Changes

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, X, to considering a larger universe of discourse, EX.

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>

Each of these operators, in general terms having the form F : X -> EX, acts on each proposition p : X -> B of the source universe X to produce a proposition Fp : EX -> B of the target universe EX.

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>

The two main operators that we have worked with up to this point are the enlargement operator E : X -> EX and the difference operator D : X -> EX.

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>

E and D take a proposition in X, that is, a proposition p : X -> B that is said to be "about" the subject matter of X, and produce the extended propositions Ep, Dp : EX -> B, which may be interpreted as being about specified collections of changes that might occur in X.

<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>

Here we have need of visual representations, some array of concrete pictures to anchor our

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.

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.

One good picture comes to us by way of the "field" concept.  Given a space X, a "field" of a specified type T over X is formed by assigning to each point of X an object of type T.  If that sounds like the same thing as a function from X to the space of things of type T, it is, but it does seems to help to vary the mental pictures and the figures of speech that naturally spring to mind within these fertile fields.

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>\mathcal{T}\!</math> over <math>X\!</math> is formed by assigning to each point of <math>X\!</math> an object of type <math>\mathcal{T}.\!</math> If that sounds like the same thing as a function from <math>X\!</math> to the space of things of type <math>\mathcal{T},\!</math> it is, but it does seems to help to vary the mental pictures and the figures of speech that naturally spring to mind within these fertile fields.

In the field picture, a proposition p : X -> B becomes a "scalar" field, that is, a field of values in B, or a "field of true-false indications".

In the field picture a proposition <math>p : X \to \mathbb{B}\!</math> becomes a ''scalar field'', that is, a field of values in <math>\mathbb{B},\!</math> or a ''field of true-false indications''.

Let us take a moment to view an old proposition in this new light, for example, the conjunction

Let us take a moment to view an old proposition in this new light, for example, the conjunction <math>uv : X \to \mathbb{B}\!</math> that is depicted in Figure&nbsp;1.

uv : X -> B that is depicted in Figure 1.

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