Computability and Logic

21.1. SOLVABLE AND UNSOLVABLE DECISION PROBLEMS 271
Church’s theorem need be recalled for purposes of this chapter. We are going to prove
sharper results than Church’s theorem, to the effect that the satisfiability problem is
unsolvable for narrower classes K than the class of all first-order sentences; but in
no case will we prove these sharper results by going back to the proof of Church’s
theorem and sharpening the proof. Instead, we are simply going to prove that if the
satisfiability problem for K were solvable, then the satisfiability problem for full
first-order logic would be solvable, as Church’s theorem tells us it is not. And we
are going to prove this simply by showing how one can effectively associate to any
arbitrary sentence a sentence in K that is equivalent to it for satisfiability.
We have in fact already done this in one case in section 19.4, where we showed
how one can effectively associate to any arbitrary sentence a sentence of predicate
logic (that is, one not involving constants or function symbols), and indeed one of
predicate logic without identity, that is equivalent to it for satisfiability. Thus we have
already proved the following slight sharpening of Church’s theorem.
21.1 Lemma. The satisfiability problem for predicate logic without identity is unsolvable.
Sharper results will be obtained by considering narrower classes of sentences:
dyadic logic, the part of predicate logic without identity where only two-place predicates
are allowed; the logic of a triadic predicate, where only a single three-place
predicate is allowed; and finally the logic of a dyadic predicate, where only a single
two-place predicate is allowed. Section 21.3 will be devoted to proving the following
three results.
21.2 Lemma. The satisfiability problem for dyadic logic is unsolvable.
21.3 Lemma. The satisfiability problem for the logic of a triadic predicate is unsolvable.
21.4 Theorem (The Church–Herbrand theorem). The satisfiability problem for the
logic of a dyadic predicate is unsolvable.
Let us now turn to positive results. Call a sentence n-satisfiable if has a model of
some size m ≤ n. Now note three things. First, we know from section 12.2 that if a
sentence comes out true in some interpretation of size m, then it comes out true in
some interpretation whose domain is the set of natural numbers from 1 to m. Second,
for a given finite language, there are only finitely many interpretations whose domain
is the set of natural numbers from 1 to m. Third, for any given one of them we
can effectively determine for any sentence whether or not it comes out true in that
interpretation.
[It is easy to see this last claim holds for quantifier-free sentences: the specification
of the model tells us which atomic sentences are true, and then we can easily work out
whether a given truth-functional compound of them is true. Perhaps the easiest way to
see the claim holds for all sentences is to reduce the general case to the special case of
quantifier-free sentences. To do so, for each 1 ≤ k ≤ m add to the language a constant k
denoting k. To any sentence A of the expanded language we can effectively associate
a quantifier-free sentence A* as follows. If A is atomic, A* isA. IfA is a truthfunctional
compound, then A* is the same compound of the quantifier-free sentences