Willard Van Orman Quine

280 joseph s. ullian
pinch-hit for itself. The result of just dropping that outermost quantifier
will always pass muster as an instance.
Now the main method: Transform each of the schemata whose
joint inconsistency is sought into prenex form, and do so in such a
way that no variable free in one of them is quantified in another (thus
forestalling interference with needed instantiations). Then take instances
of schemata that you have and that you generate, subject
only to one vital restriction. When you take an instance of an existential
quantification, the pinch-hitting variable must not be free in
any earlier line; for short, it must be new. So no moving from ‘∃xFx’
and ‘∃x − Fx’ to‘Fy’ and ‘−Fy’, for this requires ‘y’ to be new at two
different junctures. A good thing, too, in light of the compatibility
of the two quantifications. That is all you do, generate instances
in accord with the restriction (which does not limit instantiations
of universal quantifications). When and only when you assemble a
truth-functionally inconsistent collection of unquantified lines have
you succeeded in proving joint inconsistency of the original batch.
And of course whether such a truth-functionally inconsistent set
(call it a tfi combo) has been assembled is routinely checkable by any
method suitable for testing truth-functional consistency. Thus the
main method reduces inconsistency of quantificational schemata to
something that can be verified.
But their consistency is not thereby so reduced, since for the initial
batch to be consistent is for no tfi combo ever to accrue. In general,
it cannot be inferred that because none has surfaced so far none can
be obtained. Proofs of inconsistency, even for short schemata, can
run as long as you wish. So the main method, though shown to be
complete (adequate to proving inconsistency wherever it lurks), is
no decision procedure, no algorithm always emitting a right answer.
From Church, of course, we know that no such algorithm exists.
Since validity is just inconsistency of the negation, any complete
method for showing inconsistency of quantificational schemata confers
on us a complete method for establishing validity as well, but
still without hope of an algorithm for the purpose. Given a quantificational
schema we know that a complete proof procedure will
allow verification of its validity if it is valid, yet will in general reveal
nothing if it fails of validity.
Keeping our focus on validity, there are some natural subclasses
of the quantificational schemata that do have validity tests. As
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