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5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic

[logical assertions that that two objects are identical]

16 ideas
Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey]
The sign of identity is not allowed in 'Tractatus' [Bostock on Wittgenstein]
The identity sign is not essential in logical notation, if every sign has a different meaning [Wittgenstein]
Quantification theory can still be proved complete if we add identity [Quine]
Predicate logic has to spell out that its identity relation '=' is an equivalent relation [Sommers]
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
If we are to express that there at least two things, we need identity [Bostock]
Identity is clearly a logical concept, and greatly enhances predicate calculus [Boolos]
In logic identity involves reflexivity (x=x), symmetry (if x=y, then y=x) and transitivity (if x=y and y=z, then x=z) [Baillie]
In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn]
Both non-contradiction and excluded middle need identity in their formulation [McGinn]
Identity is unitary, indefinable, fundamental and a genuine relation [McGinn]
Identity is a level one relation with a second-order definition [Hodes]
Identity is invariant under arbitrary permutations, so it seems to be a logical term [McGee]
Unlike most other signs, = cannot be eliminated [Engelbretsen/Sayward]