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