* **Propositions and the Leibniz rule**
**@ 2022-09-09 5:50 Posina Venkata Rayudu**
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From: Posina Venkata Rayudu @ 2022-09-09 5:50 UTC (permalink / raw)
To: categories
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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