Willard Van Orman Quine

282 joseph s. ullian
equivalent to ‘∃x1 ···∃xn∀y1 ···∀ym − B(x1, ..., xn, y1, ..., ym)’, must
be satisfied in some nonempty universe. Therein there must be
a1, ..., an for which ‘−B(a1, ..., an, y1, ..., ym)’ holds for every choice
of y, ..., yn. 7 But then if the universe were chopped down to just
a1, ..., an, that last schema would still hold for all remaining choices
of y1, ..., ym, there being, if anything, fewer such choices than before.
So the schema fails in the universe with just a1, ..., an, which is what
we sought to show. Had all quantifiers been universal a 1-element
universe would have sufficed.
The pure existentials test itself is a simple truth-functional one:
drop all initial universal quantifiers, obtaining a schema S all of
whose quantifiers are existential; form the alternation of all the ways
of substituting free variables of S for those governed by its existential
quantifiers; and test that alternation for truth-functional validity. (If
there are no free variables in S to substitute you just supply one of
your own.) Now Quine never says it, but the alternation to be tested
is precisely the expansion of S over a universe consisting of objects
named by its free variables. Quine has observed in Chapter 22 that
existential quantifications amount, in fixed finite universes, to such
alternations. Fixing the universe as just indicated, what we have
formed is the relevant expansion, the truth-functional articulation
over that universe of what S says. Since the universe has the pivotal
number of objects, we know that validity of the expansion assures
validity of the schema. And even without that realization, we know
that nonvalidity of the expansion guarantees nonvalidity of the original.
For, as Quine’s argument about (5’) on page 184 illustrates, from
a nonvalid expansion we can construct at once a falsifying interpretation
for the quantification with which we began.
Quine’s argument for the method’s soundness turns not on the
pivotal size of the expansion’s universe, but on the fact that an existential
quantification is implied by any of its instances. Every alternant
of the expansion stands in the ancestral of the “instance of”
relation to S, so validity of the expansion assures that of S, and in
turn that of the original schema.
It is noteworthy that Quine says so little about interpretations that
falsify (or satisfy) quantificational schemata. It is mainly in his chapters
on metatheory that they are mentioned. On page 215 we find a
satisfiable schema with no finite model, an “infinity schema”, with
the observation that if we could confirm this status for all schemata
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