[CG] Re: Differential Logic

Differential Logic • 18 • https://inquiryintoinquiry.com/2024/11/23/differential-logic-18-a/ Tangent and Remainder Maps — If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition f = pq : X → B in the following way. The next venn diagram shows the differential proposition df = d(pq) : EX → B we get by extracting the linear approximation to the difference map Df = D(pq) : EX → B at each cell or point of the universe X. What results is the logical analogue of what would ordinarily be called “the differential” of pq but since the adjective “differential” is being attached to just about everything in sight the alternative name “tangent map” is commonly used for df whenever it's necessary to single it out. Tangent Map d(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-dif… To be clear about what's being indicated here, it's a visual way of summarizing the following data. d(pq) = p ∙ q ∙ (dp , dq) + p ∙ (q) ∙ dq + (p) ∙ q ∙ dp + (p) ∙ (q) ∙ 0 To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences. • (dp , dq) = dp ∙ (dq) + (dp) ∙ dq • dp = dp ∙ dq + dp ∙ (dq) • dq = dp ∙ dq + (dp) ∙ dq Capping the analysis of the proposition pq in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the “remainder map” r(pq) : EX → B, which happens to be linear in pairs of variables. Remainder r(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-rem… Reading the arrows off the map produces the following data. r(pq) = p ∙ q ∙ dp ∙ dq + p ∙ (q) ∙ dp ∙ dq + (p) ∙ q ∙ dp ∙ dq + (p) ∙ (q) ∙ dp ∙ dq In short, r(pq) is a constant field, having the value dp ∙ dq at each cell. 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/lypmDv cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…