[FOM] Quantification and identity

[Matthew McKeon]

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'.    
 
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