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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
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Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
|}
|}
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential True.jpg|500px]]
| [[Image:Cactus Graph Existential True.jpg|500px]]
|}
|}
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential False.jpg|500px]]
| [[Image:Cactus Graph Existential False.jpg|500px]]
|}
|}
Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Venn Diagram P And Q.jpg|500px]]
| [[Image:Venn Diagram P And Q.jpg|500px]]
|}
|}
Don't think about it — just compute:
Don't think about it — just compute:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
|}
|}
The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (P,dP).jpg|500px]]
| [[Image:Cactus Graph (P,dP).jpg|500px]]
|}
|}
proposition <math>pq\!</math> over there, at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
proposition <math>pq\!</math> over there, at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
|}
|}
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>pq\!</math> is true? Well, substituting <math>1\!</math> for <math>p\!</math> and <math>1\!</math> for <math>q\!</math> in the graph amounts to erasing the labels <math>p\!</math> and <math>q\!,</math> as shown here:
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>pq\!</math> is true? Well, substituting <math>1\!</math> for <math>p\!</math> and <math>1\!</math> for <math>q\!</math> in the graph amounts to erasing the labels <math>p\!</math> and <math>q\!,</math> as shown here:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
|}
|}
And this is equivalent to the following graph:
And this is equivalent to the following graph:
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]
| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]
|}
|}
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
{| align="center" cellspacing="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
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<math>\begin{matrix}
<math>\begin{matrix}
|}
|}
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph PQ Diff ((dP)(dQ)).jpg|500px]]
| [[Image:Cactus Graph PQ Diff ((dP)(dQ)).jpg|500px]]
|}
|}
We begin with a proposition or a boolean function <math>f(p, q) = pq.\!</math>
We begin with a proposition or a boolean function <math>f(p, q) = pq.\!</math>
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Venn Diagram F = P And Q.jpg|500px]]
| [[Image:Venn Diagram F = P And Q.jpg|500px]]
|-
|-
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.
{| align="center" cellspacing="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>
| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>
|-
|-
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
|-
|-
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellpadding="10" width="90%"
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<math>\begin{array}{rcc}
<math>\begin{array}{rcc}
where:
where:
{| align="center" cellspacing="10" width="90%"
{| align="center" cellpadding="10" width="90%"
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<math>\begin{array}{rcc}
<math>\begin{array}{rcc}
Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.
Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.
{| align="center" cellspacing="10"
{| align="center" cellpadding="10"
| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]
| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]
|}
|}
following formula:
following formula:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
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<math>\begin{matrix}
<math>\begin{matrix}
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
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<math>\begin{matrix}
<math>\begin{matrix}
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
|
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<math>\begin{matrix}
<math>\begin{matrix}
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
|
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<math>\begin{matrix}
<math>\begin{matrix}
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>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
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>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
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<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
|
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<math>\begin{matrix}
<math>\begin{matrix}
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
{| align="center" cellspacing="20"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
|-
|-
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
{| align="center" cellspacing="20"
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Lobe Rule.jpg|500px]]
| [[Image:Cactus Graph Lobe Rule.jpg|500px]]
|-
|-
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
{| align="center" cellspacing="20"
{| align="center" cellpadding="10"
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
|}
|}
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
{| align="center" cellspacing="20"
{| align="center" cellpadding="10"
|
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<math>\begin{array}{rcccc}
<math>\begin{array}{rcccc}
Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute over the extended universe of discourse <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,</math> the difference map <math>\operatorname{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,</math> as shown in the following Figure.
Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute over the extended universe of discourse <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,</math> the difference map <math>\operatorname{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,</math> as shown in the following Figure.
{| align="center" cellspacing="20" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
|-
|-
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
{| align="center" cellspacing="20" width="90%"
{| align="center" cellpadding="10" width="90%"
|
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<math>\begin{array}{lccl}
<math>\begin{array}{lccl}
For a proposition of the form <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}X \to \mathbb{B}</math> that is defined by the following equation:
For a proposition of the form <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}X \to \mathbb{B}</math> that is defined by the following equation:
{| align="center" cellspacing="20" width="90%"
{| align="center" cellpadding="10" width="90%"
|
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<math>\begin{array}{l}
<math>\begin{array}{l}
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
{| align="center" cellspacing="20" width="90%"
{| align="center" cellpadding="10" width="90%"
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to "multiply things out" in the usual manner to arrive at the following result:
{| align="center" cellspacing="20" width="90%"
{| align="center" cellpadding="10" width="90%"
|
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<math>\begin{matrix}
<math>\begin{matrix}
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math> Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math> Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
{| align="center" cellspacing="20" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
|-
|-
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
{| align="center" cellspacing="20" width="90%"
{| align="center" cellpadding="10" width="90%"
|
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<math>\begin{matrix}
<math>\begin{matrix}
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
{| align="center" cellspacing="20" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
|-
|-
Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure 22-a.
Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure 22-a.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
|-
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure 22-b.
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure 22-b.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
|-
|-
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure 22-c.
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure 22-c.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
|-
|-
A generic enough picture at this stage of the game, and one that will remind us of these fundamental features of the cybernetic system even as things get far more complex, is afforded by Figure 23.
A generic enough picture at this stage of the game, and one that will remind us of these fundamental features of the cybernetic system even as things get far more complex, is afforded by Figure 23.
{| align="center" cellspacing="10" style="text-align:center; width:90%"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
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<pre>
<pre>
Figure 24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field — analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.
Figure 24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field — analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
|-
Figure 24-2 shows the tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
Figure 24-2 shows the tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
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| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
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Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure 25-1 shows the enlargement or shift map <math>\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure 25-1 shows the enlargement or shift map <math>\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
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In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other. The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true. The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure 25-2.
In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other. The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true. The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure 25-2.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
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Figure 26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\operatorname{d}f.\!</math>
Figure 26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\operatorname{d}f.\!</math>
{| align="center" cellspacing="10" style="text-align:center"
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| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
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Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
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<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}
To understand the extended interpretations, that is, the conjunctions of basic and differential features that are being indicated here, it may help to note the following equivalences:
To understand the extended interpretations, that is, the conjunctions of basic and differential features that are being indicated here, it may help to note the following equivalences:
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<math>\begin{matrix}
<math>\begin{matrix}
Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure 26-2 shows the remainder map <math>\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.
Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure 26-2 shows the remainder map <math>\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
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Reading the arrows off the map produces the following data:
Reading the arrows off the map produces the following data:
{| align="center" cellspacing="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
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<math>\begin{array}{rcccccc}
<math>\begin{array}{rcccccc}