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

Peter Apostoli apostoli at cs.toronto.edu
Thu Feb 27 13:19:37 EST 2003

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.

The precedent of constructivism is appropriate, I think, and may shed some
light on the role of identity in classical set theory. There are two ways to
accommodate the viewpoint of the constructive observer in logic and set
theory. One is the familiar rejection of bivalence associated with
intuitionism. The other is to pass to a non T_0 topology of sets by
granulating the universe of discourse under constructive or "continuous"
indiscernibility. The apparent failure of bivalence on the former approach
is explained in *classical* topological terms on the later approach.
Identity theoretic notions - constructive indiscernibility - constitute the
very notion of a mathematical object here in the much the same way the
admissible operations of invariant theory constitute the objects of physical

Exercise: Find the passages in the Grundlagen where Frege anticipates the
rise of invariant theory.

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.

Yes, thanks for the very interesting examples. I am not sure what to say
about the dispensability/indispensability of identity in formal theories.
The approach I describe does not
presuppose a formal language (another important theme from intuitionism).

(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

The very notion of counterfactuality is at base identity (indiscernibility)

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.)

One only has to glance at Harry Deutch's recent article on relative identity
(Stanford Online Encyc. of Philosophy) to see that most of the philosophical
puzzles that have absorbed philosohers of language over the last 40 years
can be given a single diagnosis and solution by acknowledging the role
relative identity or indiscernibility plays in constituting our notion of
objectivity (I'm
 not suggesting Harry would agree with my description of his project).

Thanks for the comments,

Peter A.

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