Changes

==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>

It would be good to 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

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>

of discourse, each of which takes us from considering one

universe of discourse, X%, to considering a larger universe

of discourse, EX%.

The two main operators that we have worked with up to this point 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 two main operators that we have worked with up to this

point are the enlargement or shift operator E : X% -> EX%

and the difference operator D : X% -> EX%.

E and D take a proposition in X%, that is, a proposition f : X -> B

E and D take a proposition in X%, that is, a proposition f : X -> B

that is said to be "about" the subject matter of X, and produce the

that is said to be "about" the subject matter of X, and produce the