Computability and Logic

14.3. OTHER PROOF PROCEDURES AND HILBERT’S THESIS 179
(S6) If Ɣ ∪{∼B(t)}⇒∅ is derivable for some closed term t, then
Ɣ ∪ {∼∃xB(x)}⇒∅ is derivable:
.
Ɣ ∪{∼B(t)} ⇒∅
Ɣ ⇒{B(t)}
Ɣ ⇒{∃xB(x)}
Ɣ ∪ {∼∃xB(x)} ⇒∅.
Given
(R9a)
(R5)
(R2a)
(S7) Ɣ ∪{t = t}⇒∅ derivable for some closed term t, then Ɣ ⇒ ∅ is derivable:
.
Ɣ ∪{t = t} ⇒∅
Ɣ ⇒ ∅.
Given
(R7)
(S8) If Ɣ ∪{B(t)}⇒∅ is derivable, then Ɣ ∪{B(s), s = t}⇒∅ is derivable:
.
Ɣ ∪{B(t)} ⇒∅
Ɣ ⇒{∼B(t)}
Ɣ ∪{s = t} ⇒{∼B(s)}
Ɣ ∪{s = t, B(s)} ⇒∅.
Given
(R2b)
(R8a)
(R9b)
This verification finishes the proof of completeness.
14.3* Other Proof Procedures and Hilbert’s Thesis
A great many other sound and complete proof procedures are known. We begin by
considering modifications of our own procedure that involve only adding or dropping
a rule or two, and first of all consider the result of dropping (R9). The following
lemma says that it will not be missed. Its proof gives just a taste of the methods of
proof theory, a branch of logical studies that otherwise will be not much explored in
this book.
14.15 Lemma (Inversion lemma). Using (R0)–(R8):
(a) If there is a derivation of Ɣ ∪{∼A}⇒, then there is a derivation of
Ɣ ⇒{A}∪.
(b) If there is a derivation of Ɣ ⇒{∼A}∪, then there is a derivation of
Ɣ ∪{A}⇒.
Proof: The two parts are similarly proved, and we do only (a). A counterexample to
the lemma would be a derivation of a sequent Ɣ ∪{∼A}⇒ for which no derivation
of Ɣ ⇒{A}∪ is possible. We want to show there can be no counterexample by
showing that a contradiction follows from the supposition that there is one. Now if
there are any counterexamples, among them there must be one that is as short as