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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]
     Full Idea: In 'a = b' either 'a' and 'b' are names of the same thing, in which case the proposition says nothing, or of different things, in which case it is absurd. In neither case is it an assertion of a fact; it only asserts when a or b are descriptions.
     From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1)
     A reaction: This is essentially Frege's problem with Hesperus and Phosphorus. How can identities be informative? So 2+2=4 is extensionally vacuous, but informative because they are different descriptions.
Identity is invariant under arbitrary permutations, so it seems to be a logical term [Tarski, by McGee]
     Full Idea: Tarski showed that the only binary relations invariant under arbitrary permutations are the universal relation, the empty relation, identity and non-identity, thus giving us a reason to include '=' among the logical terms.
     From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Vann McGee - Logical Consequence 6
     A reaction: Tarski was looking for a criterion to distinguish logical from non-logical terms, since his account of logical validity depended on it. This idea lies behind whether a logic is or is not specified to be 'with identity' (i.e. using '=').
The sign of identity is not allowed in 'Tractatus' [Wittgenstein, by Bostock]
     Full Idea: The 'Tractatus' does not allow the introduction of a sign for identity.
     From: report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by David Bostock - Philosophy of Mathematics 9.B.4
The identity sign is not essential in logical notation, if every sign has a different meaning [Wittgenstein, by Ramsey]
     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.
     From: report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by Frank P. Ramsey - The Foundations of Mathematics p.139
     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).
Quantification theory can still be proved complete if we add identity [Quine]
     Full Idea: Complete proof procedures are available not only for quantification theory, but for quantification theory and identity together. Gödel showed that the theory is still complete if we add self-identity and the indiscernability of identicals.
     From: Willard Quine (Philosophy of Logic [1970], Ch.5)
     A reaction: Hence one talks of first-order logic 'with identity', even though, as Quine observes, it is unclear whether identity is actually a logical or a mathematical notion.
Predicate logic has to spell out that its identity relation '=' is an equivalent relation [Sommers]
     Full Idea: Because predicate logic contrues identities dyadically, its account of inferences involving identity propositions needs laws or axioms of identity, explicitly asserting that the dyadic realtion in 'x=y' possesses symmetry, reflexivity and transitivity.
     From: Fred Sommers (Intellectual Autobiography [2005], 'Syllogistic')
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
     Full Idea: We shall use 'a=b' as short for 'a is the same thing as b'. The sign '=' thus expresses a particular two-place predicate. Officially we will use 'I' as the identity predicate, so that 'Iab' is as formula, but we normally 'abbreviate' this to 'a=b'.
     From: David Bostock (Intermediate Logic [1997], 8.1)
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
     Full Idea: We usually take these two principles together as the basic principles of identity: |= α=α and α=β |= φ(α/ξ) ↔ φ(β/ξ). The second (with scant regard for history) is known as Leibniz's Law.
     From: David Bostock (Intermediate Logic [1997], 8.1)
If we are to express that there at least two things, we need identity [Bostock]
     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.
     From: David Bostock (Intermediate Logic [1997], 8.1)
     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 '='?
Identity is clearly a logical concept, and greatly enhances predicate calculus [Boolos]
     Full Idea: Indispensable to cross-reference, lacking distinctive content, and pervading thought and discourse, 'identity' is without question a logical concept. Adding it to predicate calculus significantly increases the number and variety of inferences possible.
     From: George Boolos (To be is to be the value of a variable.. [1984], p.54)
     A reaction: It is not at all clear to me that identity is a logical concept. Is 'existence' a logical concept? It seems to fit all of Boolos's criteria? I say that all he really means is that it is basic to thought, but I'm not sure it drives the reasoning process.
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]
     Full Idea: In logic identity is an equivalence relation, which involves reflexivity (x=x), symmetry (if x=y, then y=x), and transitivity (if x=y and y=z, then x=z).
     From: James Baillie (Problems in Personal Identity [1993], Intr p.4)
In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn]
     Full Idea: If we say 'for some x, x is F and x is G' we are making tacit appeal to the idea of identity in using 'x' twice here: it has to be the same object that is both F and G.
     From: Colin McGinn (Logical Properties [2000], Ch.1)
     A reaction: This may well be broadened to any utterances whatsoever. The only remaining question is to speculate about whether it is possible to think without identities. The Hopi presumably gave identity to processes rather objects. How does God think?
Both non-contradiction and excluded middle need identity in their formulation [McGinn]
     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').
     From: Colin McGinn (Logical Properties [2000], Ch.1)
     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.
Identity is unitary, indefinable, fundamental and a genuine relation [McGinn]
     Full Idea: I have endorsed four main theses about identity: it is unitary, it is indefinable, it is fundamental, and it is a genuine relation
     From: Colin McGinn (Logical Properties [2000], Ch.1)
     A reaction: That it is fundamental to our thinking seems certain (but to all possible thought?). That it is a relation looks worth questioning. One might challenge unitary by comparing the identity of numbers, values, electrons and continents. I can't define it.
Identity is a level one relation with a second-order definition [Hodes]
     Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition.
     From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])
Unlike most other signs, = cannot be eliminated [Engelbretsen/Sayward]
     Full Idea: Unlike ∨, →, ↔, and ∀, the sign = is not eliminable from a logic.
     From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Ch.3)