Willard Van Orman Quine

284 joseph s. ullian
Identity is added to the machinery late; axioms for it are specified
that swell the main method to one sound and complete for the enlarged
realm, quantification theory with identity. Neither soundness
nor completeness of the augmented method is argued, though soundness
requires just one step beyond the soundness result obtained for
the main method, inasmuch as all the identity axioms are themselves
valid.
Quine gives perspicuous treatment to definite descriptions and the
means of eliminating them. As noted earlier, this lets us pare down
the singular terms to just the variables. Function symbols are not
broached; they are often reckoned among the paraphernalia of logic,
but predicates and identity can be utilized to do their jobs. There is
a foray into predicate functors and how they enable elimination of
variables. Finally, there is just a sample, though a brisk one, of some
set theory.
Branching quantifiers would not have been likely grist for Methods,
even if Quine had seen them as belonging within the province
of logic. But, largely because of shortcomings with regard to completeness,
he does not see them as belonging there. The form (3)
exhibits the simplest set of quantificational dependencies not readily
expressible by a first-order schema. 8 The form is what Quine
calls functionally existential: “all its function quantifiers are out in
front and existential” (EQ 110). Now Quine observes that the Skolem
method of functional normal forms is easily adapted to apply to such
schemata, and is then a complete method for establishing their inconsistency.
So far so good. Correspondingly, this gives a complete
method for establishing validity – but now of functionally universal
schemata, those with function quantifiers out front and universal.
In general the negation of a schema in one of these newly dubbed
classes has an equivalent not in that same class but in the other.
The method cited, then, fails to supply the functionally existential
class with a complete means of proving validity; nor does it offer a
complete means of showing inconsistency for the functionally universal
class. In fact a result of Ehrenfeucht shows that there can be
no complete procedure for showing either of these. 9
Strictly, Ehrenfeucht’s argument was for functionally existential
schemata in logic with identity. In affirming that the functionally
existential schemata are not “covered by any proof procedure”,
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