Computability and Logic

14.1. SEQUENT CALCULUS 167
equivalence has two directions. The result that whenever D is deducible from Ɣ, D
is a consequence of Ɣ, isthesoundness theorem. The result that whenever D is a
consequence of Ɣ, then D is deducible from Ɣ,istheGödel completeness theorem.
Our goal in this chapter will be to present a particular system of deduction for which
soundness and completeness can be established. The proof of completeness uses the
main lemma from the preceding chapter. Our system, which is of the general sort used
in more advanced, theoretical studies, will be different from that used in virtually any
introductory textbook—or to put a positive spin on it, virtually no reader will have an
advantage over any other reader of previous acquaintance with the particular kind of
system we are going to be using. Largely for the benefit of readers who have been or
will be looking at other books, in the final section of the chapter we briefly indicate the
kinds of variations that are possible and are actually to be met with in the literature.
But as a matter of fact, it is not the details of any particular system that really matter,
but rather the common features shared by all such systems, and except for a brief
mention at the end of the next chapter (in a section that itself is optional reading), we
will when this chapter is over never again have occasion to mention the details of our
particular system or any other. The existence of some proof procedure or other with
the properties of soundness and completeness will be the result that will matter.
[Let us indicate one consequence of the existence of such a procedure that will be
looked at more closely in the next chapter. It is known that the consequence relation
is not effectively decidable: that there cannot be a procedure, governed by definite
and explicit rules, whose application would, in every case, in principle enable one to
determine in a finite amount of time whether or not a given finite set Ɣ of sentences
implies a given sentence D. Two proofs of this fact appear in sections 11.1 and
11.2, with another to come in chapter 17. But the existence of a sound and complete
proof procedure shows that the consequence relation is at least (positively) effectively
semidecidable. There is a procedure whose application would, in case Ɣ does imply
D, in principle enable one to determine in a finite amount of time that it does so.
The procedure is simply to search systematically through all finite objects of the
appropriate kind, determining for each whether or not it constitutes a deduction of
D from Ɣ. For it is part of the notion of a proof procedure that there are definite and
explicit rules for determining whether a given finite object of the appropriate sort does
or does not constitute such a deduction. If Ɣ does imply D, then checking through all
possible deductions one by one, one would by completeness eventually find one that
is a deduction of D from Ɣ, thus by soundness showing that Ɣ does imply D; but if
Ɣ does not imply D, checking through all possible deductions would go on forever
without result. As we said, these matters will be further discussed in the next chapter.]
At the same time one looks for a syntactic notion of deduction to capture and
make recognizable the semantic notion of consequence, one would like to have also
a syntactic notion of refutation to capture the semantic notion of unsatisfiability,
and a syntactic notion of demonstration to capture the semantic notion of validity.
At the cost of some very slight artificiality, the three notions of consequence,
unsatisfiability, and validity can be subsumed as special cases under a single, more
general notion. We say that one set of sentences Ɣ secures another set of sentences
if every interpretation that makes all sentences in Ɣ true makes some sentence in