Computability and Logic

266 THE CRAIG INTERPOLATION THEOREM
and other concepts. The supposition of Beth’s theorem, then, is that α and β 1 ,...,β n
are nonlogical symbols of the language L of some theory T and that α is not among
the β i .
The first explication is straightforward and embodies the idea that a theory defines
a concept in terms of others when ‘a definition of that concept in terms of the others
is a consequence of the theory’. This sort of definition is called an explicit definition:
we say that α is explicitly definable in terms of the β i in T if a definition of α from
the β i is one of the sentences in T. What precisely is meant by a definition of α in
terms of the β i depends on whether α is a predicate or a function symbol. In the case
ofa(k + 1)-place predicate, such a definition is a sentence of the form
∀x 0 ∀x 1 ···∀x k (α(x 0 , x 1 ,...,x k ) ↔ B(x 0 ,...,x k ))
and in case of a k-place function symbol, such a definition is a sentence of the form
∀x 0 ∀x 1 ···∀x k (x 0 = α(x 1 ,...,x k ) ↔ B(x 0 ,...,x k ))
where in either case B is a formula whose only nonlogical symbols are among the β i .
(Constants may be regarded as 0-place function symbols, and do not require separate
discussion. In this case the right side of the biconditional would simply be x 0 = α.)
The general form of a definition may be represented as
∀x 0 ···∀x k (—α, x 0 ,...,x k — ↔ B(x 0 ,...,x k )).
The second explication is rather more subtle, and incorporates the idea that a theory
defines a concept in terms of others if ‘any specification of the universe of discourse
of the theory and the meanings of the symbols representing the other concepts (that
is compatible with the truth of all the sentences in the theory) uniquely determines
the meaning of the symbol representing that concept’. This sort of definition is called
implicit definition: we say that α is implicitly definable from the β i in T if any two
models of T that have the same domain and agree in what they assign to the β i also
agree in what they assign to α.
It will be useful to develop a more ‘syntactic’ reformulation of this ‘semantic’
definition of implicit definability. To this end, we introduce a new language L ′ obtained
from L by replacing every nonlogical symbol γ of L, other than the β i ,bya
new symbol γ ′ of the same kind: 17-place function symbols are replaced by 17-place
function symbols, 59-place predicates by 59-place predicates, and so on.
Given two models M and N of T that have the same domain and agree on what
they assign to the β i ,weletM+ N be the interpretation of L ∪ L ′ that has the same
domain, and assigns the same denotations to the β i , and, for any other nonlogical
symbol γ of L, assigns to γ what M assigns to γ , and assigns to γ ′ what N assigns
to γ . Then M+ N is a model of T ∪ T ′ .
Conversely, if K is a model of T ∪ T ′ , then K can clearly be ‘decomposed’ into
two models M and N of T , which have the same domain (as each other, and as K)
and agree (with each other and with K) on what they assign to the β i , where for any
other nonlogical symbol γ of L, what M assigns to γ is what K assigns to γ , and
what N assigns to γ is what K assigns to γ ′ .