Computability and Logic

20
The Craig Interpolation Theorem
Suppose that a sentence A implies a sentence C. The Craig interpolation theorem tells us
that in that case there is a sentence B such that A implies B, B implies C, and B involves
no nonlogical symbols but such as occur both in A and in B. This is one of the basic
results of the theory of models, almost on a par with, say, the compactness theorem.
The proof is presented in section 20.1. The proof for the special case where identity and
function symbols are absent is an easy further application of the same lemmas that we
have applied to prove the compactness theorem in Chapter 13, and could have been
presented there. But the easiest proof for the general case is by reduction to this special
case, using the machinery for the elimination of function symbols and identity developed
in section 19.4. Sections 20.2 and 20.3, which are independent of each other, take up
two significant corollaries of the interpolation theorem, Robinson’s joint consistency
theorem and Beth’s definability theorem.
We begin with a simple observation.
20.1 Craig’s Theorem and Its Proof
20.1 Proposition. If a sentence A implies a sentence C, then there is a sentence B that
A implies, that implies C, and that contains only such constants as are contained in both of
A and C.
Proof: The reason is clear: If there are no constants in A not in C, we may take A
for our B; otherwise, let a 1 ,...,a n be all the constants in A and not in C, and let A*be
the result of replacing each a i by some new variable v i . Then, since A → C is valid,
so is ∀v 1 ···∀v n (A* → C), and hence so is ∃v 1 ···∃v n A* → C. Then ∃v 1 ···∃v n A*
is a suitable B, for A implies it, it implies C, and all constants in it are in both A and C.
It might occur to one to ask whether the fact just proved about constants can be
subsumed under one about constants, function symbols, and predicates; that is, to ask
whether if A implies C, there is always a sentence B that A implies, that implies C,
and that contains only constants, function symbols, and predicates that are in both A
and C. The answer to the question, as stated, is no.
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