Propositions and the Leibniz rule

[Posina Venkata Rayudu <posinavrayudu@gmail.com>]

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

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