Computability and Logic

14
Proofs and Completeness
Introductory textbooks in logic devote much space to developing one or another kind
of proof procedure, enabling one to recognize that a sentence D is implied by a set of
sentences Ɣ, with different textbooks favoring different procedures. In this chapter we
introduce the kind of proof procedure, called a Gentzen system or sequent calculus, that
is used in more advanced work, where in contrast to introductory textbooks the emphasis
is on general theoretical results about the existence of proofs, rather than practice in
constructing specific proofs. The details of any particular procedure, ours included,
are less important than some features shared by all procedures, notably the features
that whenever there is a proof of D from Ɣ, D is a consequence of Ɣ, and conversely,
whenever D is a consequence of Ɣ, there is a proof of D from Ɣ. These features are called
soundness and completeness, respectively. (Another feature is that definite, explicit rules
can be given for determining in any given case whether a purported proof or deduction
really is one or not; but we defer detailed consideration of this feature to the next chapter.)
Section 14.1 introduces our version or variant of sequent calculus. Section 14.2 presents
proofs of soundness and completeness. The former is easy; the latter is not so easy, but
all the hard work for it has been done in the previous chapter. Section 14.3, which is
optional, comments briefly on the relationship of our formal notion to other such formal
notions, as might be found in introductory textbooks or elsewhere, and of any formal
notion to the unformalized notion of a deduction of a conclusion from a set of premisses,
or proof of a theorem from a set of axioms.
14.1 Sequent Calculus
The idea in setting up a proof procedure is that even when it is not obvious that
Ɣ implies D, we may hope to break the route from Ɣ to D down into a series of small
steps that are obvious, and thus render the implication relationship recognizable.
Every introductory textbook develops some kind of formal notion of proof or deduction.
Though these take different shapes in different books, in every case a formal
deduction is some kind of finite array of symbols, and there are definite, explicit rules
for determining whether a given finite array of symbols is or is not a formal deduction.
The notion of deduction is ‘syntactic’ in the sense that these rules mention the
internal structure of formulas, but do not mention interpretations. In the end, though,
the condition that there exists a deduction of D from Ɣ turns out to be equivalent
to the condition that every interpretation making all sentences in Ɣ true makes the
sentence D true, which was the original ‘semantic’ definition of consequence. This
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