[FOM] Quantification and identity
Matthew McKeon
mckeonm at msu.edu
Tue Feb 25 10:26:29 EST 2003
A. Hazen writes,
In his postings to the "Understanding Universal Quantification" string,
Peter Apostoli has emphasized the requirement of a "criterion of identity"
for items of the domain. This is certainly natural from the standpoint of
constructive mathematics and constructive type theory, where specification
of identity conditions for objects of the type is typically part of what is
involved in setting up a type. I'd like to register a doubt, though: from
a classical point of view, there are indications that identity is an
"optional extra" rather than an essential part of quantificational logic.
Three indications:
(1) Identity doesn't seem to add much to the pure logic: completeness,
compactness, Löwenheim-Skolem theorems for FOL (=First Order Logic) with
Identity are obtained by routine elaborations of the proofs for
identity-less FOL. And, as Quine pointed out, in a First Order language
with a finite vocabulary, identity can be DEFINED (as indiscernibility with
repect to each predicate (etc)).
But Quine explicitly states in his review of Geach that quantification
requires a notion of identity (in the meta-language),and he says it is a
notion of "absolute" identity. I take Quine to mean, in part, that we
need a symbol in the meta-language functionally equivalent with '=' which
is defined stronger than 'indiscernibility with
respect to each predicate of the metalanguage'.
Department of Philosophy
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