Computability and Logic

20.1. CRAIG’S THEOREM AND ITS PROOF 261
20.2 Example (A failure of interpolation). Let A be ∃xFx& ∃x ∼ Fx, and let C be
∃x∃yx≠ y. Then A implies C, but there is no sentence at all that contains only constants,
function symbols, and predicates that are in both A and C, and therefore there is no
such sentence that A implies and that implies C.
Suppose we do not count the logical predicate of identity, and ask whether, if
A implies C, there is always a sentence B that A implies, that implies C, and that
contains no nonlogical symbols (that is, no constants, no function symbols, and no
nonlogical predicates) except such as are both in A and in C. The Craig interpolation
theorem is the assertion that the answer to our question, thus restated, is yes.
20.3 Theorem (Craig interpolation theorem). If A implies C, then there is a sentence
B that A implies, that implies C, and that contains no nonlogical symbols except such as
are both in A and in C.
Such a B is called an interpolant between A and C. Before launching into the
proof, let us make one clarificatory observation.
20.4 Example (Degenerate cases). It may happen that we need to allow the identity symbol
to appear in the interpolant even though it appears in neither A nor C. Such a situation can
arise if A is unsatisfiable. For instance, if A is ∃x(Fx & ∼Fx) and C is ∃xGx, then ∃x x≠ x
will do for B, but there are no sentences at all containing only predicates that occur in both
A and C, since there are no such predicates. A similar situation can arise if C is valid.
For instance, if A is ∃xFx and C is ∃x(Gx ∨∼Gx), then ∃x x= x will do for B, but
again there are no predicates that occur in both A and C. Note that ∃x x≠ x will do for
an interpolant in any case where A is unsatisfiable, and ∃x x= x in any case where C is
valid. (We could avoid the need for identity if we admitted the logical constants ⊤ and ⊥ of
section 19.1.)
Proof, Part I: This noted, we may restrict our attention to cases where A is
satisfiable and C is not valid. The proof that, under this assumption, an interpolant
B exists will be, like so many other proofs, divided into two parts. First we consider
the case where identity and function symbols are absent, then reduce the general case
where they are present to this special case. (The proof, unlike that of Proposition 20.1,
will be nonconstructive. It will prove the existence of an interpolant, without showing
how to find one. More constructive proofs are known, but are substantially longer and
more difficult.)
Let us begin immediately on the proof of the special case. Considering only sentences
and formulas without identity or function symbols, let A be a sentence that
is satisfiable and C a sentence that is not valid (which is equivalent to saying that
∼C is satisfiable), such that A implies C (which is equivalent to saying {A, ∼C}
is unsatisfiable). We want to show there is a sentence B containing only predicates
common to A and C, such that A implies B and B implies C (which is equivalent to
saying ∼C implies ∼B).
What we are going do is to show that if there is no such interpolant B, then
{A, ∼C} is after all satisfiable. To show this we apply the model existence theorem