[CG] Re: Differential Logic

Differential Logic • 6 • https://inquiryintoinquiry.com/2024/11/05/differential-logic-6-a/ Differential Expansions of Propositions — Panoptic View • Difference Maps — In the previous post we computed what is variously described as the “difference map”, the “difference proposition”, or the “local proposition” Dfₓ of the proposition f(p, q) = pq at the point x where p = 1 and q = 1. In the universe of discourse X = P × Q the four propositions pq, p(q), (p)q, (p)(q) can be taken to indicate the so‑called “cells” or smallest distinguished regions of the universe, otherwise indicated by their coordinates as the “points” (1,1), (1,0), (0,1), (0,0), respectively. In that regard the four propositions are called “singular propositions” because they serve to single out the minimal regions of the universe of discourse. Thus we can write Dfₓ = Df|_x = Df|_(1, 1) = Df|_pq, so long as we know the frame of reference in force. In the example f(p,q) = pq, the value of the difference proposition Dfₓ at each of the four points x in X may be computed in graphical fashion as shown below. Cactus Graph Df = ((p,dp)(q,dq),pq) • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpq… Cactus Graph Difference pq @ pq = ((dp)(dq)) • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-differe… Cactus Graph Difference pq @ p(q) = (dp)dq • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-differe… Cactus Graph Difference pq @ (p)q = dp(dq) • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-differe… Cactus Graph Difference pq @ (p)(q) = dp dq • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-differe… The easy way to visualize the values of the above graphical expressions is just to notice the following graphical equations. Cactus Graph Lobe Rule • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe… Cactus Graph Spike Rule • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-spik… Adding the arrows to the venn diagram gives us the picture of a “differential vector field”. Venn Diagram Difference pq • https://inquiryintoinquiry.files.wordpress.com/2020/04/venn-diagram-differe… The Figure shows the points of the extended universe EX = P×Q × dP×dQ indicated by the difference map Df : EX→B, namely, the following six points or singular propositions. 1. p q dp dq 2. p q dp (dq) 3. p q (dp) dq 4. p (q) (dp) dq 5. (p) q dp (dq) 6. (p)(q) dp dq The information borne by Df should be clear enough from a survey of these six points — they tell you what you have to do from each point of X in order to change the value borne by f(p, q), that is, the move you have to make in order to reach a point where the value of the proposition f(p, q) is different from what it is where you started. We have been studying the action of the difference operator D on propositions of the form f : P×Q→B, as illustrated by the example f(p,q) = pq = the conjunction of p and q. The resulting difference map Df is a “(first order) differential proposition”, that is, a proposition of the form Df : P×Q × dP×dQ → B. The augmented venn diagram shows how the “models” or “satisfying interpretations” of Df distribute over the extended universe of discourse EX = P×Q × dP×dQ. Abstracting from that picture, the difference map Df can be represented in the form of a “digraph” or “directed graph”, one whose points are labeled with the elements of X = P×Q and whose arrows are labeled with the elements of dX = dP×dQ, as shown in the following Figure. Directed Graph Difference pq • https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-diffe… The same 6 points of the extended universe EX = P×Q × dP×dQ given by the difference map Df : EX→B can be described by the following formula. Df = p q · ((dp)(dq)) + p (q) · (dp) dq + (p) q · dp (dq) + (p)(q) · dp dq Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal previously unsuspected aspects of the proposition's meaning. We will encounter more and more such alternative readings as we go. 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/lOQPpB cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…