[FOM] Quantification and identity (re: P. Apostoli)

[A.P. Hazen]

    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)).
   (2)  As Leibniz and David Lewis have famously suggested, the modal
operators of necessity and possibility can be thought of as universal and
existential quantifiers over a domain of *possible worlds.*  When
additional operators are added to the modal logic (I'm thinking about
actuality operators of variable "strength": see discussion at the end of my
"Actuality and Quantification," NDJoFL 31 (1990), pp. 498-508 (pp. 506-507
for this point)) the language becomes as expressive as ordinary
quantification theory.  In the absence of explicit variables bound to the
modal "quantifiers," however, identity of possible worlds isn't even
expressible.
   (3)  The best-known First Order theory in the Foundations of
Mathematics, ZF set theory, can be formulated as a theory in FOL WITHOUT
identity: the axiom of extensionality merely says that any set containing a
given set as a member will also contain every set with the same members as
the given set, and the identity predicate is introduced as an abbreviatory
definition.
   Since so many great logicians stress the importance of identity
criteria, I may be over-looking something, I suppose....  (Anil Gupta's
"The Logic of Common Nouns" (Yale U.P. 1980, ISBN 0-300-02346-4) gives an
interesting and sophisticated account of identity in quantified modal
logic: I don't know whether it would suggest anything useful for the model
theory of constructive type theory.)
----
Allen Hazen
Philosophy Department
University of Melbourne