Computability and Logic

14.1. SEQUENT CALCULUS 169
We naturally say that D is deducible from Ɣ if there is a deduction of D from Ɣ,
that Ɣ is refutable if there is a refutation of Ɣ, and that D is demonstrable if there
is a demonstration of D, where deduction, refutation, and demonstration are defined
in terms of derivation as in Table 14-2. An irrefutable set of sentences is also called
consistent, and a refutable one inconsistent. Our main goal will be so to define the
notion of derivation that we can prove the following two theorems.
14.1 Theorem (Soundness theorem). Every derivable sequent is secure.
14.2 Theorem (Gödel completeness theorem). Every secure sequent is derivable.
It will then immediately follow (on comparing Tables 14-1 and 14-2) that there is
an exact coincidence between two parallel sets of metalogical notions, the semantic
and the syntactic, as shown in Table 14-3.
Table 14-3. Correspondences between metalogical notions
D is deducible from Ɣ if and only if D is a consequence of Ɣ
Ɣ is inconsistent if and only if Ɣ is unsatisfiable
D is demonstrable if and only if D is valid
To generalize to the case of infinite sets of sentences, we simply define to be
derivable from Ɣ if and only if some finite subset 0 of is derivable from some
finite subset Ɣ 0 of Ɣ, and define deducibility and inconsistency in the infinite case
similarly. As an easy corollary of the compactness theorem, Ɣ secures if and only
if some finite subset Ɣ 0 of Ɣ secures some finite subset 0 of . Thus Theorems 14.1
and 14.2 will extend to the infinite case: will be derivable from Ɣ if and only if
is secured by Ɣ, even when Ɣ and are infinite.
So much by way of preamble. It remains, then, to specify what conditions a
sequence of sequents must fulfill in order to count as a derivation. In order for a
sequence of steps to qualify as a derivation, each step must either be of the form
{A} ⇒{A} or must follow from earlier steps according to one of another of several
rules of inference permitting passage from one or more sequents taken as premisses
to some other sequent taken as conclusion. The usual way of displaying rules is to
write the premiss or premisses of the rule, a line below them, and the conclusion
of the rule. The provision that a step may be of the form {A}⇒{A} may itself be
regarded as a special case of a rule of inference with zero premisses; and in listing the
rules of inference, we in fact list this one first. In general, in the case of any rule, any
sentence that appears in a premiss but not the conclusion of a rule is said to be exiting,
any that appears in the conclusion but not the premisses is said to be entering, and
any that appears in both a premiss and the conclusion is said to be standing. In the
special case of the zero-premiss rule and steps of the form {A}⇒{A}, the sentence
A counts as entering. It will be convenient in this chapter to work as in the preceding
chapter with a version of first-order logic in which the only logical symbols are ∼,
∨, ∃, =, that is, in which & and ∀ are treated as unofficial abbreviations. (If we
admitted & and ∀, there would be a need for four more rules, two for each. Nothing