Willard Van Orman Quine

Quine and Logic 285
Quine thus added “at any rate when identity is included” (EQ 112).
But the demurral is not needed, since a complete proof procedure for
the class without identity would yield one for the class with identity;
from any schema or sentence of the latter we can find, by standard
construction, one of the former that is valid if and only if the original
was.
So “classical, unsupplemented quantification theory is...maximal:
it is as far out as you can go and still have complete coverage
of validity and inconsistency by the Skolem proof procedure.” Or, it
is safe to say, by any proof procedure. That Quine draws his boundary
for logic where he does is no accident. He continues, “Classical
quantification theory enjoys an extraordinary combination of depth
and simplicity, beauty and utility. It is bright within and bold in its
boundaries”(EQ 111–113). Not only does logic serve the purpose of
helping to moor Quine’s epistemology; it is clear that his work in it
has been a labor of love.
notes
1. Roger Gibson, Jr., The Philosophy of W. V. Quine (U. of South Florida,
1982), 109.
2. For more details on Quine’s work in set theory see my “Quine and the
Field of Mathematical Logic,” in The Philosophy of W. V. Quine, ed.
Hahn and Schilpp, Open Court, 1986 (expanded ed. 1998), 569–89.
3. See Leon Henkin, “Some Remarks on Infinitely Long Formulas,” in
Infinitistic Methods, Pergamon Press, 1961, especially pp. 181ff. The
branched schema (2) and the two standard ones alluded to may be seen
to be pairwise inequivalent by considering interpretations in a universe
of just two members.
4. See Jon Barwise, “On Branching Quantifiers in English,” Journal of
Philosophical Logic, vol. 8, no. 1, Feb., 1979, pp. 47–80.
5. As is remarked later, application of the main method need not be limited
to finite collections.
6. A class is finitely controllable if and only if any nonvalid schema in it
is falsifiable in some finite universe. A study of such classes may be
found in Dreben and Goldfarb, The Decision Problem, Addison-Wesley,
1979. For any finitely controllable class there is a validity test. The next
natural finitely controllable class beyond the Bernays–Schönfinkel is
the Ackermann class, consisting of all prenex schemata with at most
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