[CG] Re: Differential Logic

Differential Logic • 14 • https://inquiryintoinquiry.com/2024/11/16/differential-logic-14-a/ Field Picture — Let us summarize the outlook on differential logic we've 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 Xº to considering a larger universe of discourse EXº. An operator W of that general type, namely, W : Xº → EXº, acts on each proposition f : X → B of the source universe Xº to produce a proposition Wf : EX → B of the target universe EXº. The operators we've examined so far are the enlargement or shift operator E : Xº → EXº and the difference operator D : Xº → EXº. The operators E and D act on propositions in Xº, that is, propositions of the form f : X → B which may be taken as being statements “about” the subject matter of X, and they produce propositions of the forms Ef, Df : EX → B which may be taken as being statements “about” specified collections of changes conceivably occurring in X. At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction. One good picture comes to us by way of the “field” concept. Given a space X, a “field” of a specified type Y over X is formed by associating with each point of X an object of type Y. If that sounds like the same thing as a function from X to the space of things of type Y — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea. In the field picture a proposition f : X → B becomes a “scalar field”, that is, a field of values in B. For example, consider the logical conjunction pq : X → B shown in the following venn diagram. Conjunction pq : X → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-con… Each of the operators E, D : Xº → EXº takes us from considering propositions f : X → B, here viewed as “scalar fields” over X, to considering the corresponding “differential fields” over X, analogous to what in real analysis are usually called “vector fields” over X. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/ Regards, Jon cc: https://www.academia.edu/community/5k9wGN cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…