Computability and Logic

19.3. HERBRAND’S THEOREM 253
enumerator of the set of all sets of numbers. By the Löwenheim–Skolem theorem, there is
an enumerable subinterpretation K of J in which the sentence is also true. (Note that all
numbers will be in its domain, since each is the denotation of some constant.) Thus there is
an interpretation K whose domain contains only enumerably many sets of numbers, and in
which S is true of just the sets of numbers in its domain. This is the ‘Skolem paradox’.
How is the paradox to be resolved? Well, though the set of all sets of numbers in the
domain of K does indeed have an enumerator, since the domain is enumerable, none of its
enumerators can be in the domain of K. [Otherwise, it would satisfy F(x), and ∃wF(x)
would be true in K, as it is not.] So part of the explanation of how the sentence ∼∃wF(x)
can be true in K is that those sets that ‘witness’ that the set of sets of numbers in the domain
of K is enumerable are not themselves members of the domain of K.
A further part of the explanation is that what a sentence should be understood as saying
or meaning or denying is at least as much as function of the domain over which the sentence
is interpreted (and even of the way in which that interpretation is described or referred to)
as of the symbols that constitute the sentence. ∼∃wF(x) can be understood as saying
‘nonenumerably many sets of numbers exist’ when its quantifiers are understood as ranging
over a collection containing all numbers and all sets of numbers, as with the domain of the
standard interpretation J ; but it cannot be so understood when its quantifiers range over other
domains, and in particular not when they range over the members of enumerable domains.
The sentence ∼∃wF(x)—that sequence of symbols—‘says’ something only when supplied
with an interpretation. It may be surprising and even amusing that it is true in all sorts of
interpretations, including perhaps some subinterpretations K of J that have enumerable
domains, but it should not a priori seem impossible for it to be true in these. Interpreted
over such a K, it will only say ‘the domain of K contains no enumerator of the set of sets
of numbers in K ′ . And this, of course, is true.
19.3 Herbrand’s Theorem
The applications of Skolem normal form with which we are going to be concerned
in this section require some preliminary machinery, with which we begin. We work
throughout in logic without identity. (Extensions of the results of this section to logic
with identity are possible using the machinery to be developed in the next section,
but we do not go into the matter.)
Let A 1 , ..., A n be atomic sentences. A (truth-functional) valuation of them is
simply a function ω assigning each of them one of the truth values, true or false
(represented by, say, 1 and 0). The valuation can be extended to truth-functional
compounds of the A i (that is, quantifier-free sentences built up from the A i using ∼
and & and ∨) in the same way that the notion of truth in an interpretation is extended
from atomic to quantifier-free sentences:
ω(∼B) = 1 if and only if ω(B) = 0
ω(B & C) = 1 if and only if ω(B) = 1 and ω(C) = 1
ω(B ∨ C) = 1 if and only if ω(B) = 1orω(C) = 1.
A set Ɣ of quantifier-free sentences built up from the A i is said to be truth-functionally
satisfiable if there is some valuation ω giving every sentence S in Ɣ the value 1.