Computability and Logic

19.4. ELIMINATING FUNCTION SYMBOLS AND IDENTITY 255
that is satisfiable by compactness, instead one would apply compactness for truthfunctional
satisfiability to conclude that is truth-functionally satisfiable, from which
it follows that is satisfiable.) Herbrand’s theorem actually then implies the compactness
theorem: Given a set Ɣ of sentences, let Ɣ # be the set of Skolem forms of sentences
in Ɣ. We know from the preceding section that if every finite subset of Ɣ is satisfiable,
then every finite subset of Ɣ # is satisfiable and hence truth-functionally satisfiable,
and so by Herbrand’s theorem Ɣ # is satisfiable, whence the original Ɣ is satisfiable.
Herbrand’s theorem also implies the soundness and Gödel completeness theorems
for an appropriate kind of proof procedure (different from that used earlier in this book
and from those used in introductory textbooks), which we next describe. Suppose we
are given a finite set Ɣ of sentences and wish to know if Ɣ is unsatisfiable. We first
replace the sentences in Ɣ by Skolem forms: the proofs of the normal form theorems
given in the preceding two sections implicitly provide an effective method of doing
so. Now having the finite set S 1 , ..., S n ,of∀-sentences that are the Skolem forms
of our original sentences, and any effective enumeration t 1 , t 2 , t 3 ,...of terms of the
language, we set about effectively generating all possible substitution instances. (We
could do this by first substituting in each formula for each of its variables the term
t 1 , then substituting for each variable in each formula either t 1 or t 2 , then substituting
for each variable in each formula either t 1 or t 2 or t 3 , and so on. At each stage we get
only finitely many substitution instances, namely, at stage m just k m , where k is the
total number of variables; but in the end we get them all.)
Each time we generate a new substitution instance, we check whether the finitely
many instances we have generated so far are truth-functionally satisfiable. This can
be done effectively, since on the one hand at any given stage we will have generated
so far only finitely many substitution instances, so that there are only finitely many
valuations to be considered (if the substitution instances generated so far involve m
distinct atomic sentences, the number of possible valuations will be 2 m ); while on the
other hand, given a valuation ω and a truth-functional compound B of given atomic
sentences A i , we can effectively work out the value ω(B) required (the method of
truth tables expounded in introductory textbooks being a way of setting out the work).
If any finite set of Skolem instances (that is, of substitution instances of Skolem
forms) turns out to be truth-functionally unsatisfiable, then the original set Ɣ is
unsatisfiable: producing such a set of Skolem instances is a kind of refutation of
Ɣ. Conversely, if Ɣ is unsatisfiable, the above-described procedure will eventually
produce such a refutation. This is because we know from the preceding section that Ɣ
is unsatisfiable if and only if Ɣ # is, and so, by Herbrand’s theorem, Ɣ is unsatisfiable
if and only if some finite set of substitution instances of Skolem forms is truthfunctionally
unsatisfiable. The refutation procedure just described is thus sound and
complete (hence so would be the proof procedure that proves Ɣ implies D by refuting
Ɣ ∪{∼D}).
19.4 Eliminating Function Symbols and Identity
While the presence of identity and function symbols is often a convenience, their
absence can often be a convenience, too, and in this section we show how they can,