Computability and Logic

182 PROOFS AND COMPLETENESS
Proofs: We begin with Corollary14.20. It is easily seen that rule (R10) is sound,
so soundness for (R0)–(R10) follows from the soundness theorem for (R0)–(R9).
Completeness for (R0)–(R10) follows from the completeness theorem for (R0)–(R9),
since adding rules cannot make a complete system incomplete.
Now Corollary14.19 follows, since the same sequents are derivable in any two
sound and complete proof procedures: by Corollary14.17 a sequent will be derivable
using (R0)–(R10) if and only if it is secure, and by Theorems 14.1 and 14.2 it will be
secure if and only if it is derivable using (R0)–(R9).
And now Lemma 14.18 follows also, since if there are derivations of Ɣ ⇒{(A → B)}
∪ and of Ɣ ⇒{A}∪ using (R0)–(R9), then there is certainly a derivation of
Ɣ ⇒{B}∪ using (R0)–(R10) [namely, the one consisting simply of concatenating
the two given derivations and adding a last line inferring Ɣ ⇒{B}∪ by (R10)],
and by Corollary 14.19, this implies there must be a derivation of Ɣ ⇒{B}∪ using
only (R0)–(R9).
Note the contrast between the immediately foregoing proof of the cut elimination
lemma, Lemma 14.18, and the earlier proof of the inversion lemma, Lemma
14.15. The inversion proof is constructive: it actually contains implicit instructions
for converting a derivation of Ɣ ∪{∼A}⇒ into a derivation of Ɣ ⇒{A}∪.
The cut elimination proof we have given is nonconstructive: it gives no hint how
to find a derivation of Ɣ ⇒{B}∪ given derivations of Ɣ ⇒{A}∪ and Ɣ ⇒
{(A → B)}∪, though it promises us that such a derivation exists.
A constructive proof of the corollary is known, Gentzen’s proof, but it is very
much more complicated than the proof of the inversion lemma, and the result is that
while the derivation of Ɣ ⇒{A}∪ obtained from the proof of the inversion lemma
is about the same length as the given derivation of Ɣ ∪{∼A}⇒, the derivation
of Ɣ ⇒{B}∪ obtained from the constructive proof of the foregoing corollary
may be astronomically longer than the given derivations of Ɣ ⇒{(A → B)}∪ and
Ɣ ⇒{A}∪ combined.
So much for dropping (R9) or adding (R10). A great deal more adding and dropping
of rules could be done. If enough new rules are added, some of our original rules
(R0)–(R8) could then be dropped, since the effect of them could be achieved using
the new rules. If we allowed & and ∀ officially, we would want rules for them, and the
addition of these rules might make it possible to drop some of the rules for ∨ and ∃,
if indeed we did not choose to drop ∨ and ∃ altogether from our official language,
treating them as abbreviations. Similarly for → and ↔.
In all the possible variations mentioned in the preceding paragraph, we were
assuming that the basic objects would still be sequents Ɣ ⇒ . But variation is
possible in this respect as well. It is possible, with the right selection of rules, to get
by working only with sequents of form Ɣ ⇒{D} (in which case one would simply
write Ɣ ⇒ D), making deduction the central notion. It is even possible to get by working
only with sequents of form Ɣ ⇒ ∅ (in which case one would simply write Ɣ),
making refutation the central notion. Indeed, it is even possible to get by working
only with sequents of form ∅ ⇒{D} (which in one would simply write D), making
demonstration the central notion.