[CG] Re: Differential Logic

Differential Logic • 7 • https://inquiryintoinquiry.com/2024/11/06/differential-logic-7-a/ Differential Expansions of Propositions — Panoptic View • Enlargement Maps — The “enlargement” or “shift” operator E exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features playing out on the surface of our initial example, f(p, q) = pq. A suitably generic definition of the extended universe of discourse is afforded by the following set‑up. • Let X = X₁ × … × Xₖ. • Let dX = dX₁ × … × dXₖ. • Then EX = X × dX = X₁ × … × Xₖ × dX₁ × … × dXₖ For a proposition of the form f : X₁ × … × Xₖ → B, the “(first order) enlargement” of f is the proposition Ef : EX→B defined by the following equation. • Ef(x₁, …, xₖ, dx₁, …, dxₖ) = f(x₁ + dx₁, …, xₖ + dxₖ) = f(x₁ xor dx₁, …, xₖ xor dxₖ) The “differential variables” dx_j are boolean variables of the same type as the ordinary variables x_j. Although it is conventional to distinguish the (first order) differential variables with the operational prefix “d”, that way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables. In the example of logical conjunction, f(p, q) = pq, the enlargement Ef is formulated as follows. • Ef(p, q, dp, dq) = (p + dp)(q + dq) = (p xor dp)(q xor dq) Given that the above expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to “multiply things out” in the usual manner to arrive at the following result. • Ef(p, q, dp, dq) = p·q + p·dq + q·dp + dp·dq To understand what the “enlarged” or “shifted” proposition means in logical terms, it serves to go back and analyze the above expression for Ef in the same way we did for Df. To that end, the value of Efₓ at each x in X may be computed in graphical fashion as shown below. Cactus Graph Ef = (p,dp)(q,dq) • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpq… Cactus Graph Enlargement pq @ pq = (dp)(dq) • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Cactus Graph Enlargement pq @ p(q) = (dp)dq • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Cactus Graph Enlargement pq @ (p)q = dp(dq) • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Cactus Graph Enlargement pq @ (p)(q) = dp dq • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Collating the data of that analysis yields a boolean expansion or “disjunctive normal form” (DNF) equivalent to the enlarged proposition Ef. • Ef = p q · Ef| p q + p (q) · Ef| p (q) + (p) q · Ef|(p) q} + (p)(q) · Ef|{p)(q) Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element (dp)(dq) is drawn as a loop at the point pq. Directed Graph Enlargement pq • https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-enlar… Logical Formula Enlargement pq • https://inquiryintoinquiry.files.wordpress.com/2024/11/logical-formula-enla… We may understand the enlarged proposition Ef as telling us all the ways of reaching a model of the proposition f from the points of the universe 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/5NAZ9q cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…