Willard Van Orman Quine

Quine and Logic 275
second-order forms as (3)? Quine paused to examine this question in
“Existence and Quantification.” We will return to it later.
Mathematics is part of total science for Quine, and continuous
with it. “Pure mathematics...isfirmlyimbedded as an integral part
of our system of the world” (RP 400). As Dreben put it, it is not
sui generis for Quine. Clearly Quine does not regard mathematics –
or even logic, for that matter – as distinguished by its a priori necessity.
A “score” on which logic and mathematics can be contrasted
with the rest of science is “their versatility: their vocabulary
pervades all branches of science” (RP 399). Yet, Quine continues,
“Where I see a major discontinuity is not between mathematical
theory and physical theory, but between terms that can be taught
strictly by ostension and terms that cannot.” Thus “the objects of
pure mathematics and theoretical physics are epistemologically on
apar....Epistemologically the primary cleavage is between these on
the one hand and observables on the other” (RP 402). Quine further
insists that “all ascriptions of reality must come...from within
one’s theory of the world” (TPT 21) and acknowledges that “I see
all objects as theoretical” (TPT 20). In particular, for him the likes
of classes and numbers are reified because of their usefulness for
scientific theory (see TPT 15).
In Quine’s underappreciated book Set Theory and Its Logic, he
takes great pains to show how much of what passes as set theory
is translatable into quantificational logic and so devoid of commitment
to sets. His device of virtual classes, dating back to his 1944
book O Sentido da Nova Lógica, allows us to “enjoy a good deal of
the benefit of a class without its existing” (STL xii). The trick is a
triclausal contextual definition that allows elimination of abstracts
(expressions with form ‘{x:...x ...}’), withholding imputations of
reference to them except when they precede ‘ɛ’. The device is implicit
in other treatments of the subject, but since Set Theory and
Its Logic it has been recognized increasingly as the best treatment
of abstracts when the universe is limited to sets. Seeking both ontological
and notational economies was always high among Quine’s
priorities.
Set Theory and Its Logic builds its development of set theory on an
amazingly sparse basis. It adopts only a small core of axioms, mainly
neutral between divergent theories. When a proof requires more sets
than the meager assumptions provide for, their existence is taken as
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