From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10797 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Posina Venkata Rayudu Newsgroups: gmane.science.mathematics.categories Subject: Propositions and the Leibniz rule Date: Fri, 9 Sep 2022 11:20:42 +0530 Message-ID: Reply-To: Posina Venkata Rayudu Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="26762"; mail-complaints-to="usenet@ciao.gmane.io" To: categories Original-X-From: majordomo@rr.mta.ca Sun Sep 11 02:25:18 2022 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.75]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1oXAmk-0006o8-Be for gsmc-categories@m.gmane-mx.org; Sun, 11 Sep 2022 02:25:18 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:59486) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1oXAlY-0001ls-CY; Sat, 10 Sep 2022 21:24:04 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1oXAl7-0004p3-KY for categories-list@rr.mta.ca; Sat, 10 Sep 2022 21:23:37 -0300 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10797 Archived-At: Dear All, I hope and pray you and your families are all well. If I may, let's define negation of a part A (of a whole W) as the smallest part not'(A) of W, whose union with A is: A v not'(A) = W (Lawvere and Rosebrugh (2003) Sets for Mathematics, p. 201). (This definition of negation is dual to the usual definition of negation of a part A (of a whole W) as the largest part not(A) of W, whose intersection with A is empty. In the category of sets, not'(A) = not(A).) Next, define boundary b(A) of A as: b(A) = A ^ not'(A) The topological notion of boundary A and not'(A) corresponds to contradictions in logical terms. What I find interesting is that the boundary b(A ^ B) of the product (A ^ B) of two objects A and B is given by the Leibniz rule: b(A ^ B) = (b(A) ^ B) v (A ^ b(B)) which seems to hold in the case of rectangle-shaped planes (A ^ B), with A and B as line segments and b(A) and b(B) as pairs of endpoints of the line segments, in the sense we get the rectangle-shape b(A ^ B) as the union of two pairs of parallel line segments (b(A) ^ B) and (A ^ b(B)). I am wondering about the meaning of Leibniz's rule, with A and B as propositions; and boundaries b(A) and b(B) as contradictions (and/or concepts, construed as domain/codomain objects of arrows denoting prepositions?). Your time permitting, please correct any mistakes I have made in the above. I look forward to your corrections and clarifications. thanking you, yours truly, posina https://www.reddit.com/r/ConceptualMathematics/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]