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Single Idea 13429

[filed under theme 5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic ]

Full Idea

Wittgenstein discovered that the sign of identity is not a necessary constituent of logical notation, but can be replaced by the convention that different signs must have different meanings.

Gist of Idea

The identity sign is not essential in logical notation, if every sign has a different meaning

Source

report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by Frank P. Ramsey - The Foundations of Mathematics p.139

Book Ref

Ramsey,Frank: 'Philosophical Papers', ed/tr. Mellor,D.H. [CUP 1990], p.194

A Reaction

[Ramsey cites p.139 - need to track down the modern reference] Hence in modern logic it is usually necessary to say that we are using 'classical logic with identity', since the use of identity is very convenient, and reasonably harmless (I think).

The 16 ideas with the same theme [logical assertions that that two objects are identical]:

13427 Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey]
18759 Identity is invariant under arbitrary permutations, so it seems to be a logical term [Tarski, by McGee]
18154 The sign of identity is not allowed in 'Tractatus' [Wittgenstein, by Bostock]
13429 The identity sign is not essential in logical notation, if every sign has a different meaning [Wittgenstein, by Ramsey]
10012 Quantification theory can still be proved complete if we add identity [Quine]
18897 Predicate logic has to spell out that its identity relation '=' is an equivalent relation [Sommers]
13800 |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
13803 If we are to express that there at least two things, we need identity [Bostock]
13799 The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
10697 Identity is clearly a logical concept, and greatly enhances predicate calculus [Boolos]
3299 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]
6051 In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn]
6055 Both non-contradiction and excluded middle need identity in their formulation [McGinn]
6059 Identity is unitary, indefinable, fundamental and a genuine relation [McGinn]
10011 Identity is a level one relation with a second-order definition [Hodes]
13851 Unlike most other signs, = cannot be eliminated [Engelbretsen/Sayward]