Computability and Logic

254 NORMAL FORMS
Now if Ɣ is satisfiable in the ordinary sense, that is, if there is an interpretation A
in which every sentence in Ɣ comes out true, then certainly Ɣ is truth-functionally
satisfiable. Simply take for ω the function that gives a sentence the value 1 if and
only if it is true in A.
The converse is also true. In other words, if there is a valuation ω that gives every
sentence in Ɣ the value 1, then there is an interpretation A in which every sentence in
Ɣ comes out true. To show this, it is enough to show that for any valuation ω, there is
an interpretation A such that each A i comes out true in A just in case ω assigns it the
value 1. This is in fact the case even if we start with an infinite set of atomic formulas
A i . To specify A, we must specify a domain, and assign a denotation to each constant,
function symbol, and predicate occurring in the A i . Well, simply take for each closed
term t in the language some object t*, with distinct terms corresponding to distinct
objects. We take the domain of our interpretation to consist of these objects t*. We
take the denotation of a constant c to be c*, and we take the denotation of a function
symbol f to be the function that given the objects t 1 *, ..., t n * associated with terms
t 1 , ..., t n as arguments, yields as value the object f (t 1 , ..., t n )* associated with the
term f (t 1 , ..., t n ). It follows by induction on complexity that the denotation of an
arbitrary term t is the object t* associated with it. Finally, we take as the denotation
of a predicate P the relation that holds of objects the objects t 1 *, ..., t n * associated
with terms t 1 , ..., t n if and only if the sentence P(t 1 , ..., t n ) is one of the A i and
ω assigns it the value 1. Thus truth-functional satisfiability and satisfiability in the
ordinary sense come to the same thing for quantifier-free sentences.
Let now Ɣ be a set of ∀-formulas of some language L, and consider the set of all
instances P(t 1 , ..., t n ) obtained by substituting in sentences ∀x 1 ...∀x n P(x 1 , ..., x n )
of Ɣ terms t 1 , ..., t n of L for the variables. If every finite subset of is truthfunctionally
satisfiable, then every finite subset of is satisfiable, and hence so is ,
by the compactness theorem.
Moreover, by Proposition 12.7, if A is an interpretation in which every sentence in
comes out true, and B is the subinterpretation of A whose domain is the set of all
denotations of closed terms, then every sentence in also comes out true in B. Since
in B every element of the domain is the denotation of some term, from the fact that
every instance P(t 1 , ..., t n ) comes out true it follows that the ∀-formula ∀x 1 ...∀x n
P(x 1 , ..., x n ) comes out true, and thus B is a model of Ɣ. Hence Ɣ is satisfiable.
Conversely, if Ɣ is satisfiable, then since a sentence implies all its substitution instances,
every finite or infinite set of substitution instances of sentences in Ɣ will be satisfiable
and hence truth-functionally satisfiable. Thus we have proved the following.
19.11 Theorem (Herbrand’s theorem). Let Ɣ be a set of ∀-sentences. Then Ɣ is satisfiable
if and only if every finite set of substitution instances of sentences in Ɣ is truthfunctionally
satisfiable.
It is possible to avoid dependence on the compactness theorem in the foregoing
proof, by proving a kind of compactness theorem for truth-functional valuations,
which is considerably easier than proving the ordinary compactness theorem. (Then,
starting with the assumption that every finite subset of is truth-functionally satisfiable,
instead of arguing that each finite subset is therefore satisfiable, and hence