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==Quick Review==

==Quick Review : Field Picture==

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>

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 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>

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>

==Tacit Extension==

===Proposition and Tacit Extension===

Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B},</math> and its difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B}</math> can now be drawn.

Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B},</math> and its difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B}</math> can now be drawn.

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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> 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>

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==Tangent and Remainder Maps==

===Tangent and Remainder Maps===

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.