Equality as Identity: Convergent; Divergent; Distinction; Recursion
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B is different than C.
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B contains a distinction within it that C contains, this distinction is A.
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B equals C through the distinction A as A is equal to itself across the contexts of B and C which contain it.
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Equality is one degree the equivocation of distinction across context, this equivocation is repetition.
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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.
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In these respects A does not equal A as the identity of A is the context by which it occurs.
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The differing contexts change into eachother by A, thus B → C and C → B; B and C are thus biconditional B<->C.
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The differing of A results in the difference of the context B and C as
A → (B,C).
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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.
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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.
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Identity through equality is identity as process where convergence and divergence happen simultaneously thus leaving only the distinct identity of a thing.
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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;
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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)
- 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)
- 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)
- As foundational distinctions increase so do the following emergent identities increase in equivocation.