Computability and Logic

14.3. OTHER PROOF PROCEDURES AND HILBERT’S THESIS 183
Just by way of illustration, the rules for a variant approach in which ∼ and → and
∀ and = are the official logical operators, and in which one works only with sequents
of form Ɣ ⇒ D, are listed in Table 14-5.
Table 14-5. Rules of a variant sequent calculus
(Q0) Ɣ ⇒ A A in Ɣ
(Q1)
Ɣ ⇒ A → B
Ɣ ⇒ A
(Q2)
(Q3)
(Q4)
(Q5)
(Q6)
(Q7)
Ɣ ⇒ B
Ɣ, A ⇒ B
Ɣ ⇒ A → B
Ɣ ⇒∼∼A
Ɣ ⇒ A
Ɣ, A ⇒ B
Ɣ, A ⇒∼B
Ɣ ⇒∼A
Ɣ ⇒∀xA(x)
Ɣ ⇒ A(t)
Ɣ ⇒ A(c)
Ɣ ⇒∀xA(x)
Ɣ ⇒ s = t
Ɣ ⇒ A(s)
Ɣ ⇒ A(t)
c not in Ɣ or A(x)
(Q8)
Ɣ ⇒ t = t
This variation can be proved sound and complete in the sense that a sequent Ɣ ⇒ D
will be obtainable by these rules if and only if D is a consequence of Ɣ.Wegiveone
sample deduction to give some idea how the rules work.
14.21 Example. A deduction.
(1) ∼A →∼B, B, ∼A ⇒∼A →∼B (Q0), (i)
(2) ∼A →∼B, B, ∼A ⇒∼A (Q0), (iii)
(3) ∼A →∼B, B, ∼A ⇒∼B (Q1), (1), (2)
(4) ∼A →∼B, B, ∼A ⇒ B (Q0), (ii)
(5) ∼A →∼B, B ⇒∼∼A (Q4), (3), (4)
(6) ∼A →∼B, B ⇒ A (Q3), (5)
(7) ∼A →∼B ⇒ B → A (Q2), (6)
The lowercase Roman numerals (i)–(iii) associated with (Q0) indicate whether it is the
first, second, or third sentence in Ɣ= {∼A →∼B, B, ∼A} that is playing the role of A in
the rule (Q0).