[CG] Re: Differential Logic

Differential Logic • 16 • https://inquiryintoinquiry.com/2024/11/19/differential-logic-16-a/ Propositions and Tacit Extensions — Now that we've introduced the field picture as an aid to visualizing propositions and their analytic series, a pleasing way to picture the relationship of a proposition f : X → B to its enlargement or shift map Ef : EX → B and its difference map Df : EX → B can now be drawn. To illustrate the possibilities, let's return to the differential analysis of the conjunctive proposition f(p, q) = pq and give its development a slightly different twist at the appropriate point. The proposition pq : X → B is shown again in the venn diagram below. In the field picture it may be seen as a scalar field — analogous to a “potential hill” in physics but in logic amounting to a “potential plateau” — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0. Proposition pq : X → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-con… Given a proposition f : X → B, the “tacit extension” of f to EX is denoted εf : EX → B and defined by the equation εf = f, so it's really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a given set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a “don't care” condition on the new variables. The tacit extension of the scalar field pq : X → B to the differential field ε(pq) : EX → B is shown in the following venn diagram. Tacit Extension ε(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-tac… ε(pq) = p ∙ q ∙ (dp)(dq) + p ∙ q ∙ (dp) dq + p ∙ q ∙ dp (dq) + p ∙ q ∙ dp dq 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/V00dPp cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…