Computability and Logic

172 PROOFS AND COMPLETENESS
is because & is not in the official notation, and we do not directly have rules for it, that
the derivation in this example needs more steps than that in the preceding example.
Since the two examples so far have both been of derivations constituting deductions,
let us give two equally short examples of derivations constituting refutations
and demonstrations.
14.5 Example. Demonstration of a tautology
(1) A ⇒ A (R0)
(2) ⇒ A, ∼A (R2b), (1)
(3) ⇒ A ∨∼A (R3), (2)
14.6 Example. Refutation of a contradiction
(1) ∼A ⇒∼A (R0)
(2) ⇒∼A, ∼∼A (R2b), (1)
(3) ⇒∼A ∨∼∼A (R3), (2)
(4) ∼(∼A ∨∼∼A) ⇒ (R2a), (3)
(5) A & ∼A ⇒ (4)
The remarks above about the immateriality of the order in which sentences are
written are especially pertinent to the next example.
14.7 Example. Commutativity of disjunction
(1) A ⇒ A (R0)
(2) A ⇒ B, A (R1), (1)
(3) A ⇒ B ∨ A (R3), (2)
(4) B ⇒ B (R0)
(5) B ⇒ B, A (R1), (4)
(6) B ⇒ B ∨ A (R3), (5)
(7) A ∨ B ⇒ B ∨ A (R4), (3), (6)
The commutativity of conjunction would be obtained similarly, though there would
be more steps, for the same reason that there are more steps in Examples 14.4 and
14.6 than in Examples 14.3 and 14.5. Next we give a couple of somewhat more
substantial examples, illustrating how the quantifier rules are to be used, and a couple
of counter-examples to show how they are not to be used.
14.8 Example. Use of the first quantifier rule
(1) Fc⇒ Fc (R0)
(2) ⇒ Fc, ∼Fc (R2b), (1)
(3) ⇒∃xFx, ∼Fc (R5), (2)
(4) ⇒∃xFx, ∃x ∼Fx (R5), (3)
(5) ∼∃x ∼Fx ⇒∃xFx (R2a), (4)
(6) ∀xFx⇒∃xFx (5)