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Single Idea 13803

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

Full Idea

To say that there is at least one thing x such that Fx we need only use an existential quantifier, but to say that there are at least two things we need identity as well.

Gist of Idea

If we are to express that there at least two things, we need identity

Source

David Bostock (Intermediate Logic [1997], 8.1)

Book Ref

Bostock,David: 'Intermediate Logic' [OUP 1997], p.328

A Reaction

The only clear account I've found of why logic may need to be 'with identity'. Without it, you can only reason about one thing or all things. Presumably plural quantification no longer requires '='?

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]