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Single Idea 6055

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

Full Idea

To formulate the law of non-contradiction ('nothing can be both F and non-F') and the law of excluded middle ('everything is either F or it is not-F'), we need the concept of identity (in 'nothing' and 'everything').

Gist of Idea

Both non-contradiction and excluded middle need identity in their formulation

Source

Colin McGinn (Logical Properties [2000], Ch.1)

Book Ref

McGinn,Colin: 'Logical Properties' [OUP 2003], p.11

A Reaction

Two good examples in McGinn's argument that identity is basic to all thinking. But the argument also works to say that necessity is basic (since both laws claim it) and properties are basic. Let's just declare everything 'basic', and we can all go home.

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]