Equality as Identity: Convergent; Divergent; Distinction; Recursion
1. B is different than C.
2. B contains a distinction within it that C contains, this distinction is A.
3. B equals C through the distinction A as A is equal to itself across the contexts of B and C which contain it.
4. Equality is one degree the equivocation of distinction across context, this equivocation is repetition.
5. The repetition of distinction A, across contexts B and C reveals A as having a different identity to itself by degree of the contexts which contain it.
6. In these respects A does not equal A as the identity of A is the context by which it occurs.
7. The differing contexts change into eachother by A, thus B -> C and C -> B; B and C are thus biconditional B<->C.
8. The differing of A results in the difference of the context B and C as
A -> (B,C).
9. A as a single distinction is equal to itself, its repetition is what allows different contexts to equate while dually the negation of this equivocation of A itself.
10. What remains is equality as an act of distinction, distinction equaling distinction is distinction through distinction as a distinction, distinction not equal to distinction is distinct through distinction as distinction.
11. Identity through equality is identity as process where convergence and divergence happen simultaneously thus leaving only the distinct identity of a thing.
12. What remains is (A -> A) -> (A, B, C) thus identity is recursion thus C is self-scaling by means of the variables which emerge and dissolve from it;
13. where each variable is but a scale of A and each scale becomes a variable itself thus is subject to scaling:
(B->B) -> (B,D,F)
(D = F) <-> (B -> D, B -> F)
(C->C) -> (C,F,I)
(F = I) <-> (C -> F, C -> I)
14. Now the recursion of the variable upon its scales results in a deeper degree of scaling;
(A -> B) -> (A,B,C,D)
(B -> C) -> (B,C,E,F)
15. where multiple distinctions exist across contexts of scale thus resulting in a higher order of equality:
(A -> B) -> (A,B,C,D)
(C = D) <-> ((A,B) -> (C,D))
(B -> C) -> (B,C,E,F)
(E = F) <-> ((B,C) -> (E,F))
(A -> B -> C) -> (A,B,C,D,E,F)
(D=E=F) <-> ((A,B,C) -> (D,E,F)
16. As foundational distinctions increase so do the following emergent identities increase in equivocation.