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/ ]