Willard Van Orman Quine

Quine and Logic 283
that have it we would have a complete procedure for establishing
consistency, and so even a test for it. We can confirm inconsistency,
and we can confirm finite satisfiability. It is the third category that
derails us.
Quine has made a decision about interpretations that reflects this
theoretical consideration. There can be no complete method for generating
satisfying (or falsifying) interpretations with assurance that
they are such. Methods is a book that concentrates on complete
methods; here there is none to offer. I think a good case can be built
for making a different decision about interpretations. It has been
my experience that one of the great aids to grasping what validity
amounts to is the construction of models. It is, first of all, how we
see nonvalidity of arguments that we map by Venn-type diagrams.
We set down the information given by the premises and then find an
allowable placement of objects that belies the purported conclusion.
Or, the inequivalence of (4) and (5) above, to take a handy example,
is most compellingly grasped by consideration of a two-element
universe in which (4) holds and (5) fails. When the universe of an
interpretation is finite, expansion serves as a canonical way of evaluating
schemata. If there is one strand in the teaching of elementary
logic that Quine might have pursued further it is, by my lights, this
matter of looking at interpretations.
Quine observes that when, in a finite set of prenex schemata, no
universal quantifier precedes any existential one, the main method
provides a test of the set’s inconsistency. Quine’s argument for the
main method’s completeness makes this apparent: for this is the case
where his “rigid routine” produces just a finite stock of instances. It
is also, of course, precisely the situation in which pure existentials
can be applied for the same purpose, to the negation of the conjunction
of the schemata in the set.
Methods is loaded with other materials, more than can be
squeezed into a semester-long course. There is metatheory, proof
of the main method’s soundness and completeness and, from the
latter, a form of the Skolem-Löwenheim Theorem. It is shown that
if systematic application of the method does not issue in any tfi
combo, there will be an interpretation in a universe of positive integers
that satisfies all the schemata in the entire train, including those
in the initial collection. Indeed, the collection need not be finite
here, so that a proof of quantificational compactness is also at hand.
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