Computability and Logic

168 PROOFS AND COMPLETENESS
true. (Note that when the sets are finite, Ɣ ={C 1 ,...,C m } and ={D 1 ,...,D n },
this amounts to saying that every interpretation that makes C 1 & ... & C m true makes
D 1 ∨···∨ D n true: the elements of Ɣ are being taken jointly as premisses, but the
elements of are being taken alternatively as conclusions, so to speak.) When a set
contains but a single sentence, then of course making some sentence in the set true
and making every sentence in the set true come to the same thing, namely, making the
sentence in the set true; and in this case we naturally speak of the sentence as doing
the securing or as being secured. When the set is empty, then of course the condition
that some sentence in it is made true is not fulfilled, since there is no sentence in
it to be made true; and we count the condition that every sentence in the set is made
true as being ‘vacuously’ fulfilled. (After all, there is no sentence in the set that is not
made true.) With this understanding, consequence, unsatisfiability, and validity can
be seen to be special cases of security in the way listed in Table 14-1.
Table 14-1. Metalogical notions
D is a consequence of Ɣ if and only if Ɣ secures {D}
Ɣ is unsatisfiable if and only if Ɣ secures ∅
D is valid if and only if ∅ secures {D}
Correspondingly, our approach to deductions will subsume them along with refutations
and demonstrations under a more general notion of derivation. Thus for us the
soundness and completeness theorems will be theorems relating a syntactic notion of
derivability to a semantic notion of security, from which relationship various other
relationships between syntactic and semantic notions will follow as special cases.
The objects with which we are going to work in this chapter—the objects of which
derivations will be composed—are called sequents. A sequent Ɣ ⇒ consists of
a finite set of sentences Ɣ on the left, the symbol ⇒ in the middle, and a finite set
of sentences on the right. We call the sequent secure if its left side Ɣ secures its
right side . The goal will be to define a notion of derivation so that there will be a
derivation of a sequent if and only if it is secure.
Deliberately postponing the details of the definition, we just for the moment say
that a derivation will be a kind of finite sequence of sequents, called the steps
(or lines) of the derivation, subject to certain syntactic conditions or rules that remain
to be stated. A derivation will be a derivation of a sequent Ɣ ⇒ if and only if
that sequent is its last step (or bottom line). A sequent will be derivable if and only
if there is some derivation of it. It is in terms of this notion of derivation that we will
define other syntactic notions of interest, as in Table 14-2.
Table 14-2. Metalogical notions
A deduction of D from Ɣ is a derivation of Ɣ ⇒{D}
A refutation of Ɣ is a derivation of Ɣ ⇒ ∅
A demonstration of D is a derivation of ∅ ⇒{D}