Computability and Logic

14.9 Example. Proper use of the two quantifier rules
14.1. SEQUENT CALCULUS 173
(1) Fc ⇒ Fc (R0)
(2) Fc ⇒ Fc, Gc (R1), (1)
(3) Gc ⇒ Gc (R0)
(4) Gc ⇒ Fc, Gc (R1), (3)
(5) Fc ∨ Gc ⇒ Fc, Gc (R4), (2), (4)
(6) Fc ∨ Gc ⇒∃xF x, Gc (R5), (5)
(7) Fc ∨ Gc ⇒∃xF x, ∃xGx (R5), (6)
(8) Fc ∨ Gc ⇒∃xF x ∨∃xGx (R3), (7)
(9) ∃x(F x ∨ Gx) ⇒∃xF x ∨∃xGx (R6), (8)
14.10 Example. Improper use of the second quantifier rule
(1) Fc ⇒ Fc (R0)
(2) Fc, ∼Fc ⇒ (R2b), (1)
(3) ∃xF x, ∼Fc ⇒ (R6), (2)
(4) ∃xF x, ∃x ∼F x ⇒ (R6), (3)
(5) ∃xF x ⇒∼∃x ∼F x (R2b), (4)
(6) ∃xF x ⇒∀xF x (5)
Since ∃xFx does not imply ∀xFx, there must be something wrong in this last
example, either with our rules, or with the way they have been deployed in the
example. In fact, it is the deployment of (R6) at line (3) that is illegitimate. Specifically,
the side condition ‘c not in Ɣ’ in the official statement of the rule is not met, since
the relevant Ɣ in this case would be {∼Fc}, which contains c. Contrast this with a
legitimate application of (R6) as at the last line in the preceding example. Ignoring
the side condition ‘c not in ’ can equally lead to trouble, as in the next example.
(Trouble can equally arise from ignoring the side condition ‘c not in A(x)’, but we
leave it to the reader to provide an example.)
14.11 Example. Improper use of the second quantifier rule
(1) Fc ⇒ Fc (R0)
(2) ∃xF x ⇒ Fc (R6), (1)
(3) ∃xF x, ∼Fc ⇒ (R2b), (2)
(4) ∃xF x, ∃x ∼F x ⇒ (R6), (3)
(5) ∃xF x ⇒∼∃x ∼F x (R2a), (4)
(6) ∃xF x ⇒∀xF x (5)
Finally, let us illustrate the use of the identity rules.
14.12 Example. Reflexivity of identity
(1) c = c ⇒ c = c (R0)
(2) ⇒ c = c (R7), (1)
(3) ∼c = c ⇒ (R2b), (2)
(4) ∃x ∼x = x ⇒ (R6), (3)
(5) ⇒∼∃x ∼x = x (R2a), (4)
(6) ⇒∀xx= x (5)