[FOM] Quantification and identity
Peter Apostoli
apostoli at cs.toronto.edu
Wed Feb 26 01:24:34 EST 2003
MK:
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'.
PA:
In the axiomatization of identity theory in fol based upon Wangs single
axiom scheme
(E) P(a) <-> (Ex)(x=a & P(x))
is equivalent to that based upon
(A) P(a) <-> (Allx)(x=a -> P(x)).
The semantic equivalence of (E) and (A) requires the "standard"
interpretation of the identity sign in model theory. However, Geach asks us
to consider (E) as a scheme in *two* variables: formulas "P" and the
interpretation of "=". Interpreting "=" as formal indiscernibility, or any
relation coarser than strict identity, results in a semantics in which (E)
and (A) diverge in a manner quite familiar to Aristotle. This may be the
reason Quine requires access to strict identity in the meta language for the
definition of truth on a model.
Geach's gestalt switch reveals the roots of counterpart theory in "ordinary"
identity theory. Let U be a nonempty domain of discourse and R an
equivalence relation on U. Interpreting "=" as R, one obtains a model
(U,R) of fol in which (E) and (A) converge for all "complete" formulas
P(x). A formula of fol is *complete* iff its truth-set in U is closed under
R.
Now a thought most chilling to Quine. R, formerly a mere equivalence
relation, now represents itself as a full blown congruence relation with
Leibnizian credentials: indiscernibility with respect to falling under
precisely the same complete concepts. Call two objects counterparts just in
case the are indiscernible. Now relax our restriction on the range of "P".
Then (A) becomes the definition of a modality of *predication* : a is
possibly P iff some counterpart of a is P. (B) becomes: a is necessarily P
iff all of a's counterparts are P. This hidden de re modality fails to
appear in fol because, in effect, we are limiting values of "P" to complete
concepts. Without this limitation, fol with identity is an S5 de re
modality.
This de re modality is the basis of an adequate understanding of Russell's
paradox.
AH:
(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.
The full statement of the axiom of extensionality is given in Quine and
Fraenkel and Bar Hillel as "if two sets have precisely the same members then
they are elements of precisely the same sets". The converse, tantamount to
the Principle of the Identity of Indiscernibles, is a trivial consequence of
the presence of "singleton" sets, where recall we have agreed to define the
singleton set of a set a as {x: x is co-extensive with a}. However, if
instead for each set a we posit {x: x is indiscernible from a}, the PII is
no longer a theorem, and must be posited to recover ZF. Without this posit
of strict singletons, but rather of R classes, we enter a universe of sets
with a non T_0 topology in which naive comprehension for complete concepts
is consistent. This is the consistent universe of "naive" sets predicted
using formal methods by Nino Cocchiarella.
Peter A.
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