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==Note 22==
==Note 22==
Let us summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far. We've been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, <math>X^\circ,</math> to considering a larger universe of discourse, <math>\operatorname{E}X^\circ.</math> An operator <math>\operatorname{W}</math> of this general type, namely,<math>\operatorname{W} : X^\circ \to \operatorname{E}X^\circ,</math> acts on each proposition <math>f : X \to \mathbb{B}</math> of the source universe <math>X^\circ</math> to produce a proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}</math> of the target universe <math>\operatorname{E}X^\circ.</math>
the outlook on differential logic that we have reached so far.
The two main operators that we've examined so far are the enlargement or shift operator <math>\operatorname{E} : X^\circ \to \operatorname{E}X^\circ</math> and the difference operator <math>\operatorname{D} : X^\circ \to \operatorname{E}X^\circ.</math> The operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> act on propositions in <math>X^\circ,</math> that is, propositions of the form <math>f : X \to \mathbb{B}</math> that are said to be ''about'' the subject matter of <math>X,\!</math> and they produce extended propositions of the forms <math>\operatorname{E}f, \operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in <math>X.\!</math>
of discourse, each of which takes us from considering one
of discourse, EX%.
At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us as we venture higher into the ever more rarefied air of abstractions.
of the target universe EX%.
One good picture comes to us by way of the ''field'' concept. Given a space <math>X,\!</math> a ''field'' of a specified type <math>Y\!</math> over <math>X\!</math> is formed by associating with each point of <math>X\!</math> an object of type <math>Y.\!</math> If that sounds like the same thing as a function from <math>X\!</math> to the space of things of type <math>Y\!</math> — it is nothing but — and yet it does seem helpful to vary the mental images and to take advantage of the figures of speech that spring to mind under the emblem of this field idea.
point are the enlargement or shift operator E : X% -> EX%
and the difference operator D : X% -> EX%.
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>
that is said to be "about" the subject matter of X, and produce the
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"
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
| <math>\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}</math>
|}
Each of the operators <math>\operatorname{E}, \operatorname{D} : X^\circ \to \operatorname{E}X^\circ</math> takes us from considering propositions <math>f : 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'' over <math>X.\!</math>
The structure of these differential fields can be described this way. With each point of <math>X\!</math> there is associated 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 frame of reference, if <math>X^\circ</math> is the universe that is generated by the set of coordinate propositions <math>\{ p, q \},\!</math> then <math>\operatorname{d}X^\circ</math> is the differential universe that is generated by the set of differential propositions <math>\{ \operatorname{d}p, \operatorname{d}q \}.</math> These differential propositions may be interpreted as indicating <math>{}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},</math> respectively.
in this new light, for example, the conjunction
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 22-b.
{| align="center" cellspacing="10" style="text-align:center"
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
|-
| <math>\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
|
<math>\begin{array}{rcccccc}
\operatorname{E}(pq)
& = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q)
\\[4pt]
& + & p & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~\end{array}</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 22-c.
{| align="center" cellspacing="10" style="text-align:center"
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
|-
| <math>\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
|
<math>\begin{array}{rcccccc}
\operatorname{D}(pq)
& = & p & \cdot & q & \cdot & ((\operatorname{d}p)(\operatorname{d}q))
\\[4pt]
& + & p & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~
\end{array}</math>
|}
The differential field <math>\operatorname{D}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to feel a change in the felt value of the field <math>pq.\!</math>
that need to be made from each point of X in order
to feel a change in the felt value of the field pq.
</pre>
==Note 23==
==Note 23==
