[CG] Re: Differential Logic

Differential Logic • 15 • https://inquiryintoinquiry.com/2024/11/18/differential-logic-15-a/ Differential Fields — The structure of a differential field may be described as follows. With each point of X there is associated an object of the following type: a proposition about changes in X, that is, a proposition g : dX → B. In that frame of reference, if Xº is the universe generated by the set of coordinate propositions {p, q} then dXº is the differential universe generated by the set of differential propositions {dp, dq}. The differential propositions dp and dq may thus be interpreted as indicating “change in p” and “change in q”, respectively. A differential operator W, of the first order type we are currently considering, takes a proposition f : X → B and gives back a differential proposition Wf : EX → B. In the field view of the scene, we see the proposition f : X → B as a scalar field and we see the differential proposition Wf : EX → B as a vector field, specifically, a field of propositions about contemplated changes in X. The field of changes produced by E on pq is shown in the following venn diagram. Enlargement E(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-enl… E(pq) = p ∙ q ∙ (dp)(dq) + p ∙ (q) ∙ (dp) dq + (p) ∙ q ∙ dp (dq) + (p) ∙ (q) ∙ dp dq The differential field E(pq) specifies the changes which need to be made from each point of X in order to reach one of the models of the proposition pq, that is, in order to satisfy the proposition pq. The field of changes produced by D on pq is shown in the following venn diagram. Differential D(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-dif… D(pq) = p ∙ q ∙ ((dp)(dq)) + p ∙ (q) ∙ (dp) dq + (p) ∙ q ∙ dp (dq) + (p) ∙ (q) ∙ dp dq The differential field D(pq) specifies the changes which need to be made from each point of X in order to feel a change in the felt value of the field pq. 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/LGv6Q0 cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…