Willard Van Orman Quine

Quine and Logic 279
to truth-functional logic than to the full-blown logic of quantification
of which it is a precursor. The method is another illustration of
Quine’s emphasis on efficiency of technique in an area’s coverage.
Rules for the manipulation of quantifiers follow, together with
reasons for them. Canons and examples of translation into the symbolic
idiom are offered. There can be no complete method here, and
students seem ever to want additional examples. In practice I have
found it useful not only to proliferate illustrations of translation into
logical notation, but also to emphasize translations from symbols
back into English. Symbolic logic is a language, and students tend to
need extensive drill to succeed in acquiring it.
Since no formal system is afoot, there is no need to distinguish between
what are usually contrasted as consistency, a syntactic property,
and satisfiability. Quine uses the former term exclusively in
Methods, but he gives it the semantic sense. To be consistent is
to be sometimes true. Ultimately he shows that his principal proof
procedure is both sound and complete, whence the distinction lapses
anyway. What cannot be shown to generate contradiction is just what
has a satisfying interpretation.
That principal formal method for pure quantification theory is
what Quine calls the main method, a variant of what is sometimes
called semantic tableaux. It is one of several methods of the same
strength included in the book. Three others, an extension of Skolem’s
method of functional normal forms, a variant due to Dreben, and
Herbrand’s method, are easily shown to be equal in power to the main
method; the rest are somewhat further afield: a system of natural
deduction for which Quine discusses strategies and two complete
axiom systems outlined in passing. Quine’s stress continues to be
on the method most efficient for the purpose at hand. The present
purpose is cast as that of establishing joint inconsistency of finite
collections of quantificational schemata.
The main method is unsurpassed in both simplicity and applicability
for this end. 5 First you need the notion of an instance of a
quantification, of any schema with an initial quantifier that governs
the whole schema. An instance is obtained when you drop that initial
quantifier and supplant all the variable occurrences just freed by
free occurrences of some one variable, called the instantial variable.
I like to call it the pinch-hitting variable, though this has the peculiarity
of allowing a variable – the one of the dropped quantifier – to
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