Computability and Logic

21
Monadic and Dyadic Logic
We have given in earlier chapters several different proofs of Church’s theorem to the
effect that first-order logic is undecidable: there is no effective procedure that applied
to any first-order sentence will in a finite amount of time tell us whether or not it is
valid. This negative result leaves room on the one hand for contrasting positive results,
and on the other hand for sharper negative results. The most striking of the former is
the Löwenheim–Behmann theorem, to the effect that the logic of monadic (one-place)
predicates is decidable, even when the two-place logical predicate of identity is admitted.
The most striking of the latter is the Church–Herbrand theorem that the logic of a single
dyadic (two-place) predicate is undecidable. These theorems are presented in sections
21.2 and 21.3 after some general discussion of solvable and unsolvable cases of the
decision problem for logic in section 21.1. While the proof of Church’s theorem requires
the use of considerable computability theory (the theory of recursive functions, or of
Turing machines), that is not so for the proof of the Löwenheim–Behmann theorem or for
the proof that Church’s theorem implies the Church–Herbrand theorem. The former uses
only material developed by Chapter 11. The latter uses also the elimination of function
symbols and identity from section 19.4, but nothing more than this. The proofs of these
two results, positive and negative, are independent of each other.
21.1 Solvable and Unsolvable Decision Problems
Let K be some syntactically defined class of first-order sentences. By the decision
problem for K is meant the problem of devising an effective procedure that, applied
to any sentence S in K , will in a finite amount of time tell us whether or not S is
valid. Since S is valid if and only if ∼S is not satisfiable, and S is satisfiable if and
only if ∼S is not valid, for any class K that contains the negation of any sentence it
contains, the decision problem for K is equivalent to the satisfiability problem for K ,
the problem of devising an effective procedure that, applied to any sentence S in K ,
will in a finite amount of time tell us whether or not S is satisfiable, or has a model.
The formulation in terms of satisfiability turns out to be the more convenient for our
purposes in this chapter.
The most basic result in this area is a negative one, Church’s theorem, which asserts
the unsolvability of the satisfiability problem full first-order logic, where K is the class
of all sentences. We have given three different proofs of this result, two in Chapter 11
and another in section 17.1; but none of the machinery from any of these proofs of
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