Willard Van Orman Quine

Quine and Logic 273
Nonetheless, it should be noted that there are at most six set
theories that are immediately recognizable to those who study the
foundations of mathematics, and two of them are Quine’s. Along
with ZF, ML, and NF, there are von Neumann-Bernays-Gödel (VBG),
Morse-Kelley (MK), and Principia’s type theory. ML and NF are of
course related, but less closely than ZF and VBG and no more closely
than ZF and MK. Actually NF is studied much more than ML now,
largely with an eye to finding proof of consistency relative to some
safer theory. There is no doubt that Quine’s proposed theories have
advanced the discussion of the subject even while they have turned
out not to be fully viable.
The logic of the framework for science is quantificational logic,
possibly with identity. This is where Quine marks the boundary between
logic and mathematics; as soon as one enters into genuine
set theory, one has crossed into the farther territory. Set theory is
mathematics. Logic “has no objects it can call its own; its variables
admit all values indiscriminately” (FSS 52). Set theory postulates
specific sets, abstract objects determined by what bears them the
membership relation. This typifies standard mathematics, with its
numbers, functions, and topological spaces. Logic has at most identity
as its own predicate, and Quine is sometimes grudging about
allowing even it. Often he confines his attention to logic without
identity, where he likes to point out that identity is definable in
terms of a theory’s other predicates if there are only finitely many
of them. A compelling reason that Quine draws the boundary where
he does is the fact that predicate logic has a complete proof procedure,
one adequate to establishing validity wherever it lies. Set
theory, like any part of mathematics into which number theory can
be embedded, can have no such complete proof procedure; whatever
our method of proof, there must, by Gödel’s Incompleteness
Theorem, be sentences that can be neither proved nor refuted by
it – unless the method is so indiscriminate as to allow proof of
everything.
Quine sees what is called second-order logic as lying on set theory’s
side of the border; in quantifying over functions and predicates,
it assumes existence of subsets of and functions on the domain. And
perhaps the assumptions are the more perilous for being made less
overtly than in axiomatic set theory.
Quantificational logic for Quine is, of course, the classical kind.
While acknowledging that he has found some attractive features in
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