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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]
|
13429
|
The identity sign is not essential in logical notation, if every sign has a different meaning
[Wittgenstein, by Ramsey]
|
18154
|
The sign of identity is not allowed in 'Tractatus'
[Wittgenstein, by Bostock]
|
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]
|
13799
|
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b)
[Bostock]
|
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]
|
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]
|