe that is generated by the set of coordinate propositions <math>\{ u, v \},\!</math> then <math>\mathrm{d}X\!</math> is the differential universe that is generated by the set of differential propositions <math>\{ \mathrm{d}u, \mathrm{d}v \}.\!</math> These differential propositions may be interpreted as indicating "change in <math>u\!</math>" and "change in <math>v\!</math>", respectively.
A differential operator <math>\operatorname{F},\!</math> of the first order sort that we have been considering, takes a proposition <math>p : X \to \mathbb{B}\!</math> and gives back a differential proposition <math>\operatorname{F}p : \operatorname{E}X \to \mathbb{B}.\!</math>
A differential operator <math>\mathrm{F},\!</math> of the first order sort that we have been considering, takes a proposition <math>p : X \to \mathbb{B}\!</math> and gives back a differential proposition <math>\mathrm{F}p : \mathrm{E}X \to \mathbb{B}.\!</math>
In the field view, we see the proposition <math>p : X \to \mathbb{B}\!</math> as a scalar field and we see the differential proposition <math>\operatorname{F}p : \operatorname{E}X \to \mathbb{B}\!</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>
In the field view, we see the proposition <math>p : X \to \mathbb{B}\!</math> as a scalar field and we see the differential proposition <math>\mathrm{F}p : \mathrm{E}X \to \mathbb{B}\!</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>
The field of changes produced by <math>\operatorname{E}\!</math> on <math>uv\!</math> is shown in Figure 2.
The field of changes produced by <math>\mathrm{E}\!</math> on <math>uv\!</math> is shown in Figure 2.
<pre>
<pre>
</pre>
</pre>
The differential field <math>\operatorname{E}[uv]\!</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to reach one of the models of the proposition <math>uv,\!</math> that is, in order to satisfy the proposition <math>uv.\!</math>
The differential field <math>\mathrm{E}[uv]\!</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to reach one of the models of the proposition <math>uv,\!</math> that is, in order to satisfy the proposition <math>uv.\!</math>
The field of changes produced by <math>\operatorname{D}\!</math> on <math>uv\!</math> is shown in Figure 3.
The field of changes produced by <math>\mathrm{D}\!</math> on <math>uv\!</math> is shown in Figure 3.
<pre>
<pre>
</pre>
</pre>
The differential field <math>\operatorname{D}[uv]\!</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to change the value of the proposition <math>uv.\!</math>
The differential field <math>\mathrm{D}[uv]\!</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to change the value of the proposition <math>uv.\!</math>
__TOC__
__TOC__
\end{centering}\end{figure}
\end{centering}\end{figure}
This new quality, $\operatorname{d}q,$ is an example of a \textit{differential quality}, since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a ``circle" that distinguishes two halves of the universe of discourse, in this case, the portions of $X$ outside and inside the region $\operatorname{d}Q.$
This new quality, $\mathrm{d}q,$ is an example of a \textit{differential quality}, since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a ``circle" that distinguishes two halves of the universe of discourse, in this case, the portions of $X$ outside and inside the region $\mathrm{d}Q.$
Figure 1 represents a universe of discourse, $X,$ together with a basis of discussion, $\{ q \},$ for expressing propositions about the contents of that universe. Once the quality $q$ is given a name, say, the symbol $``q"$, we have a basis for a formal language that is specifically cut out for discussing $X$ in terms of $q,$ and this formal language is more formally known as the \textit{propositional calculus} with alphabet $\{ ``q" \}.$
Figure 1 represents a universe of discourse, $X,$ together with a basis of discussion, $\{ q \},$ for expressing propositions about the contents of that universe. Once the quality $q$ is given a name, say, the symbol $``q"$, we have a basis for a formal language that is specifically cut out for discussing $X$ in terms of $q,$ and this formal language is more formally known as the \textit{propositional calculus} with alphabet $\{ ``q" \}.$
In the context marked by $X$ and $\{ q \}$ there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: $\textsl{false},\ \lnot q,\ q,\ \textsl{true}.$ Referring to the sample of points in Figure 1, $\textsl{false}$ holds of no points, $\lnot q$ holds of $h$ and $k$, $q$ holds of $i$ and $j$, and $\textsl{true}$ holds of all points in the sample.
In the context marked by $X$ and $\{ q \}$ there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: $\textsl{false},\ \lnot q,\ q,\ \textsl{true}.$ Referring to the sample of points in Figure 1, $\textsl{false}$ holds of no points, $\lnot q$ holds of $h$ and $k$, $q$ holds of $i$ and $j$, and $\textsl{true}$ holds of all points in the sample.
Figure $1^\prime$ preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, $\{ q,\ \operatorname{d}q \}.$ In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, $\{ ``q", ``\operatorname{d}q" \}.$ Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:
Figure $1^\prime$ preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, $\{ q,\ \mathrm{d}q \}.$ In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, $\{ ``q", ``\mathrm{d}q" \}.$ Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:
\begin{itemize}
\begin{itemize}
\item
\item
$\overline{q}\ \overline{\operatorname{d}q}$ describes $h$
$\overline{q}\ \overline{\mathrm{d}q}$ describes $h$
\item
\item
$\overline{q}\ \operatorname{d}q$ describes $k$
$\overline{q}\ \mathrm{d}q$ describes $k$
\item
\item
$q\ \overline{\operatorname{d}q}$ describes $i$
$q\ \overline{\mathrm{d}q}$ describes $i$
\item
\item
$q\ \operatorname{d}q$ describes $j$
$q\ \mathrm{d}q$ describes $j$
\end{itemize}
\end{itemize}
Table 3 exhibits the rules of inference that give the differential quality $\operatorname{d}q$ its meaning in practice.
Table 3 exhibits the rules of inference that give the differential quality $\mathrm{d}q$ its meaning in practice.
\begin{center}\begin{tabular}{ccccccc}
\begin{center}\begin{tabular}{ccccccc}
\multicolumn{7}{c}{\textbf{Table 3. Differential Inference Rules}} \\[12pt]
\multicolumn{7}{c}{\textbf{Table 3. Differential Inference Rules}} \\[12pt]
From & $\overline{q}$ & and & $\overline{\operatorname{d}q}$ & infer & $\overline{q}$ & next. \\[6pt]
From & $\overline{q}$ & and & $\overline{\mathrm{d}q}$ & infer & $\overline{q}$ & next. \\[6pt]
From & $\overline{q}$ & and & $\operatorname{d}q$ & infer & $q$ & next. \\[6pt]
From & $\overline{q}$ & and & $\mathrm{d}q$ & infer & $q$ & next. \\[6pt]
From & $q$ & and & $\overline{\operatorname{d}q}$ & infer & $q$ & next. \\[6pt]
From & $q$ & and & $\overline{\mathrm{d}q}$ & infer & $q$ & next. \\[6pt]
From & $q$ & and & $\operatorname{d}q$ & infer & $\overline{q}$ & next. \\[6pt]
From & $q$ & and & $\mathrm{d}q$ & infer & $\overline{q}$ & next. \\[6pt]
\end{tabular}\end{center}
\end{tabular}\end{center}
\hline
\hline
$~$ &
$~$ &
$\operatorname{True}$ &
$\mathrm{True}$ &
$1$ \\[4pt]
$1$ \\[4pt]
\hline
\hline
$(~)$ &
$(~)$ &
$\operatorname{False}$ &
$\mathrm{False}$ &
$0$ \\[4pt]
$0$ \\[4pt]
\hline
\hline
\hline
\hline
$(x)$ &
$(x)$ &
$\operatorname{Not}\ x$ &
$\mathrm{Not}\ x$ &
$\begin{matrix}
$\begin{matrix}
x' \\
x' \\
\hline
\hline
$x\ y\ z$ &
$x\ y\ z$ &
$x\ \operatorname{and}\ y\ \operatorname{and}\ z$ &
$x\ \mathrm{and}\ y\ \mathrm{and}\ z$ &
$x \land y \land z$ \\[4pt]
$x \land y \land z$ \\[4pt]
\hline
\hline
$((x)(y)(z))$ &
$((x)(y)(z))$ &
$x\ \operatorname{or}\ y\ \operatorname{or}\ z$ &
$x\ \mathrm{or}\ y\ \mathrm{or}\ z$ &
$x \lor y \lor z$ \\[4pt]
$x \lor y \lor z$ \\[4pt]
\hline
\hline
$(x\ (y))$ &
$(x\ (y))$ &
$\begin{matrix}
$\begin{matrix}
x\ \operatorname{implies}\ y \\
x\ \mathrm{implies}\ y \\
\operatorname{If}\ x\ \operatorname{then}\ y \\
\mathrm{If}\ x\ \mathrm{then}\ y \\
\end{matrix}$ &
\end{matrix}$ &
$x \Rightarrow y$ \\[4pt]
$x \Rightarrow y$ \\[4pt]
$(x, y)$ &
$(x, y)$ &
$\begin{matrix}
$\begin{matrix}
x\ \operatorname{not~equal~to}\ y \\
x\ \mathrm{not~equal~to}\ y \\
x\ \operatorname{exclusive~or}\ y \\
x\ \mathrm{exclusive~or}\ y \\
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
$((x, y))$ &
$((x, y))$ &
$\begin{matrix}
$\begin{matrix}
x\ \operatorname{is~equal~to}\ y \\
x\ \mathrm{is~equal~to}\ y \\
x\ \operatorname{if~and~only~if}\ y \\
x\ \mathrm{if~and~only~if}\ y \\
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
$(x, y, z)$ &
$(x, y, z)$ &
$\begin{matrix}
$\begin{matrix}
\operatorname{Just~one~of} \\
\mathrm{Just~one~of} \\
x, y, z \\
x, y, z \\
\operatorname{is~false}. \\
\mathrm{is~false}. \\
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
$((x),(y),(z))$ &
$((x),(y),(z))$ &
$\begin{matrix}
$\begin{matrix}
\operatorname{Just~one~of} \\
\mathrm{Just~one~of} \\
x, y, z \\
x, y, z \\
\operatorname{is~true}. \\
\mathrm{is~true}. \\
& \\
& \\
\operatorname{Partition~all} \\
\mathrm{Partition~all} \\
\operatorname{into}\ x, y, z. \\
\mathrm{into}\ x, y, z. \\
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
\operatorname{Oddly~many~of} \\
\mathrm{Oddly~many~of} \\
x, y, z \\
x, y, z \\
\operatorname{are~true}. \\
\mathrm{are~true}. \\
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
$(w, (x),(y),(z))$ &
$(w, (x),(y),(z))$ &
$\begin{matrix}
$\begin{matrix}
\operatorname{Partition}\ w \\
\mathrm{Partition}\ w \\
\operatorname{into}\ x, y, z. \\
\mathrm{into}\ x, y, z. \\
& \\
& \\
\operatorname{Genus}\ w\ \operatorname{comprises} \\
\mathrm{Genus}\ w\ \mathrm{comprises} \\
\operatorname{species}\ x, y, z. \\
\mathrm{species}\ x, y, z. \\
\end{matrix}$ &
\end{matrix}$ &
$\begin{matrix}
$\begin{matrix}
$A^*$ &
$A^*$ &
$(\operatorname{hom} : A \to \mathbb{B})$ &
$(\mathrm{hom} : A \to \mathbb{B})$ &
Linear functions &
Linear functions &
$(\mathbb{B}^n)^* \cong \mathbb{B}^n$ \\[4pt]
$(\mathbb{B}^n)^* \cong \mathbb{B}^n$ \\[4pt]
= &
= &
e_1 + \ldots + e_n &
e_1 + \ldots + e_n &
\operatorname{where} &
\mathrm{where} &
e_i = a_i &
e_i = a_i &
\operatorname{or} &
\mathrm{or} &
e_i = 0 &
e_i = 0 &
\operatorname{for}\ i = 1\ \operatorname{to}\ n. \\
\mathrm{for}\ i = 1\ \mathrm{to}\ n. \\
\end{matrix}$\end{quote}
\end{matrix}$\end{quote}
= &
= &
e_1 \cdot \ldots \cdot e_n &
e_1 \cdot \ldots \cdot e_n &
\operatorname{where} &
\mathrm{where} &
e_i = a_i &
e_i = a_i &
\operatorname{or} &
\mathrm{or} &
e_i = 1 &
e_i = 1 &
\operatorname{for}\ i = 1\ \operatorname{to}\ n. \\
\mathrm{for}\ i = 1\ \mathrm{to}\ n. \\
\end{matrix}$\end{quote}
\end{matrix}$\end{quote}
= &
= &
e_1 \cdot \ldots \cdot e_n &
e_1 \cdot \ldots \cdot e_n &
\operatorname{where} &
\mathrm{where} &
e_i = a_i &
e_i = a_i &
\operatorname{or} &
\mathrm{or} &
e_i = (a_i) &
e_i = (a_i) &
\operatorname{for}\ i = 1\ \operatorname{to}\ n. \\
\mathrm{for}\ i = 1\ \mathrm{to}\ n. \\
\end{matrix}$\end{quote}
\end{matrix}$\end{quote}
\subsection{Differential extensions}
\subsection{Differential extensions}
An initial universe of discourse, $A^\circ$, supplies the groundwork for any number of further extensions, beginning with the \textit{first order differential extension}, $\operatorname{E}A^\circ.$ The construction of $\operatorname{E}A^\circ$ can be described in the following stages:
An initial universe of discourse, $A^\circ$, supplies the groundwork for any number of further extensions, beginning with the \textit{first order differential extension}, $\mathrm{E}A^\circ.$ The construction of $\mathrm{E}A^\circ$ can be described in the following stages:
\begin{itemize}
\begin{itemize}
\item
\item
The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in a \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows:
The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\mathrm{d}\mathfrak{A} = \{ ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \},$ resulting in a \textit{first order extended alphabet}, $\mathrm{E}\mathfrak{A},$ defined as follows:
\begin{quote}
\begin{quote}
$\operatorname{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A} = \{ ``a_1", \ldots, ``a_n", ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \}.$
$\mathrm{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \mathrm{d}\mathfrak{A} = \{ ``a_1", \ldots, ``a_n", ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \}.$
\end{quote}
\end{quote}
\item
\item
The initial basis, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ is extended by a \textit{first order differential basis}, $\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},$ resulting in a \textit{first order extended basis}, $\operatorname{E}\mathcal{A},$ defined as follows:
The initial basis, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ is extended by a \textit{first order differential basis}, $\mathrm{d}\mathcal{A} = \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \},$ resulting in a \textit{first order extended basis}, $\mathrm{E}\mathcal{A},$ defined as follows:
\begin{quote}
\begin{quote}
$\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.$
$\mathrm{E}\mathcal{A} = \mathcal{A}\ \cup\ \mathrm{d}\mathcal{A} = \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.$
\end{quote}
\end{quote}
\item
\item
The initial space, $A = \langle a_1, \ldots, a_n \rangle,$ is extended by a \textit{first order differential space} or \textit{tangent space}, $\operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,$ at each point of $A,$ resulting in a \textit{first order extended space} or \textit{tangent bundle space}, $\operatorname{E}A,$ defined as follows:
The initial space, $A = \langle a_1, \ldots, a_n \rangle,$ is extended by a \textit{first order differential space} or \textit{tangent space}, $\mathrm{d}A = \langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,$ at each point of $A,$ resulting in a \textit{first order extended space} or \textit{tangent bundle space}, $\mathrm{E}A,$ defined as follows:
\begin{quote}
\begin{quote}
$\operatorname{E}A = A\ \times\ \operatorname{d}A = \langle \operatorname{E}\mathcal{A} \rangle = \langle \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.$
$\mathrm{E}A = A\ \times\ \mathrm{d}A = \langle \mathrm{E}\mathcal{A} \rangle = \langle \mathcal{A}\ \cup\ \mathrm{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.$
\end{quote}
\end{quote}
\item
\item
Finally, the initial universe, $A^\circ = [ a_1, \ldots, a_n ],$ is extended by a \textit{first order differential universe} or \textit{tangent universe}, $\operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ],$ at each point of $A^\circ,$ resulting in a \textit{first order extended universe} or \textit{tangent bundle universe}, $\operatorname{E}A^\circ,$ defined as follows:
Finally, the initial universe, $A^\circ = [ a_1, \ldots, a_n ],$ is extended by a \textit{first order differential universe} or \textit{tangent universe}, $\mathrm{d}A^\circ = [ \mathrm{d}a_1, \ldots, \mathrm{d}a_n ],$ at each point of $A^\circ,$ resulting in a \textit{first order extended universe} or \textit{tangent bundle universe}, $\mathrm{E}A^\circ,$ defined as follows:
\begin{quote}
\begin{quote}
$\operatorname{E}A^\circ = [ \operatorname{E}\mathcal{A} ] = [ \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} ] = [ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n ].$
$\mathrm{E}A^\circ = [ \mathrm{E}\mathcal{A} ] = [ \mathcal{A}\ \cup\ \mathrm{d}\mathcal{A} ] = [ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n ].$
\end{quote}
\end{quote}
This gives $\operatorname{E}A^\circ$ the type:
This gives $\mathrm{E}A^\circ$ the type:
\begin{quote}
\begin{quote}
\end{itemize}
\end{itemize}
A proposition in a differential extension of a universe of discourse is called a \textit{differential proposition} and forms the analogue of a system of differential equations in \PMlinkname{ordinary calculus}{Calculus}. With these constructions, the first order extended universe $\operatorname{E}A^\circ$ and the first order differential proposition $f : \operatorname{E}A \to \mathbb{B},$ we have arrived, in concept at least, at the foothills of differential logic.
A proposition in a differential extension of a universe of discourse is called a \textit{differential proposition} and forms the analogue of a system of differential equations in \PMlinkname{ordinary calculus}{Calculus}. With these constructions, the first order extended universe $\mathrm{E}A^\circ$ and the first order differential proposition $f : \mathrm{E}A \to \mathbb{B},$ we have arrived, in concept at least, at the foothills of differential logic.
Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.
Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.
\hline
\hline
$\operatorname{d}\mathfrak{A}$ &
$\mathrm{d}\mathfrak{A}$ &
$\{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \}$ &
$\{ ``\mathrm{d}a_1", \ldots, ``\mathrm{d}a_n" \}$ &
Alphabet of differential symbols &
Alphabet of differential symbols &
$[n] = \mathbf{n}$ \\[4pt]
$[n] = \mathbf{n}$ \\[4pt]
\hline
\hline
$\operatorname{d}\mathcal{A}$ &
$\mathrm{d}\mathcal{A}$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
$\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}$ &
Basis of differential features &
Basis of differential features &
$[n] = \mathbf{n}$ \\[4pt]
$[n] = \mathbf{n}$ \\[4pt]
\hline
\hline
$\operatorname{d}A_i$ &
$\mathrm{d}A_i$ &
$\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}$ &
$\{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \}$ &
Differential dimension $i$ &
Differential dimension $i$ &
$\mathbb{D}$ \\[4pt]
$\mathbb{D}$ \\[4pt]
\hline
\hline
$\operatorname{d}A$ &
$\mathrm{d}A$ &
$\langle \operatorname{d}\mathcal{A} \rangle$ &
$\langle \mathrm{d}\mathcal{A} \rangle$ &
Tangent space at a point: &
Tangent space at a point: &
$\mathbb{D}^n$ \\[4pt]
$\mathbb{D}^n$ \\[4pt]
&
&
$\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle$ &
$\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle$ &
Set of changes, &
Set of changes, &
\\[4pt]
\\[4pt]
&
&
$\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}$ &
$\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}$ &
motions, steps, &
motions, steps, &
\\[4pt]
\\[4pt]
&
&
$\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n$ &
$\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n$ &
tangent vectors &
tangent vectors &
\\[4pt]
\\[4pt]
&
&
$\textstyle \prod_{i=1}^n \operatorname{d}A_i$ &
$\textstyle \prod_{i=1}^n \mathrm{d}A_i$ &
at a point &
at a point &
\\[4pt]
\\[4pt]
\hline
\hline
$\operatorname{d}A^*$ &
$\mathrm{d}A^*$ &
$(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})$ &
$(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})$ &
Linear functions on $\operatorname{d}A$ &
Linear functions on $\mathrm{d}A$ &
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$ \\[4pt]
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$ \\[4pt]
\hline
\hline
$\operatorname{d}A^\uparrow$ &
$\mathrm{d}A^\uparrow$ &
$(\operatorname{d}A \to \mathbb{B})$ &
$(\mathrm{d}A \to \mathbb{B})$ &
Boolean functions on $\operatorname{d}A$ &
Boolean functions on $\mathrm{d}A$ &
$\mathbb{D}^n \to \mathbb{B}$ \\[4pt]
$\mathbb{D}^n \to \mathbb{B}$ \\[4pt]
\hline
\hline
$\operatorname{d}A^\circ$ &
$\mathrm{d}A^\circ$ &
$[ \operatorname{d}\mathcal{A} ]$ &
$[ \mathrm{d}\mathcal{A} ]$ &
Tangent universe &
Tangent universe &
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$ \\[4pt]
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$ \\[4pt]
&
&
$(\operatorname{d}A, \operatorname{d}A^\uparrow)$ &
$(\mathrm{d}A, \mathrm{d}A^\uparrow)$ &
at a point of $A^\circ,$ &
at a point of $A^\circ,$ &
$(\mathbb{D}^n\ +\!\to \mathbb{B})$ \\[4pt]
$(\mathbb{D}^n\ +\!\to \mathbb{B})$ \\[4pt]
&
&
$(\operatorname{d}A\ +\!\to \mathbb{B})$ &
$(\mathrm{d}A\ +\!\to \mathbb{B})$ &
based on the &
based on the &
$[\mathbb{D}^n]$ \\[4pt]
$[\mathbb{D}^n]$ \\[4pt]
&
&
$(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))$ &
$(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))$ &
tangent features &
tangent features &
\\[4pt]
\\[4pt]
&
&
$[ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ]$ &
$[ \mathrm{d}a_1, \ldots, \mathrm{d}a_n ]$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
$\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}$ &
\\[4pt]
\\[4pt]
\hline
\hline
0 0 0 0 &
0 0 0 0 &
$(~)$ &
$(~)$ &
$\operatorname{false}$ &
$\mathrm{false}$ &
$0$ \\
$0$ \\
$f_{1}$ &
$f_{1}$ &
0 0 0 1 &
0 0 0 1 &
$(x)(y)$ &
$(x)(y)$ &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\mathrm{neither}\ x\ \mathrm{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$\lnot x \land \lnot y$ \\
$f_{2}$ &
$f_{2}$ &
0 0 1 0 &
0 0 1 0 &
$(x)\ y$ &
$(x)\ y$ &
$y\ \operatorname{without}\ x$ &
$y\ \mathrm{without}\ x$ &
$\lnot x \land y$ \\
$\lnot x \land y$ \\
$f_{3}$ &
$f_{3}$ &
0 0 1 1 &
0 0 1 1 &
$(x)$ &
$(x)$ &
$\operatorname{not}\ x$ &
$\mathrm{not}\ x$ &
$\lnot x$ \\
$\lnot x$ \\
$f_{4}$ &
$f_{4}$ &
0 1 0 0 &
0 1 0 0 &
$x\ (y)$ &
$x\ (y)$ &
$x\ \operatorname{without}\ y$ &
$x\ \mathrm{without}\ y$ &
$x \land \lnot y$ \\
$x \land \lnot y$ \\
$f_{5}$ &
$f_{5}$ &
0 1 0 1 &
0 1 0 1 &
$(y)$ &
$(y)$ &
$\operatorname{not}\ y$ &
$\mathrm{not}\ y$ &
$\lnot y$ \\
$\lnot y$ \\
$f_{6}$ &
$f_{6}$ &
0 1 1 0 &
0 1 1 0 &
$(x,\ y)$ &
$(x,\ y)$ &
$x\ \operatorname{not~equal~to}\ y$ &
$x\ \mathrm{not~equal~to}\ y$ &
$x \ne y$ \\
$x \ne y$ \\
$f_{7}$ &
$f_{7}$ &
0 1 1 1 &
0 1 1 1 &
$(x\ y)$ &
$(x\ y)$ &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\mathrm{not~both}\ x\ \mathrm{and}\ y$ &
$\lnot x \lor \lnot y$ \\
$\lnot x \lor \lnot y$ \\
\hline
\hline
1 0 0 0 &
1 0 0 0 &
$x\ y$ &
$x\ y$ &
$x\ \operatorname{and}\ y$ &
$x\ \mathrm{and}\ y$ &
$x \land y$ \\
$x \land y$ \\
$f_{9}$ &
$f_{9}$ &
1 0 0 1 &
1 0 0 1 &
$((x,\ y))$ &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x\ \mathrm{equal~to}\ y$ &
$x = y$ \\
$x = y$ \\
$f_{10}$ &
$f_{10}$ &
1 0 1 1 &
1 0 1 1 &
$(x\ (y))$ &
$(x\ (y))$ &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$\mathrm{not}\ x\ \mathrm{without}\ y$ &
$x \Rightarrow y$ \\
$x \Rightarrow y$ \\
$f_{12}$ &
$f_{12}$ &
1 1 0 1 &
1 1 0 1 &
$((x)\ y)$ &
$((x)\ y)$ &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$\mathrm{not}\ y\ \mathrm{without}\ x$ &
$x \Leftarrow y$ \\
$x \Leftarrow y$ \\
$f_{14}$ &
$f_{14}$ &
1 1 1 0 &
1 1 1 0 &
$((x)(y))$ &
$((x)(y))$ &
$x\ \operatorname{or}\ y$ &
$x\ \mathrm{or}\ y$ &
$x \lor y$ \\
$x \lor y$ \\
$f_{15}$ &
$f_{15}$ &
1 1 1 1 &
1 1 1 1 &
$((~))$ &
$((~))$ &
$\operatorname{true}$ &
$\mathrm{true}$ &
$1$ \\
$1$ \\
\hline
\hline
0 0 0 0 &
0 0 0 0 &
$(~)$ &
$(~)$ &
$\operatorname{false}$ &
$\mathrm{false}$ &
$0$ \\
$0$ \\
\hline
\hline
0 0 0 1 &
0 0 0 1 &
$(x)(y)$ &
$(x)(y)$ &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\mathrm{neither}\ x\ \mathrm{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$\lnot x \land \lnot y$ \\
$f_{2}$ &
$f_{2}$ &
0 0 1 0 &
0 0 1 0 &
$(x)\ y$ &
$(x)\ y$ &
$y\ \operatorname{without}\ x$ &
$y\ \mathrm{without}\ x$ &
$\lnot x \land y$ \\
$\lnot x \land y$ \\
$f_{4}$ &
$f_{4}$ &
0 1 0 0 &
0 1 0 0 &
$x\ (y)$ &
$x\ (y)$ &
$x\ \operatorname{without}\ y$ &
$x\ \mathrm{without}\ y$ &
$x \land \lnot y$ \\
$x \land \lnot y$ \\
$f_{8}$ &
$f_{8}$ &
1 0 0 0 &
1 0 0 0 &
$x\ y$ &
$x\ y$ &
$x\ \operatorname{and}\ y$ &
$x\ \mathrm{and}\ y$ &
$x \land y$ \\
$x \land y$ \\
\hline
\hline
0 0 1 1 &
0 0 1 1 &
$(x)$ &
$(x)$ &
$\operatorname{not}\ x$ &
$\mathrm{not}\ x$ &
$\lnot x$ \\
$\lnot x$ \\
$f_{12}$ &
$f_{12}$ &
0 1 1 0 &
0 1 1 0 &
$(x,\ y)$ &
$(x,\ y)$ &
$x\ \operatorname{not~equal~to}\ y$ &
$x\ \mathrm{not~equal~to}\ y$ &
$x \ne y$ \\
$x \ne y$ \\
$f_{9}$ &
$f_{9}$ &
1 0 0 1 &
1 0 0 1 &
$((x,\ y))$ &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x\ \mathrm{equal~to}\ y$ &
$x = y$ \\
$x = y$ \\
\hline
\hline
0 1 0 1 &
0 1 0 1 &
$(y)$ &
$(y)$ &
$\operatorname{not}\ y$ &
$\mathrm{not}\ y$ &
$\lnot y$ \\
$\lnot y$ \\
$f_{10}$ &
$f_{10}$ &
0 1 1 1 &
0 1 1 1 &
$(x\ y)$ &
$(x\ y)$ &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\mathrm{not~both}\ x\ \mathrm{and}\ y$ &
$\lnot x \lor \lnot y$ \\
$\lnot x \lor \lnot y$ \\
$f_{11}$ &
$f_{11}$ &
1 0 1 1 &
1 0 1 1 &
$(x\ (y))$ &
$(x\ (y))$ &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$\mathrm{not}\ x\ \mathrm{without}\ y$ &
$x \Rightarrow y$ \\
$x \Rightarrow y$ \\
$f_{13}$ &
$f_{13}$ &
1 1 0 1 &
1 1 0 1 &
$((x)\ y)$ &
$((x)\ y)$ &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$\mathrm{not}\ y\ \mathrm{without}\ x$ &
$x \Leftarrow y$ \\
$x \Leftarrow y$ \\
$f_{14}$ &
$f_{14}$ &
1 1 1 0 &
1 1 1 0 &
$((x)(y))$ &
$((x)(y))$ &
$x\ \operatorname{or}\ y$ &
$x\ \mathrm{or}\ y$ &
$x \lor y$ \\
$x \lor y$ \\
\hline
\hline
1 1 1 1 &
1 1 1 1 &
$((~))$ &
$((~))$ &
$\operatorname{true}$ &
$\mathrm{true}$ &
$1$ \\
$1$ \\
\hline
\hline
\end{tabular}\end{quote}
\end{tabular}\end{quote}
\subsection{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
\subsection{Table A3. $\mathrm{E}f$ Expanded Over Differential Features $\{ \mathrm{d}x, \mathrm{d}y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
\multicolumn{6}{c}{\textbf{Table A3. $\mathrm{E}f$ Expanded Over Differential Features $\{ \mathrm{d}x, \mathrm{d}y \}$}} \\
\hline
\hline
& &
& &
$\operatorname{T}_{11}$ &
$\mathrm{T}_{11}$ &
$\operatorname{T}_{10}$ &
$\mathrm{T}_{10}$ &
$\operatorname{T}_{01}$ &
$\mathrm{T}_{01}$ &
$\operatorname{T}_{00}$ \\
$\mathrm{T}_{00}$ \\
& $f$ &
& $f$ &
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
$\mathrm{E}f|_{\mathrm{d}x\ \mathrm{d}y}$ &
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
$\mathrm{E}f|_{\mathrm{d}x (\mathrm{d}y)}$ &
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
$\mathrm{E}f|_{(\mathrm{d}x) \mathrm{d}y}$ &
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
$\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}$ \\
\hline
\hline
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
\end{tabular}\end{quote}
\end{tabular}\end{quote}
\subsection{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
\subsection{Table A4. $\mathrm{D}f$ Expanded Over Differential Features $\{ \mathrm{d}x, \mathrm{d}y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\
\multicolumn{6}{c}{\textbf{Table A4. $\mathrm{D}f$ Expanded Over Differential Features $\{ \mathrm{d}x, \mathrm{d}y \}$}} \\
\hline
\hline
& $f$ &
& $f$ &
$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
$\mathrm{D}f|_{\mathrm{d}x\ \mathrm{d}y}$ &
$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
$\mathrm{D}f|_{\mathrm{d}x (\mathrm{d}y)}$ &
$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
$\mathrm{D}f|_{(\mathrm{d}x) \mathrm{d}y}$ &
$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
$\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}$ \\
\hline
\hline
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
\end{tabular}\end{quote}
\end{tabular}\end{quote}
\subsection{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
\subsection{Table A5. $\mathrm{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\multicolumn{6}{c}{\textbf{Table A5. $\mathrm{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\hline
\hline
& $f$ &
& $f$ &
$\operatorname{E}f|_{x\ y}$ &
$\mathrm{E}f|_{x\ y}$ &
$\operatorname{E}f|_{x (y)}$ &
$\mathrm{E}f|_{x (y)}$ &
$\operatorname{E}f|_{(x) y}$ &
$\mathrm{E}f|_{(x) y}$ &
$\operatorname{E}f|_{(x)(y)}$ \\
$\mathrm{E}f|_{(x)(y)}$ \\
\hline
\hline
$f_{0}$ &
$f_{0}$ &
$f_{1}$ &
$f_{1}$ &
$(x)(y)$ &
$(x)(y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ \\
$(\mathrm{d}x)(\mathrm{d}y)$ \\
$f_{2}$ &
$f_{2}$ &
$(x)\ y$ &
$(x)\ y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\mathrm{d}x)(\mathrm{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$(\mathrm{d}x)\ \mathrm{d}y$ \\
$f_{4}$ &
$f_{4}$ &
$x\ (y)$ &
$x\ (y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\mathrm{d}x)(\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$\mathrm{d}x\ (\mathrm{d}y)$ \\
$f_{8}$ &
$f_{8}$ &
$x\ y$ &
$x\ y$ &
$(\operatorname{d}x)(\operatorname{d}y)$ &
$(\mathrm{d}x)(\mathrm{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ \\
$\mathrm{d}x\ \mathrm{d}y$ \\
\hline
\hline
$f_{3}$ &
$f_{3}$ &
$(x)$ &
$(x)$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$(\operatorname{d}x)$ &
$(\mathrm{d}x)$ &
$(\operatorname{d}x)$ \\
$(\mathrm{d}x)$ \\
$f_{12}$ &
$f_{12}$ &
$x$ &
$x$ &
$(\operatorname{d}x)$ &
$(\mathrm{d}x)$ &
$(\operatorname{d}x)$ &
$(\mathrm{d}x)$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ \\
$\mathrm{d}x$ \\
\hline
\hline
$f_{6}$ &
$f_{6}$ &
$(x,\ y)$ &
$(x,\ y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$((\mathrm{d}x,\ \mathrm{d}y))$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$((\mathrm{d}x,\ \mathrm{d}y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
$(\mathrm{d}x,\ \mathrm{d}y)$ \\
$f_{9}$ &
$f_{9}$ &
$((x,\ y))$ &
$((x,\ y))$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ &
$((\mathrm{d}x,\ \mathrm{d}y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$((\operatorname{d}x,\ \operatorname{d}y))$ \\
$((\mathrm{d}x,\ \mathrm{d}y))$ \\
\hline
\hline
$f_{5}$ &
$f_{5}$ &
$(y)$ &
$(y)$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$(\operatorname{d}y)$ &
$(\mathrm{d}y)$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$(\operatorname{d}y)$ \\
$(\mathrm{d}y)$ \\
$f_{10}$ &
$f_{10}$ &
$y$ &
$y$ &
$(\operatorname{d}y)$ &
$(\mathrm{d}y)$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$(\operatorname{d}y)$ &
$(\mathrm{d}y)$ &
$\operatorname{d}y$ \\
$\mathrm{d}y$ \\
\hline
\hline
$f_{7}$ &
$f_{7}$ &
$(x\ y)$ &
$(x\ y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\mathrm{d}x)\ \mathrm{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\mathrm{d}x\ (\mathrm{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ \\
$(\mathrm{d}x\ \mathrm{d}y)$ \\
$f_{11}$ &
$f_{11}$ &
$(x\ (y))$ &
$(x\ (y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\mathrm{d}x)\ \mathrm{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ &
$(\mathrm{d}x\ \mathrm{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ \\
$(\mathrm{d}x\ (\mathrm{d}y))$ \\
$f_{13}$ &
$f_{13}$ &
$((x)\ y)$ &
$((x)\ y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\mathrm{d}x\ (\mathrm{d}y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ &
$(\mathrm{d}x\ \mathrm{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ \\
$((\mathrm{d}x)\ \mathrm{d}y)$ \\
$f_{14}$ &
$f_{14}$ &
$((x)(y))$ &
$((x)(y))$ &
$(\operatorname{d}x\ \operatorname{d}y)$ &
$(\mathrm{d}x\ \mathrm{d}y)$ &
$(\operatorname{d}x\ (\operatorname{d}y))$ &
$(\mathrm{d}x\ (\mathrm{d}y))$ &
$((\operatorname{d}x)\ \operatorname{d}y)$ &
$((\mathrm{d}x)\ \mathrm{d}y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
$((\mathrm{d}x)(\mathrm{d}y))$ \\
\hline
\hline
$f_{15}$ &
$f_{15}$ &
\end{tabular}\end{quote}
\end{tabular}\end{quote}
\subsection{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
\subsection{Table A6. $\mathrm{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{\textbf{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\multicolumn{6}{c}{\textbf{Table A6. $\mathrm{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
\hline
\hline
& $f$ &
& $f$ &
$\operatorname{D}f|_{x\ y}$ &
$\mathrm{D}f|_{x\ y}$ &
$\operatorname{D}f|_{x (y)}$ &
$\mathrm{D}f|_{x (y)}$ &
$\operatorname{D}f|_{(x) y}$ &
$\mathrm{D}f|_{(x) y}$ &
$\operatorname{D}f|_{(x)(y)}$ \\
$\mathrm{D}f|_{(x)(y)}$ \\
\hline
\hline
$f_{0}$ &
$f_{0}$ &
$f_{1}$ &
$f_{1}$ &
$(x)(y)$ &
$(x)(y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
$((\mathrm{d}x)(\mathrm{d}y))$ \\
$f_{2}$ &
$f_{2}$ &
$(x)\ y$ &
$(x)\ y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$(\mathrm{d}x)\ \mathrm{d}y$ \\
$f_{4}$ &
$f_{4}$ &
$x\ (y)$ &
$x\ (y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$\mathrm{d}x\ (\mathrm{d}y)$ \\
$f_{8}$ &
$f_{8}$ &
$x\ y$ &
$x\ y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ \\
$\mathrm{d}x\ \mathrm{d}y$ \\
\hline
\hline
$f_{3}$ &
$f_{3}$ &
$(x)$ &
$(x)$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ \\
$\mathrm{d}x$ \\
$f_{12}$ &
$f_{12}$ &
$x$ &
$x$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ &
$\mathrm{d}x$ &
$\operatorname{d}x$ \\
$\mathrm{d}x$ \\
\hline
\hline
$f_{6}$ &
$f_{6}$ &
$(x,\ y)$ &
$(x,\ y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
$(\mathrm{d}x,\ \mathrm{d}y)$ \\
$f_{9}$ &
$f_{9}$ &
$((x,\ y))$ &
$((x,\ y))$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ &
$(\mathrm{d}x,\ \mathrm{d}y)$ &
$(\operatorname{d}x,\ \operatorname{d}y)$ \\
$(\mathrm{d}x,\ \mathrm{d}y)$ \\
\hline
\hline
$f_{5}$ &
$f_{5}$ &
$(y)$ &
$(y)$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$\operatorname{d}y$ \\
$\mathrm{d}y$ \\
$f_{10}$ &
$f_{10}$ &
$y$ &
$y$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$\operatorname{d}y$ &
$\mathrm{d}y$ &
$\operatorname{d}y$ \\
$\mathrm{d}y$ \\
\hline
\hline
$f_{7}$ &
$f_{7}$ &
$(x\ y)$ &
$(x\ y)$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ \\
$\mathrm{d}x\ \mathrm{d}y$ \\
$f_{11}$ &
$f_{11}$ &
$(x\ (y))$ &
$(x\ (y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ \\
$\mathrm{d}x\ (\mathrm{d}y)$ \\
$f_{13}$ &
$f_{13}$ &
$((x)\ y)$ &
$((x)\ y)$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ &
$((\mathrm{d}x)(\mathrm{d}y))$ &
$(\operatorname{d}x)\ \operatorname{d}y$ \\
$(\mathrm{d}x)\ \mathrm{d}y$ \\
$f_{14}$ &
$f_{14}$ &
$((x)(y))$ &
$((x)(y))$ &
$\operatorname{d}x\ \operatorname{d}y$ &
$\mathrm{d}x\ \mathrm{d}y$ &
$\operatorname{d}x\ (\operatorname{d}y)$ &
$\mathrm{d}x\ (\mathrm{d}y)$ &
$(\operatorname{d}x)\ \operatorname{d}y$ &
$(\mathrm{d}x)\ \mathrm{d}y$ &
$((\operatorname{d}x)(\operatorname{d}y))$ \\
$((\mathrm{d}x)(\mathrm{d}y))$ \\
\hline
\hline
$f_{15}$ &
$f_{15}$ &
\PMlinkescapephrase{Expands}
\PMlinkescapephrase{Expands}
The actions of the \PMlinkname{difference operator}{FiniteDifference} $\operatorname{D}$ and the \PMlinkname{tangent operator}{TangentMap} $\operatorname{d}$ on the 16 propositional forms in two variables are shown in the Tables below.
The actions of the \PMlinkname{difference operator}{FiniteDifference} $\mathrm{D}$ and the \PMlinkname{tangent operator}{TangentMap} $\mathrm{d}$ on the 16 propositional forms in two variables are shown in the Tables below.
Table A7 expands the resulting differential forms over a \textit{logical basis}:
Table A7 expands the resulting differential forms over a \textit{logical basis}:
\begin{center}
\begin{center}
$\{ (\operatorname{d}x)(\operatorname{d}y),\ \operatorname{d}x\,(\operatorname{d}y),\ (\operatorname{d}x)\,\operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$
$\{ (\mathrm{d}x)(\mathrm{d}y),\ \mathrm{d}x\,(\mathrm{d}y),\ (\mathrm{d}x)\,\mathrm{d}y,\ \mathrm{d}x\,\mathrm{d}y \}.$
\end{center}
\end{center}
\begin{center}
\begin{center}
$\partial x = \operatorname{d}x\,(\operatorname{d}y)$ and $\partial y = (\operatorname{d}x)\,\operatorname{d}y.$
$\partial x = \mathrm{d}x\,(\mathrm{d}y)$ and $\partial y = (\mathrm{d}x)\,\mathrm{d}y.$
\end{center}
\end{center}
\begin{center}
\begin{center}
$\{ 1,\ \operatorname{d}x,\ \operatorname{d}y,\ \operatorname{d}x\,\operatorname{d}y \}.$
$\{ 1,\ \mathrm{d}x,\ \mathrm{d}y,\ \mathrm{d}x\,\mathrm{d}y \}.$
\end{center}
\end{center}
&
&
$f$ &
$f$ &
$\operatorname{D}f$ &
$\mathrm{D}f$ &
$\operatorname{d}f$ \\
$\mathrm{d}f$ \\
\hline
\hline
$f_{0}$ &
$f_{0}$ &
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \mathrm{d}x\ (\mathrm{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\mathrm{d}x)\ \mathrm{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \mathrm{d}x\ \mathrm{d}y \\
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \mathrm{d}x\ (\mathrm{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\mathrm{d}x)\ \mathrm{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \mathrm{d}x\ \mathrm{d}y \\
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \mathrm{d}x\ (\mathrm{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\mathrm{d}x)\ \mathrm{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \mathrm{d}x\ \mathrm{d}y \\
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \mathrm{d}x\ (\mathrm{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\mathrm{d}x)\ \mathrm{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \mathrm{d}x\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}x\ (\operatorname{d}y) & + &
\mathrm{d}x\ (\mathrm{d}y) & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ (\operatorname{d}y) & + &
\mathrm{d}x\ (\mathrm{d}y) & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}x\ (\operatorname{d}y) & + &
\mathrm{d}x\ (\mathrm{d}y) & + &
(\operatorname{d}x)\ \operatorname{d}y \\
(\mathrm{d}x)\ \mathrm{d}y \\
\operatorname{d}x\ (\operatorname{d}y) & + &
\mathrm{d}x\ (\mathrm{d}y) & + &
(\operatorname{d}x)\ \operatorname{d}y \\
(\mathrm{d}x)\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
(\operatorname{d}x)\ \operatorname{d}y & + &
(\mathrm{d}x)\ \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
(\operatorname{d}x)\ \operatorname{d}y & + &
(\mathrm{d}x)\ \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \mathrm{d}x\ (\mathrm{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\mathrm{d}x)\ \mathrm{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \mathrm{d}x\ \mathrm{d}y \\
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \mathrm{d}x\ (\mathrm{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\mathrm{d}x)\ \mathrm{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \mathrm{d}x\ \mathrm{d}y \\
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \mathrm{d}x\ (\mathrm{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\mathrm{d}x)\ \mathrm{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \mathrm{d}x\ \mathrm{d}y \\
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \mathrm{d}x\ (\mathrm{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\mathrm{d}x)\ \mathrm{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \mathrm{d}x\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
&
&
$f$ &
$f$ &
$\operatorname{D}f$ &
$\mathrm{D}f$ &
$\operatorname{d}f$ \\
$\mathrm{d}f$ \\
\hline
\hline
$f_{0}$ &
$f_{0}$ &
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
(y) & \operatorname{d}x & + &
(y) & \mathrm{d}x & + &
(x) & \operatorname{d}y & + &
(x) & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
y & \operatorname{d}x & + &
y & \mathrm{d}x & + &
(x) & \operatorname{d}y & + &
(x) & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
(y) & \operatorname{d}x & + &
(y) & \mathrm{d}x & + &
x & \operatorname{d}y & + &
x & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
y & \operatorname{d}x & + &
y & \mathrm{d}x & + &
x & \operatorname{d}y & + &
x & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
y & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
(y) & \mathrm{d}x & + & x & \mathrm{d}y \\
y & \operatorname{d}x & + & x & \operatorname{d}y \\
y & \mathrm{d}x & + & x & \mathrm{d}y \\
\end{smallmatrix}$ \\
\end{smallmatrix}$ \\
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{smallmatrix}$ \\
\end{smallmatrix}$ \\
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}x & + & \operatorname{d}y \\
\mathrm{d}x & + & \mathrm{d}y \\
\operatorname{d}x & + & \operatorname{d}y \\
\mathrm{d}x & + & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}x & + & \operatorname{d}y \\
\mathrm{d}x & + & \mathrm{d}y \\
\operatorname{d}x & + & \operatorname{d}y \\
\mathrm{d}x & + & \mathrm{d}y \\
\end{smallmatrix}$ \\
\end{smallmatrix}$ \\
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{smallmatrix}$ \\
\end{smallmatrix}$ \\
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
y & \operatorname{d}x & + &
y & \mathrm{d}x & + &
x & \operatorname{d}y & + &
x & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
(y) & \operatorname{d}x & + &
(y) & \mathrm{d}x & + &
x & \operatorname{d}y & + &
x & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
y & \operatorname{d}x & + &
y & \mathrm{d}x & + &
(x) & \operatorname{d}y & + &
(x) & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
(y) & \operatorname{d}x & + &
(y) & \mathrm{d}x & + &
(x) & \operatorname{d}y & + &
(x) & \mathrm{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
y & \operatorname{d}x & + & x & \operatorname{d}y \\
y & \mathrm{d}x & + & x & \mathrm{d}y \\
(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
(y) & \mathrm{d}x & + & x & \mathrm{d}y \\
y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
y & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
\end{smallmatrix}$ \\
\end{smallmatrix}$ \\
\hline
\hline
\begin{center}\begin{tabular}{|c|c|c||c|c|c|c|}
\begin{center}\begin{tabular}{|c|c|c||c|c|c|c|}
\multicolumn{7}{c}{\textbf{Taylor Series Expansion $\operatorname{D}f = \operatorname{d}f + \operatorname{d}^2 f$}} \\
\multicolumn{7}{c}{\textbf{Taylor Series Expansion $\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2 f$}} \\
\hline
\hline
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}f = \\
\mathrm{d}f = \\
\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y \\
\partial_x f \cdot \mathrm{d}x\ +\ \partial_y f \cdot \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}^2 f = \\
\mathrm{d}^2 f = \\
\partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y \\
\partial_{xy} f \cdot \mathrm{d}x\, \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\operatorname{d}f|_{x\ y}$ &
$\mathrm{d}f|_{x\ y}$ &
$\operatorname{d}f|_{x\ (y)}$ &
$\mathrm{d}f|_{x\ (y)}$ &
$\operatorname{d}f|_{(x)\ y}$ &
$\mathrm{d}f|_{(x)\ y}$ &
$\operatorname{d}f|_{(x)(y)}$ \\
$\mathrm{d}f|_{(x)(y)}$ \\
\hline
\hline
$f_0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$f_0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
&
&
$\begin{matrix}
$\begin{matrix}
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
y & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
(y) & \mathrm{d}x & + & x & \mathrm{d}y \\
y & \operatorname{d}x & + & x & \operatorname{d}y \\
y & \mathrm{d}x & + & x & \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
0 \\
0 \\
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
0 \\
0 \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
0 \\
0 \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}x \\
\mathrm{d}x \\
0 \\
0 \\
\end{matrix}$ \\
\end{matrix}$ \\
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$ \\
\end{matrix}$ \\
\hline
\hline
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$ \\
\end{matrix}$ \\
\hline
\hline
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$ \\
\end{matrix}$ \\
\hline
\hline
&
&
$\begin{matrix}
$\begin{matrix}
y & \operatorname{d}x & + & x & \operatorname{d}y \\
y & \mathrm{d}x & + & x & \mathrm{d}y \\
(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
(y) & \mathrm{d}x & + & x & \mathrm{d}y \\
y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
y & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\operatorname{d}x\ \operatorname{d}y \\
\mathrm{d}x\ \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}x \\
\mathrm{d}x \\
0 \\
0 \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
0 \\
0 \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
0 \\
0 \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
$\begin{matrix}
$\begin{matrix}
0 \\
0 \\
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$ \\
\end{matrix}$ \\
\hline
\hline
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}f = \\
\mathrm{d}f = \\
\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y
\partial_x f \cdot \mathrm{d}x\ +\ \partial_y f \cdot \mathrm{d}y
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
y & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
(y) & \mathrm{d}x & + & x & \mathrm{d}y \\
y & \operatorname{d}x & + & x & \operatorname{d}y \\
y & \mathrm{d}x & + & x & \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x \\
\mathrm{d}x \\
\operatorname{d}x \\
\mathrm{d}x \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\operatorname{d}x + \operatorname{d}y \\
\mathrm{d}x + \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
\operatorname{d}y \\
\mathrm{d}y \\
\operatorname{d}y \\
\mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
&
&
$\begin{matrix}
$\begin{matrix}
y & \operatorname{d}x & + & x & \operatorname{d}y \\
y & \mathrm{d}x & + & x & \mathrm{d}y \\
(y) & \operatorname{d}x & + & x & \operatorname{d}y \\
(y) & \mathrm{d}x & + & x & \mathrm{d}y \\
y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
y & \mathrm{d}x & + & (x) & \mathrm{d}y \\
(y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
(y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\
\end{matrix}$
\end{matrix}$
&
&
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
\begin{quote}\begin{tabular}{||c||c|c|c|c||}
\multicolumn{5}{c}{\textbf{Detail of Calculation for $\operatorname{D}f = \operatorname{E}f + f$}} \\[6pt]
\multicolumn{5}{c}{\textbf{Detail of Calculation for $\mathrm{D}f = \mathrm{E}f + f$}} \\[6pt]
\hline\hline
\hline\hline
&
&
$\begin{array}{cr}
$\begin{array}{cr}
& \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\
& \mathrm{E}f|_{\mathrm{d}x\ \mathrm{d}y} \\
+ & f|_{\operatorname{d}x\ \operatorname{d}y} \\
+ & f|_{\mathrm{d}x\ \mathrm{d}y} \\
= & \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \\
= & \mathrm{D}f|_{\mathrm{d}x\ \mathrm{d}y} \\
\end{array}$
\end{array}$
&
&
$\begin{array}{cr}
$\begin{array}{cr}
& \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
& \mathrm{E}f|_{\mathrm{d}x\ (\mathrm{d}y)} \\
+ & f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
+ & f|_{\mathrm{d}x\ (\mathrm{d}y)} \\
= & \operatorname{D}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
= & \mathrm{D}f|_{\mathrm{d}x\ (\mathrm{d}y)} \\
\end{array}$
\end{array}$
&
&
$\begin{array}{cr}
$\begin{array}{cr}
& \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
& \mathrm{E}f|_{(\mathrm{d}x)\ \mathrm{d}y} \\
+ & f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
+ & f|_{(\mathrm{d}x)\ \mathrm{d}y} \\
= & \operatorname{D}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
= & \mathrm{D}f|_{(\mathrm{d}x)\ \mathrm{d}y} \\
\end{array}$
\end{array}$
&
&
$\begin{array}{cr}
$\begin{array}{cr}
& \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
& \mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)} \\
+ & f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
+ & f|_{(\mathrm{d}x)(\mathrm{d}y)} \\
= & \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
= & \mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)} \\
\end{array}$ \\[6pt]
\end{array}$ \\[6pt]
\hline\hline
\hline\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ y & \operatorname{d}x & \operatorname{d}y \\
& x\ y & \mathrm{d}x & \mathrm{d}y \\
+ & (x)(y) & \operatorname{d}x & \operatorname{d}y \\
+ & (x)(y) & \mathrm{d}x & \mathrm{d}y \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
= & ((x, y)) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\
& x\ (y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x) (y) & \mathrm{d}x & (\mathrm{d}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
= & (y) & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\
& (x)\ y & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x) (y) & (\mathrm{d}x) & \mathrm{d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
= & (x) & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\
& (x)(y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (x)(y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ (y) & \operatorname{d}x & \operatorname{d}y \\
& x\ (y) & \mathrm{d}x & \mathrm{d}y \\
+ & (x)\ y & \operatorname{d}x & \operatorname{d}y \\
+ & (x)\ y & \mathrm{d}x & \mathrm{d}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
= & (x, y) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ y & \operatorname{d}x & (\operatorname{d}y) \\
& x\ y & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x)\ y & \mathrm{d}x & (\mathrm{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
= & y & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\
& (x) (y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x)\ y & (\mathrm{d}x) & \mathrm{d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
= & (x) & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
& (x)\ y & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (x)\ y & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x)\ y & \operatorname{d}x & \operatorname{d}y \\
& (x)\ y & \mathrm{d}x & \mathrm{d}y \\
+ & x\ (y) & \operatorname{d}x & \operatorname{d}y \\
+ & x\ (y) & \mathrm{d}x & \mathrm{d}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
= & (x, y) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\
& (x) (y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & x\ (y) & \mathrm{d}x & (\mathrm{d}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
= & (y) & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ y & (\operatorname{d}x) & \operatorname{d}y \\
& x\ y & (\mathrm{d}x) & \mathrm{d}y \\
+ & x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & x\ (y) & (\mathrm{d}x) & \mathrm{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
= & x & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
& x\ (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & x\ (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x)(y) & \operatorname{d}x & \operatorname{d}y \\
& (x)(y) & \mathrm{d}x & \mathrm{d}y \\
+ & x\ y & \operatorname{d}x & \operatorname{d}y \\
+ & x\ y & \mathrm{d}x & \mathrm{d}y \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
= & ((x, y)) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\
& (x)\ y & \mathrm{d}x & (\mathrm{d}y) \\
+ & x\ y & \operatorname{d}x & (\operatorname{d}y) \\
+ & x\ y & \mathrm{d}x & (\mathrm{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
= & y & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\
& x\ (y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & x\ y & (\operatorname{d}x) & \operatorname{d}y \\
+ & x\ y & (\mathrm{d}x) & \mathrm{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
= & x & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
& x\ y & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & x\ y & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline\hline
\hline\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x & \operatorname{d}x & \operatorname{d}y \\
& x & \mathrm{d}x & \mathrm{d}y \\
+ & (x) & \operatorname{d}x & \operatorname{d}y \\
+ & (x) & \mathrm{d}x & \mathrm{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
= & 1 & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x & \operatorname{d}x & (\operatorname{d}y) \\
& x & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x) & \mathrm{d}x & (\mathrm{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
= & 1 & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x) & (\operatorname{d}x) & \operatorname{d}y \\
& (x) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x) & (\mathrm{d}x) & \mathrm{d}y \\
= & 0 & (\operatorname{d}x) & \operatorname{d}y \\
= & 0 & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x) & (\operatorname{d}x) & (\operatorname{d}y) \\
& (x) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (x) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x) & \operatorname{d}x & \operatorname{d}y \\
& (x) & \mathrm{d}x & \mathrm{d}y \\
+ & x & \operatorname{d}x & \operatorname{d}y \\
+ & x & \mathrm{d}x & \mathrm{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
= & 1 & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x) & \operatorname{d}x & (\operatorname{d}y) \\
& (x) & \mathrm{d}x & (\mathrm{d}y) \\
+ & x & \operatorname{d}x & (\operatorname{d}y) \\
+ & x & \mathrm{d}x & (\mathrm{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
= & 1 & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x & (\operatorname{d}x) & \operatorname{d}y \\
& x & (\mathrm{d}x) & \mathrm{d}y \\
+ & x & (\operatorname{d}x) & \operatorname{d}y \\
+ & x & (\mathrm{d}x) & \mathrm{d}y \\
= & 0 & (\operatorname{d}x) & \operatorname{d}y \\
= & 0 & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& x & (\operatorname{d}x) & (\operatorname{d}y) \\
& x & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & x & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & x & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline\hline
\hline\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x, y) & \operatorname{d}x & \operatorname{d}y \\
& (x, y) & \mathrm{d}x & \mathrm{d}y \\
+ & (x, y) & \operatorname{d}x & \operatorname{d}y \\
+ & (x, y) & \mathrm{d}x & \mathrm{d}y \\
= & 0 & \operatorname{d}x & \operatorname{d}y \\
= & 0 & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\
& ((x, y)) & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x, y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x, y) & \mathrm{d}x & (\mathrm{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
= & 1 & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\
& ((x, y)) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x, y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x, y) & (\mathrm{d}x) & \mathrm{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
= & 1 & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\
& (x, y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (x, y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
& ((x, y)) & \mathrm{d}x & \mathrm{d}y \\
+ & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
+ & ((x, y)) & \mathrm{d}x & \mathrm{d}y \\
= & 0 & \operatorname{d}x & \operatorname{d}y \\
= & 0 & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x, y) & \operatorname{d}x & (\operatorname{d}y) \\
& (x, y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\
+ & ((x, y)) & \mathrm{d}x & (\mathrm{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
= & 1 & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x, y) & (\operatorname{d}x) & \operatorname{d}y \\
& (x, y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\
+ & ((x, y)) & (\mathrm{d}x) & \mathrm{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
= & 1 & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
& ((x, y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & ((x, y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline\hline
\hline\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& y & \operatorname{d}x & \operatorname{d}y \\
& y & \mathrm{d}x & \mathrm{d}y \\
+ & (y) & \operatorname{d}x & \operatorname{d}y \\
+ & (y) & \mathrm{d}x & \mathrm{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
= & 1 & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (y) & \operatorname{d}x & (\operatorname{d}y) \\
& (y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & (y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (y) & \mathrm{d}x & (\mathrm{d}y) \\
= & 0 & \operatorname{d}x & (\operatorname{d}y) \\
= & 0 & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& y & (\operatorname{d}x) & \operatorname{d}y \\
& y & (\mathrm{d}x) & \mathrm{d}y \\
+ & (y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (y) & (\mathrm{d}x) & \mathrm{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
= & 1 & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
& (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (y) & \operatorname{d}x & \operatorname{d}y \\
& (y) & \mathrm{d}x & \mathrm{d}y \\
+ & y & \operatorname{d}x & \operatorname{d}y \\
+ & y & \mathrm{d}x & \mathrm{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
= & 1 & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& y & \operatorname{d}x & (\operatorname{d}y) \\
& y & \mathrm{d}x & (\mathrm{d}y) \\
+ & y & \operatorname{d}x & (\operatorname{d}y) \\
+ & y & \mathrm{d}x & (\mathrm{d}y) \\
= & 0 & \operatorname{d}x & (\operatorname{d}y) \\
= & 0 & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (y) & (\operatorname{d}x) & \operatorname{d}y \\
& (y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & y & (\operatorname{d}x) & \operatorname{d}y \\
+ & y & (\mathrm{d}x) & \mathrm{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
= & 1 & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& y & (\operatorname{d}x) & (\operatorname{d}y) \\
& y & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & y & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & y & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline\hline
\hline\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\
& ((x)(y)) & \mathrm{d}x & \mathrm{d}y \\
+ & (x\ y) & \operatorname{d}x & \operatorname{d}y \\
+ & (x\ y) & \mathrm{d}x & \mathrm{d}y \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
= & ((x, y)) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\
& ((x)\ y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x\ y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x\ y) & \mathrm{d}x & (\mathrm{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
= & y & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\
& (x\ (y)) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x\ y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x\ y) & (\mathrm{d}x) & \mathrm{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
= & x & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
& (x\ y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (x\ y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\
& ((x)\ y) & \mathrm{d}x & \mathrm{d}y \\
+ & (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\
+ & (x\ (y)) & \mathrm{d}x & \mathrm{d}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
= & (x, y) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\
& ((x) (y)) & \mathrm{d}x & (\mathrm{d}y) \\
+ & (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x\ (y)) & \mathrm{d}x & (\mathrm{d}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
= & (y) & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ y) & (\operatorname{d}x) & \operatorname{d}y \\
& (x\ y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x\ (y)) & (\mathrm{d}x) & \mathrm{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
= & x & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
& (x\ (y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & (x\ (y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\
& (x\ (y)) & \mathrm{d}x & \mathrm{d}y \\
+ & ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\
+ & ((x)\ y) & \mathrm{d}x & \mathrm{d}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
= & (x, y) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ y) & \operatorname{d}x & (\operatorname{d}y) \\
& (x\ y) & \mathrm{d}x & (\mathrm{d}y) \\
+ & ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & ((x)\ y) & \mathrm{d}x & (\mathrm{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
= & y & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\
& ((x) (y)) & (\mathrm{d}x) & \mathrm{d}y \\
+ & ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & ((x)\ y) & (\mathrm{d}x) & \mathrm{d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
= & (x) & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
& ((x)\ y) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & ((x)\ y) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline
\hline
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ y) & \operatorname{d}x & \operatorname{d}y \\
& (x\ y) & \mathrm{d}x & \mathrm{d}y \\
+ & ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\
+ & ((x)(y)) & \mathrm{d}x & \mathrm{d}y \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
= & ((x, y)) & \mathrm{d}x & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\
& (x\ (y)) & \mathrm{d}x & (\mathrm{d}y) \\
+ & ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+ & ((x) (y)) & \mathrm{d}x & (\mathrm{d}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
= & (y) & \mathrm{d}x & (\mathrm{d}y) \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\
& ((x)\ y) & (\mathrm{d}x) & \mathrm{d}y \\
+ & ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+ & ((x) (y)) & (\mathrm{d}x) & \mathrm{d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
= & (x) & (\mathrm{d}x) & \mathrm{d}y \\
\end{smallmatrix}$
\end{smallmatrix}$
&
&
$\begin{smallmatrix}
$\begin{smallmatrix}
& ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
& ((x)(y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
+ & ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & ((x)(y)) & (\mathrm{d}x) & (\mathrm{d}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
= & 0 & (\mathrm{d}x) & (\mathrm{d}y) \\
\end{smallmatrix}$ \\[6pt]
\end{smallmatrix}$ \\[6pt]
\hline\hline
\hline\hline
</blockquote>
</blockquote>
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \operatorname{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, 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 <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, 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.
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\epsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\epsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\epsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\epsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\epsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\epsilon\!</math>" silent.
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\epsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\epsilon\!</math>" silent.
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \operatorname{E}\mathcal{X} = \{ A, \operatorname{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, (A), A, 1 \},\!</math> the tacit extension <math>\epsilon f\!</math> of <math>f\!</math> to <math>\operatorname{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau\ ,</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\epsilon f\!</math> of <math>f\!</math> to <math>\operatorname{E}X</math> may be explicated as shown in Table 15.
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, (A), A, 1 \},\!</math> the tacit extension <math>\epsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau\ ,</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\epsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table 15.
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
|+ '''Table 15. Tacit Extension of <math>[A]\!</math> to <math>[A, \operatorname{d}A]</math>'''
|+ '''Table 15. Tacit Extension of <math>[A]\!</math> to <math>[A, \mathrm{d}A]</math>'''
|
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| <math>0\!</math>
| <math>0\!</math>
| <math>\cdot\!</math>
| <math>\cdot\!</math>
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| <math>((\mathrm{d}A),\ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>(A)\!</math>
| <math>(A)\!</math>
| <math>\cdot\!</math>
| <math>\cdot\!</math>
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| <math>((\mathrm{d}A),\ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(A)(\operatorname{d}A)\ +\ (A)\ \operatorname{d}A\!</math>
| <math>(A)(\mathrm{d}A)\ +\ (A)\ \mathrm{d}A\!</math>
|
|
|-
|-
| <math>A\!</math>
| <math>A\!</math>
| <math>\cdot\!</math>
| <math>\cdot\!</math>
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| <math>((\mathrm{d}A),\ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>A\ (\operatorname{d}A)\ +\ A\ \operatorname{d}A\!</math>
| <math>A\ (\mathrm{d}A)\ +\ A\ \mathrm{d}A\!</math>
|
|
|-
|-
| <math>1\!</math>
| <math>1\!</math>
| <math>\cdot\!</math>
| <math>\cdot\!</math>
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| <math>((\mathrm{d}A),\ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>\overline{q}\!</math>
| <math>\overline{q}\!</math>
| and
| and
| <math>\overline{\operatorname{d}q}\!</math>
| <math>\overline{\mathrm{d}q}\!</math>
| infer
| infer
| <math>\overline{q}\!</math>
| <math>\overline{q}\!</math>
| <math>\overline{q}\!</math>
| <math>\overline{q}\!</math>
| and
| and
| <math>\operatorname{d}q\!</math>
| <math>\mathrm{d}q\!</math>
| infer
| infer
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| and
| and
| <math>\overline{\operatorname{d}q}\!</math>
| <math>\overline{\mathrm{d}q}\!</math>
| infer
| infer
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| and
| and
| <math>\operatorname{d}q\!</math>
| <math>\mathrm{d}q\!</math>
| infer
| infer
| <math>\overline{q}\!</math>
| <math>\overline{q}\!</math>
| <math>(q)\!</math>
| <math>(q)\!</math>
| and
| and
| <math>(\operatorname{d}q)\!</math>
| <math>(\mathrm{d}q)\!</math>
| infer
| infer
| <math>(q)\!</math>
| <math>(q)\!</math>
| <math>(q)\!</math>
| <math>(q)\!</math>
| and
| and
| <math>\operatorname{d}q\!</math>
| <math>\mathrm{d}q\!</math>
| infer
| infer
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| and
| and
| <math>(\operatorname{d}q)\!</math>
| <math>(\mathrm{d}q)\!</math>
| infer
| infer
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| <math>q\!</math>
| and
| and
| <math>\operatorname{d}q\!</math>
| <math>\mathrm{d}q\!</math>
| infer
| infer
| <math>(q)\!</math>
| <math>(q)\!</math>
|-
|-
| <math>A^*\!</math>
| <math>A^*\!</math>
| <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| <math>(\mathrm{hom} : A \to \mathbb{B})</math>
| Linear functions
| Linear functions
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
! Type
! Type
|-
|-
| <math>\operatorname{d}\mathfrak{A}</math>
| <math>\mathrm{d}\mathfrak{A}</math>
| <math>\lbrace\!</math> “<math>\operatorname{d}a_1</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n</math>” <math>\rbrace\!</math>
| <math>\lbrace\!</math> “<math>\mathrm{d}a_1</math>” <math>, \ldots,\!</math> “<math>\mathrm{d}a_n</math>” <math>\rbrace\!</math>
| Alphabet of<br>
| Alphabet of<br>
differential<br>
differential<br>
| <math>[n] = \mathbf{n}</math>
| <math>[n] = \mathbf{n}</math>
|-
|-
| <math>\operatorname{d}\mathcal{A}</math>
| <math>\mathrm{d}\mathcal{A}</math>
| <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}</math>
| Basis of<br>
| Basis of<br>
differential<br>
differential<br>
| <math>[n] = \mathbf{n}</math>
| <math>[n] = \mathbf{n}</math>
|-
|-
| <math>\operatorname{d}A_i</math>
| <math>\mathrm{d}A_i</math>
| <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}</math>
| <math>\{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \}</math>
| Differential<br>
| Differential<br>
dimension <math>i\!</math>
dimension <math>i\!</math>
| <math>\mathbb{D}</math>
| <math>\mathbb{D}</math>
|-
|-
| <math>\operatorname{d}A</math>
| <math>\mathrm{d}A</math>
| <math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>
| <math>\langle \mathrm{d}\mathcal{A} \rangle</math><br>
<math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>
<math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle</math><br>
<math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>
<math>\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}</math><br>
<math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>
<math>\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n</math><br>
<math>\textstyle \prod_i \operatorname{d}A_i</math>
<math>\textstyle \prod_i \mathrm{d}A_i</math>
| Tangent space<br>
| Tangent space<br>
at a point:<br>
at a point:<br>
| <math>\mathbb{D}^n</math>
| <math>\mathbb{D}^n</math>
|-
|-
| <math>\operatorname{d}A^*</math>
| <math>\mathrm{d}A^*</math>
| <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})</math>
| Linear functions<br>
| Linear functions<br>
on <math>\operatorname{d}A</math>
on <math>\mathrm{d}A</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
|-
|-
| <math>\operatorname{d}A^\uparrow</math>
| <math>\mathrm{d}A^\uparrow</math>
| <math>(\operatorname{d}A \to \mathbb{B})</math>
| <math>(\mathrm{d}A \to \mathbb{B})</math>
| Boolean functions<br>
| Boolean functions<br>
on <math>\operatorname{d}A</math>
on <math>\mathrm{d}A</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
|-
|-
| <math>\operatorname{d}A^\circ</math>
| <math>\mathrm{d}A^\circ</math>
| <math>[\operatorname{d}\mathcal{A}]</math><br>
| <math>[\mathrm{d}\mathcal{A}]</math><br>
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
<math>(\mathrm{d}A, \mathrm{d}A^\uparrow)</math><br>
<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
<math>(\mathrm{d}A\ +\!\to \mathbb{B})</math><br>
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
<math>(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))</math><br>
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
<math>[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]</math>
| Tangent universe<br>
| Tangent universe<br>
at a point of <math>A^\circ,</math><br>
at a point of <math>A^\circ,</math><br>
based on the<br>
based on the<br>
tangent features<br>
tangent features<br>
<math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
<math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}</math>
| <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
| <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''<math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|+ '''<math>\mathrm{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" |
| style="width:16%" |
| style="width:16%" | <math>f\!</math>
| style="width:16%" | <math>f\!</math>
| style="width:16%" | <math>\operatorname{E}f|_{xy}</math>
| style="width:16%" | <math>\mathrm{E}f|_{xy}</math>
| style="width:16%" | <math>\operatorname{E}f|_{x(\!|y|\!)}</math>
| style="width:16%" | <math>\mathrm{E}f|_{x(\!|y|\!)}</math>
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)y}</math>
| style="width:16%" | <math>\mathrm{E}f|_{(\!|x|\!)y}</math>
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
| style="width:16%" | <math>\mathrm{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
|-
|-
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
| <math>f_{1}\!</math>
| <math>f_{1}\!</math>
| <math>(\!|x|\!)(\!|y|\!)</math>
| <math>(\!|x|\!)(\!|y|\!)</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>\mathrm{d}x\ \mathrm{d}y</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| <math>\mathrm{d}x (\!|\mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>(\!|\mathrm{d}x|\!) \mathrm{d}y</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)</math>
|-
|-
| <math>f_{2}\!</math>
| <math>f_{2}\!</math>
| <math>(\!|x|\!) y</math>
| <math>(\!|x|\!) y</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| <math>\mathrm{d}x (\!|\mathrm{d}y|\!)</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>\mathrm{d}x\ \mathrm{d}y</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>(\!|\mathrm{d}x|\!) \mathrm{d}y</math>
|-
|-
| <math>f_{4}\!</math>
| <math>f_{4}\!</math>
| <math>x (\!|y|\!)</math>
| <math>x (\!|y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>(\!|\mathrm{d}x|\!) \mathrm{d}y</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>\mathrm{d}x\ \mathrm{d}y</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| <math>\mathrm{d}x (\!|\mathrm{d}y|\!)</math>
|-
|-
| <math>f_{8}\!</math>
| <math>f_{8}\!</math>
| <math>x y\!</math>
| <math>x y\!</math>
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
| <math>(\!|\mathrm{d}x|\!) \mathrm{d}y</math>
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
| <math>\mathrm{d}x (\!|\mathrm{d}y|\!)</math>
| <math>\operatorname{d}x\ \operatorname{d}y</math>
| <math>\mathrm{d}x\ \mathrm{d}y</math>
|-
|-
| <math>f_{3}\!</math>
| <math>f_{3}\!</math>
| <math>(\!|x|\!)</math>
| <math>(\!|x|\!)</math>
| <math>\operatorname{d}x</math>
| <math>\mathrm{d}x</math>
| <math>\operatorname{d}x</math>
| <math>\mathrm{d}x</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>(\!|\mathrm{d}x|\!)</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>(\!|\mathrm{d}x|\!)</math>
|-
|-
| <math>f_{12}\!</math>
| <math>f_{12}\!</math>
| <math>x\!</math>
| <math>x\!</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>(\!|\mathrm{d}x|\!)</math>
| <math>(\!|\operatorname{d}x|\!)</math>
| <math>(\!|\mathrm{d}x|\!)</math>
| <math>\operatorname{d}x</math>
| <math>\mathrm{d}x</math>
| <math>\operatorname{d}x</math>
| <math>\mathrm{d}x</math>
|-
|-
| <math>f_{6}\!</math>
| <math>f_{6}\!</math>
| <math>(\!|x, y|\!)</math>
| <math>(\!|x, y|\!)</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x, \mathrm{d}y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x, \mathrm{d}y|\!)</math>
|-
|-
| <math>f_{9}\!</math>
| <math>f_{9}\!</math>
| <math>(\!|(\!|x, y|\!)|\!)</math>
| <math>(\!|(\!|x, y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x, \mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x, \mathrm{d}y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x, \mathrm{d}y|\!)|\!)</math>
|-
|-
| <math>f_{5}\!</math>
| <math>f_{5}\!</math>
| <math>(\!|y|\!)</math>
| <math>(\!|y|\!)</math>
| <math>\operatorname{d}y</math>
| <math>\mathrm{d}y</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}y|\!)</math>
| <math>\operatorname{d}y</math>
| <math>\mathrm{d}y</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}y|\!)</math>
|-
|-
| <math>f_{10}\!</math>
| <math>f_{10}\!</math>
| <math>y\!</math>
| <math>y\!</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}y|\!)</math>
| <math>\operatorname{d}y</math>
| <math>\mathrm{d}y</math>
| <math>(\!|\operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}y|\!)</math>
| <math>\operatorname{d}y</math>
| <math>\mathrm{d}y</math>
|-
|-
| <math>f_{7}\!</math>
| <math>f_{7}\!</math>
| <math>(\!|x y|\!)</math>
| <math>(\!|x y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x\ \mathrm{d}y|\!)</math>
|-
|-
| <math>f_{11}\!</math>
| <math>f_{11}\!</math>
| <math>(\!|x (\!|y|\!)|\!)</math>
| <math>(\!|x (\!|y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x\ \mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)</math>
|-
|-
| <math>f_{13}\!</math>
| <math>f_{13}\!</math>
| <math>(\!|(\!|x|\!) y|\!)</math>
| <math>(\!|(\!|x|\!) y|\!)</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x\ \mathrm{d}y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)</math>
|-
|-
| <math>f_{14}\!</math>
| <math>f_{14}\!</math>
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
| <math>(\!|\mathrm{d}x\ \mathrm{d}y|\!)</math>
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|\mathrm{d}x (\!|\mathrm{d}y|\!)|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!) \mathrm{d}y|\!)</math>
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
| <math>(\!|(\!|\mathrm{d}x|\!)(\!|\mathrm{d}y|\!)|\!)</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| 0 0 0 0
| 0 0 0 0
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>\operatorname{false}</math>
| <math>\mathrm{false}</math>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
| 0 0 0 1
| 0 0 0 1
| <math>(x)(y)\!</math>
| <math>(x)(y)\!</math>
| <math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math>
| <math>\mathrm{neither}\ x\ \mathrm{nor}\ y</math>
| <math>\lnot x \land \lnot y\!</math>
| <math>\lnot x \land \lnot y\!</math>
|-
|-
| 0 0 1 0
| 0 0 1 0
| <math>(x)\ y\!</math>
| <math>(x)\ y\!</math>
| <math>y\ \operatorname{without}\ x</math>
| <math>y\ \mathrm{without}\ x</math>
| <math>\lnot x \land y\!</math>
| <math>\lnot x \land y\!</math>
|-
|-
| 0 0 1 1
| 0 0 1 1
| <math>(x)\!</math>
| <math>(x)\!</math>
| <math>\operatorname{not}\ x</math>
| <math>\mathrm{not}\ x</math>
| <math>\lnot x\!</math>
| <math>\lnot x\!</math>
|-
|-
| 0 1 0 0
| 0 1 0 0
| <math>x\ (y)\!</math>
| <math>x\ (y)\!</math>
| <math>x\ \operatorname{without}\ y</math>
| <math>x\ \mathrm{without}\ y</math>
| <math>x \land \lnot y\!</math>
| <math>x \land \lnot y\!</math>
|-
|-
| 0 1 0 1
| 0 1 0 1
| <math>(y)\!</math>
| <math>(y)\!</math>
| <math>\operatorname{not}\ y</math>
| <math>\mathrm{not}\ y</math>
| <math>\lnot y\!</math>
| <math>\lnot y\!</math>
|-
|-
| 0 1 1 0
| 0 1 1 0
| <math>(x,\ y)\!</math>
| <math>(x,\ y)\!</math>
| <math>x\ \operatorname{not~equal~to}\ y</math>
| <math>x\ \mathrm{not~equal~to}\ y</math>
| <math>x \ne y\!</math>
| <math>x \ne y\!</math>
|-
|-
| 0 1 1 1
| 0 1 1 1
| <math>(x\ y)\!</math>
| <math>(x\ y)\!</math>
| <math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math>
| <math>\mathrm{not~both}\ x\ \mathrm{and}\ y</math>
| <math>\lnot x \lor \lnot y\!</math>
| <math>\lnot x \lor \lnot y\!</math>
|-
|-
| 1 0 0 0
| 1 0 0 0
| <math>x\ y\!</math>
| <math>x\ y\!</math>
| <math>x\ \operatorname{and}\ y</math>
| <math>x\ \mathrm{and}\ y</math>
| <math>x \land y\!</math>
| <math>x \land y\!</math>
|-
|-
| 1 0 0 1
| 1 0 0 1
| <math>((x,\ y))\!</math>
| <math>((x,\ y))\!</math>
| <math>x\ \operatorname{equal~to}\ y</math>
| <math>x\ \mathrm{equal~to}\ y</math>
| <math>x = y\!</math>
| <math>x = y\!</math>
|-
|-
| 1 0 1 1
| 1 0 1 1
| <math>(x\ (y))\!</math>
| <math>(x\ (y))\!</math>
| <math>\operatorname{not}\ x\ \operatorname{without}\ y</math>
| <math>\mathrm{not}\ x\ \mathrm{without}\ y</math>
| <math>x \Rightarrow y\!</math>
| <math>x \Rightarrow y\!</math>
|-
|-
| 1 1 0 1
| 1 1 0 1
| <math>((x)\ y)\!</math>
| <math>((x)\ y)\!</math>
| <math>\operatorname{not}\ y\ \operatorname{without}\ x</math>
| <math>\mathrm{not}\ y\ \mathrm{without}\ x</math>
| <math>x \Leftarrow y\!</math>
| <math>x \Leftarrow y\!</math>
|-
|-
| 1 1 1 0
| 1 1 1 0
| <math>((x)(y))\!</math>
| <math>((x)(y))\!</math>
| <math>x\ \operatorname{or}\ y</math>
| <math>x\ \mathrm{or}\ y</math>
| <math>x \lor y\!</math>
| <math>x \lor y\!</math>
|-
|-
| 1 1 1 1
| 1 1 1 1
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>\operatorname{true}</math>
| <math>\mathrm{true}</math>
| <math>1\!</math>
| <math>1\!</math>
|}
|}
| <p>0 0 0 0</p>
| <p>0 0 0 0</p>
| <p><math>(~)\!</math></p>
| <p><math>(~)\!</math></p>
| <p><math>\operatorname{false}</math></p>
| <p><math>\mathrm{false}</math></p>
| <p><math>0\!</math></p>
| <p><math>0\!</math></p>
|-
|-
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
<p><math>\mathrm{neither}\ x\ \mathrm{nor}\ y</math></p>
<p><math>y\ \operatorname{without}\ x</math></p>
<p><math>y\ \mathrm{without}\ x</math></p>
<p><math>x\ \operatorname{without}\ y</math></p>
<p><math>x\ \mathrm{without}\ y</math></p>
<p><math>x\ \operatorname{and}\ y</math></p>
<p><math>x\ \mathrm{and}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{not}\ x</math></p>
<p><math>\mathrm{not}\ x</math></p>
<p><math>x\!</math></p>
<p><math>x\!</math></p>
|}
|}
{| align="center"
{| align="center"
|
|
<p><math>x\ \operatorname{not~equal~to}\ y</math></p>
<p><math>x\ \mathrm{not~equal~to}\ y</math></p>
<p><math>x\ \operatorname{equal~to}\ y</math></p>
<p><math>x\ \mathrm{equal~to}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{not}\ y</math></p>
<p><math>\mathrm{not}\ y</math></p>
<p><math>y\!</math></p>
<p><math>y\!</math></p>
|}
|}
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
<p><math>\mathrm{not~both}\ x\ \mathrm{and}\ y</math></p>
<p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
<p><math>\mathrm{not}\ x\ \mathrm{without}\ y</math></p>
<p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
<p><math>\mathrm{not}\ y\ \mathrm{without}\ x</math></p>
<p><math>x\ \operatorname{or}\ y</math></p>
<p><math>x\ \mathrm{or}\ y</math></p>
|}
|}
|
|
| <p>1 1 1 1</p>
| <p>1 1 1 1</p>
| <p><math>((~))\!</math></p>
| <p><math>((~))\!</math></p>
| <p><math>\operatorname{true}</math></p>
| <p><math>\mathrm{true}</math></p>
| <p><math>1\!</math></p>
| <p><math>1\!</math></p>
|}
|}
| <p>0 0 0 0</p>
| <p>0 0 0 0</p>
| <p><math>(~)\!</math></p>
| <p><math>(~)\!</math></p>
| <p><math>\operatorname{false}</math></p>
| <p><math>\mathrm{false}</math></p>
| <p><math>0\!</math></p>
| <p><math>0\!</math></p>
|-
|-
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p><br>
<p><math>\mathrm{neither}\ x\ \mathrm{nor}\ y</math></p><br>
<p><math>y\ \operatorname{without}\ x</math></p><br>
<p><math>y\ \mathrm{without}\ x</math></p><br>
<p><math>x\ \operatorname{without}\ y</math></p><br>
<p><math>x\ \mathrm{without}\ y</math></p><br>
<p><math>x\ \operatorname{and}\ y</math></p>
<p><math>x\ \mathrm{and}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{not}\ x</math></p><br>
<p><math>\mathrm{not}\ x</math></p><br>
<p><math>x\!</math></p>
<p><math>x\!</math></p>
|}
|}
{| align="center"
{| align="center"
|
|
<p><math>x\ \operatorname{not~equal~to}\ y</math></p><br>
<p><math>x\ \mathrm{not~equal~to}\ y</math></p><br>
<p><math>x\ \operatorname{equal~to}\ y</math></p>
<p><math>x\ \mathrm{equal~to}\ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{not}\ y</math></p><br>
<p><math>\mathrm{not}\ y</math></p><br>
<p><math>y\!</math></p>
<p><math>y\!</math></p>
|}
|}
{| align="center"
{| align="center"
|
|
<p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p><br>
<p><math>\mathrm{not~both}\ x\ \mathrm{and}\ y</math></p><br>
<p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p><br>
<p><math>\mathrm{not}\ x\ \mathrm{without}\ y</math></p><br>
<p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p><br>
<p><math>\mathrm{not}\ y\ \mathrm{without}\ x</math></p><br>
<p><math>x\ \operatorname{or}\ y</math></p>
<p><math>x\ \mathrm{or}\ y</math></p>
|}
|}
|
|
| <p>1 1 1 1</p>
| <p>1 1 1 1</p>
| <p><math>((~))\!</math></p>
| <p><math>((~))\!</math></p>
| <p><math>\operatorname{true}</math></p>
| <p><math>\mathrm{true}</math></p>
| <p><math>1\!</math></p>
| <p><math>1\!</math></p>
|}
|}
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 3. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|+ '''Table 3. <math>\mathrm{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|- style="background:ghostwhite; height:36px"
|- style="background:ghostwhite; height:36px"
|
|
| <math>f\!</math>
| <math>f\!</math>
| <math>\operatorname{E}f|_{xy}</math>
| <math>\mathrm{E}f|_{xy}</math>
| <math>\operatorname{E}f|_{x(y)}</math>
| <math>\mathrm{E}f|_{x(y)}</math>
| <math>\operatorname{E}f|_{(x)y}</math>
| <math>\mathrm{E}f|_{(x)y}</math>
| <math>\operatorname{E}f|_{(x)(y)}</math>
| <math>\mathrm{E}f|_{(x)(y)}</math>
|-
|-
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
| <math>f_{1}\!</math>
| <math>f_{1}\!</math>
| <math>(x)(y)\!</math>
| <math>(x)(y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>\mathrm{d}x\ \mathrm{d}y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>\mathrm{d}x (\mathrm{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\mathrm{d}x) \mathrm{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}x)(\mathrm{d}y)\!</math>
|-
|-
| <math>f_{2}\!</math>
| <math>f_{2}\!</math>
| <math>(x) y\!</math>
| <math>(x) y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>\mathrm{d}x (\mathrm{d}y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>\mathrm{d}x\ \mathrm{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}x)(\mathrm{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\mathrm{d}x) \mathrm{d}y\!</math>
|-
|-
| <math>f_{4}\!</math>
| <math>f_{4}\!</math>
| <math>x (y)\!</math>
| <math>x (y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\mathrm{d}x) \mathrm{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}x)(\mathrm{d}y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>\mathrm{d}x\ \mathrm{d}y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>\mathrm{d}x (\mathrm{d}y)\!</math>
|-
|-
| <math>f_{8}\!</math>
| <math>f_{8}\!</math>
| <math>x y\!</math>
| <math>x y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}x)(\mathrm{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\mathrm{d}x) \mathrm{d}y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>\mathrm{d}x (\mathrm{d}y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>\mathrm{d}x\ \mathrm{d}y\!</math>
|-
|-
| <math>f_{3}\!</math>
| <math>f_{3}\!</math>
| <math>(x)\!</math>
| <math>(x)\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\mathrm{d}x\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\mathrm{d}x\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\mathrm{d}x)\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\mathrm{d}x)\!</math>
|-
|-
| <math>f_{12}\!</math>
| <math>f_{12}\!</math>
| <math>x\!</math>
| <math>x\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\mathrm{d}x)\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\mathrm{d}x)\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\mathrm{d}x\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\mathrm{d}x\!</math>
|-
|-
| <math>f_{6}\!</math>
| <math>f_{6}\!</math>
| <math>(x, y)\!</math>
| <math>(x, y)\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x, \mathrm{d}y)\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>((\mathrm{d}x, \mathrm{d}y))\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>((\mathrm{d}x, \mathrm{d}y))\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x, \mathrm{d}y)\!</math>
|-
|-
| <math>f_{9}\!</math>
| <math>f_{9}\!</math>
| <math>((x, y))\!</math>
| <math>((x, y))\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>((\mathrm{d}x, \mathrm{d}y))\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x, \mathrm{d}y)\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x, \mathrm{d}y)\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>((\mathrm{d}x, \mathrm{d}y))\!</math>
|-
|-
| <math>f_{5}\!</math>
| <math>f_{5}\!</math>
| <math>(y)\!</math>
| <math>(y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>\mathrm{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>\mathrm{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}y)\!</math>
|-
|-
| <math>f_{10}\!</math>
| <math>f_{10}\!</math>
| <math>y\!</math>
| <math>y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>\mathrm{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>(\mathrm{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>\mathrm{d}y\!</math>
|-
|-
| <math>f_{7}\!</math>
| <math>f_{7}\!</math>
| <math>(x y)\!</math>
| <math>(x y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>((\mathrm{d}x)(\mathrm{d}y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\mathrm{d}x) \mathrm{d}y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>(\mathrm{d}x (\mathrm{d}y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x\ \mathrm{d}y)\!</math>
|-
|-
| <math>f_{11}\!</math>
| <math>f_{11}\!</math>
| <math>(x (y))\!</math>
| <math>(x (y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\mathrm{d}x) \mathrm{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>((\mathrm{d}x)(\mathrm{d}y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x\ \mathrm{d}y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>(\mathrm{d}x (\mathrm{d}y))\!</math>
|-
|-
| <math>f_{13}\!</math>
| <math>f_{13}\!</math>
| <math>((x) y)\!</math>
| <math>((x) y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>(\mathrm{d}x (\mathrm{d}y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x\ \mathrm{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>((\mathrm{d}x)(\mathrm{d}y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\mathrm{d}x) \mathrm{d}y)\!</math>
|-
|-
| <math>f_{14}\!</math>
| <math>f_{14}\!</math>
| <math>((x)(y))\!</math>
| <math>((x)(y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>(\mathrm{d}x\ \mathrm{d}y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>(\mathrm{d}x (\mathrm{d}y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\mathrm{d}x) \mathrm{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>((\mathrm{d}x)(\mathrm{d}y))\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in <math>[\mathcal{A}]</math> may change or move with respect to the features that are noted in the initial alphabet.
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in <math>[\mathcal{A}]</math> may change or move with respect to the features that are noted in the initial alphabet.
Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\operatorname{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> (For all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\operatorname{d}\mathcal{A}.</math>)
Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A} = \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> (For all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.</math>)
In the above terms, a typical tangent space of <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\operatorname{d}A = \langle \operatorname{d}\mathcal{A} \rangle = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
In the above terms, a typical tangent space of <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\mathrm{d}A = \langle \mathrm{d}\mathcal{A} \rangle = \langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
Proceeding as we did before with the base space <math>A,\!</math> we can analyze the individual tangent space at a point of <math>A\!</math> as a product of distinct and independent factors:
Proceeding as we did before with the base space <math>A,\!</math> we can analyze the individual tangent space at a point of <math>A\!</math> as a product of distinct and independent factors:
: <math>\operatorname{d}A = \prod_{i=1}^n \operatorname{d}A_i = \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>
: <math>\mathrm{d}A = \prod_{i=1}^n \mathrm{d}A_i = \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.</math>
Here, <math>\operatorname{d}\mathcal{A}_i</math> is an alphabet of two symbols, <math>\operatorname{d}\mathcal{A}_i = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \},</math> where <math>\overline{\operatorname{d}a_i}</math> is a symbol with the logical value of <math>\operatorname{not}\ \operatorname{d}a_i.</math> Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} \cong \{ 0, 1 \}.</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D}, </math> whose intension may be indicated as follows:
Here, <math>\mathrm{d}\mathcal{A}_i</math> is an alphabet of two symbols, <math>\mathrm{d}\mathcal{A}_i = \{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \},</math> where <math>\overline{\mathrm{d}a_i}</math> is a symbol with the logical value of <math>\mathrm{not}\ \mathrm{d}a_i.</math> Each component <math>\mathrm{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D}, </math> whose intension may be indicated as follows:
: <math>\mathbb{D} = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} = \{ \operatorname{same}, \operatorname{different} \} = \{ \operatorname{stay}, \operatorname{change} \} = \{ \operatorname{stop}, \operatorname{step} \}.</math>
: <math>\mathbb{D} = \{ \overline{\mathrm{d}a_i}, \mathrm{d}a_i \} = \{ \mathrm{same}, \mathrm{different} \} = \{ \mathrm{stay}, \mathrm{change} \} = \{ \mathrm{stop}, \mathrm{step} \}.</math>
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
===Extended Universe of Discourse===
===Extended Universe of Discourse===
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as:
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as:
: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math>
: <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A} = \{a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.</math>
This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:
:{| cellpadding=2
:{| cellpadding=2
| <math>\operatorname{E}A</math>
| <math>\mathrm{E}A</math>
| =
| =
| <math>A \times \operatorname{d}A</math>
| <math>A \times \mathrm{d}A</math>
|-
|-
|
|
| =
| =
| <math>\langle \operatorname{E}\mathcal{A} \rangle</math>
| <math>\langle \mathrm{E}\mathcal{A} \rangle</math>
|-
|-
|
|
| =
| =
| <math>\langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle</math>
| <math>\langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle</math>
|-
|-
|
|
| =
| =
| <math>\langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math>
| <math>\langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,</math>
|}
|}
thus giving <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
thus giving <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
Finally, the tangent universe <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\operatorname{E}\mathcal{A}:</math>
Finally, the tangent universe <math>\mathrm{E}A^\circ = [\mathrm{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A}:</math>
: <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}] = [a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n],</math>
: <math>\mathrm{E}A^\circ = [\mathrm{E}\mathcal{A}] = [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n],</math>
thus giving the tangent universe <math>\operatorname{E}A^\circ</math> the type:
thus giving the tangent universe <math>\mathrm{E}A^\circ</math> the type:
: <math>(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).</math>
: <math>(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).</math>
A proposition in the tangent universe <math>[\operatorname{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
With these constructions, to be specific, the differential extension <math>\operatorname{E}A</math> and the differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).
With these constructions, to be specific, the differential extension <math>\mathrm{E}A</math> and the differential proposition <math>f : \mathrm{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).
==Orbit Table Template==
==Orbit Table Template==
|
|
| <math>f\!</math>
| <math>f\!</math>
| <math>\operatorname{F}f|_{xy}</math>
| <math>\mathrm{F}f|_{xy}</math>
| <math>\operatorname{F}f|_{x(y)}</math>
| <math>\mathrm{F}f|_{x(y)}</math>
| <math>\operatorname{F}f|_{(x)y}</math>
| <math>\mathrm{F}f|_{(x)y}</math>
| <math>\operatorname{F}f|_{(x)(y)}</math>
| <math>\mathrm{F}f|_{(x)(y)}</math>
|- style="height:36px"
|- style="height:36px"
| <math>f_{0}\!</math>
| <math>f_{0}\!</math>
===Tangent Operator===
===Tangent Operator===
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