Changes

In the field picture a proposition <math>f : X \to \mathbb{B}</math> becomes a ''scalar field'', that is, a field of values in <math>\mathbb{B}.</math>

In the field picture a proposition <math>f : X \to \mathbb{B}</math> becomes a ''scalar field'', that is, a field of values in <math>\mathbb{B}.</math>

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&nbsp;22-a.

For example, consider the logical conjunction <math>pq : X \to \mathbb{B}</math> that is shown in the following venn diagram:

{| align="center" cellpadding="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]]

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| <math>\text{Figure 22-a.  Conjunction}~ pq : X \to \mathbb{B}</math>

| <math>\text{Conjunction}~ pq : X \to \mathbb{B}</math>

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A differential operator <math>\operatorname{W},</math> of the first order class that we have been considering, takes a proposition <math>f : X \to \mathbb{B}</math> and gives back a differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}.</math>  In the field view, we see the proposition <math>f : X \to \mathbb{B}</math> as a scalar field and we see the differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>

A differential operator <math>\operatorname{W},</math> of the first order class that we have been considering, takes a proposition <math>f : X \to \mathbb{B}</math> and gives back a differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}.</math>  In the field view, we see the proposition <math>f : X \to \mathbb{B}</math> as a scalar field and we see the differential proposition <math>\operatorname{W}f : \operatorname{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>pq\!</math> is shown in Figure&nbsp;22-b.

The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in the next venn diagram:

{| align="center" cellpadding="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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| <math>\text{Figure 22-b.  Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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The differential field <math>\operatorname{E}(pq)</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>pq,\!</math> that is, in order to satisfy the proposition <math>pq.\!</math>

The differential field <math>\operatorname{E}(pq)</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>pq,\!</math> that is, in order to satisfy the proposition <math>pq.\!</math>

The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure&nbsp;22-c.

The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in the following venn diagram:

{| align="center" cellpadding="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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| <math>\text{Figure 22-c.  Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.

To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.

Figure&nbsp;24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field &mdash; analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' &mdash; where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.

The next venn diagram shows once again the proposition <math>pq,\!</math> which we now view as a scalar field &mdash; analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' &mdash; where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.

{| align="center" cellpadding="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]]

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| <math>\text{Figure 24-1.  Proposition}~ pq : X \to \mathbb{B}</math>

| <math>\text{Proposition}~ pq : X \to \mathbb{B}</math>

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Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is denoted <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universe.  Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is denoted <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universe.  Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

Figure&nbsp;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>

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> is shown in the following venn diagram:

{| align="center" cellpadding="10" style="text-align:center"

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]

| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]

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| <math>\text{Figure 24-2.  Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}</math>

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===Enlargement and Difference Maps===

===Enlargement and Difference Maps===

Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure&nbsp;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> the next venn digram 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" cellpadding="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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| <math>\text{Figure 25-1.  Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\operatorname{E}f,</math> both of the type <math>\operatorname{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\operatorname{E}f,</math> both of the type <math>\operatorname{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.

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&nbsp;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 below:

{| align="center" cellpadding="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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| <math>\text{Figure 25-2.  Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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If we follow the classical line that singles out linear functions as ideals of simplicity, then we may complete the analytic series of the proposition <math>f = pq : X \to \mathbb{B}</math> in the following way.

If we follow the classical line that singles out linear functions as ideals of simplicity, then we may complete the analytic series of the proposition <math>f = pq : X \to \mathbb{B}</math> in the following way.

Figure&nbsp;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>

The next venn diagram 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" cellpadding="10" style="text-align:center"

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]

| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]

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| <math>\text{Figure 26-1.  Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure&nbsp;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, the final venn diagram in this series 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" cellpadding="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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| <math>\text{Figure 26-2.  Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}</math>

| <math>\text{Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}</math>

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