Computability and Logic

184 PROOFS AND COMPLETENESS
In addition to such substantive variations as we have been discussing, considerable
variations in style are possible, and in particular in typographical layout. For instance,
if one opens an introductory textbook, one may well encounter something like what
appears in Figure 14-1.
(i) ~A → ~B
(ii)
(iii) ~A
B
(1) ~A → ~B (Q0), (i)
(2) ~A (Q0), (iii)
(3) ~B (Q1), (1), (2)
(4) B (Q0), (ii)
(5) ~~A (Q4), (3), (4)
(6) A (Q3), (5)
(7) B → A (Q2), (6)
Figure 14-1. A ‘natural deduction’.
What appears in Figure 14-1 is really the same as what appears in Example 14.21,
differently displayed. The form of display adopted in this book, as illustrated in
Example 14.21, is designed for convenience when engaged in theoretical writing
about deductions. But when engaged in the practical writing of deductions, as in
introductory texts, the form of display in Figure 14-1 is more convenient, because it
involves less rewriting of the same formula over and over again. In lines (1)–(7) in
Figure 14-1, one only writes the sentence D on the right of the sequent Ɣ ⇒ D that
occurs at the corresponding line in Example 14.21. Which of the sentences (i), (ii),
(iii) occur in the set Ɣ on the left of that sequent is indicated by the spatial position
where D is written: if it is written in the third column, all of (i)–(iii) appear; if in the
second, only (i) and (ii) appear; if in the first, only (i). Colloquially one sometimes
speaks of deducing a conclusion D ‘under’ certain hypotheses Ɣ, but in the form of
display illustrated in Figure 14-1, the spatial metaphor is taken quite literally.
It would take us too far afield to enter into a detailed description of the conventions
of this form of display, which in any case can be found expounded in many introductory
texts. The pair of examples given should suffice to make our only real point here:
that what is substantively the same kind of procedure can be set forth in very different
styles, and indeed appropriately so, given the different purposes of introductory texts
and of more theoretical books like this one. Despite the diversity of approaches possible,
the aim of any approach is to set up a system of rules with the properties that if
D is deducible from Ɣ, then D is a consequence of Ɣ (soundness), and that if D is a
consequence of Ɣ, then D is formally deducible from Ɣ (completeness). Clearly, all
systems of rules that achieve these aims will be equivalent to each other in the sense
that D will be deducible from Ɣ in the one system if and only if D is deducible from
Ɣ in the other system. Except for one optional section at the end of the next chapter,
there will be no further mention of the details of our particular proof procedure in the
rest of this book.
A word may now be said about the relationship between any formal notion, whether
ours or a variant, of deduction of a sentence from a set of sentences, and the notion in