Computability and Logic

170 PROOFS AND COMPLETENESS
Table 14-4. Rules of sequent calculus
(R0)
(R1)
(R2a)
(R2b)
(R3)
(R4)
(R5)
(R6)
(R7)
(R8a)
(R8b)
(R9a)
(R9b)
{A}⇒{A}
Ɣ ⇒
Ɣ ′ ⇒ ′ Ɣ subset of Ɣ ′ ,subset of ′
Ɣ ∪{A}⇒
Ɣ ⇒{∼A}∪
Ɣ ⇒{A}∪
Ɣ ∪{∼A}⇒
Ɣ ∪{A}⇒
Ɣ ∪{B}⇒
Ɣ ∪{A ∨ B}⇒
Ɣ ∪{A(c)}⇒
Ɣ ∪{∃xA(x)}⇒
Ɣ ∪{s = s}⇒
Ɣ ⇒
Ɣ ⇒{A, B}∪
Ɣ ⇒{(A ∨ B)}∪
Ɣ ⇒{A(s)}∪
Ɣ ⇒{∃xA(x)}∪
Ɣ ⇒{A(t)}∪
Ɣ ∪{s = t}⇒{A(s)}∪
Ɣ ∪{A(t)}⇒
Ɣ ∪{s = t, A(s)}⇒
Ɣ ∪{∼A}⇒
Ɣ ⇒{A}∪
Ɣ ⇒{∼A}∪
Ɣ ∪{A}⇒
c not in Ɣ or or A(x)
would be harder, but everything would be more tedious.) With this understanding,
the rules are those give in Table 14-4.
These rules roughly correspond to patterns of inference used in unformalized
deductive argument, and especially mathematical proof. (R2a) orright negation introduction
corresponds to ‘proof by contradiction’, where an assumption A is shown
to be inconsistent with background assumptions Ɣ and it is concluded that those
background assumptions imply its negation. (R2b)orleft negation introduction corresponds
to the inverse form of inference. (R3) or right disjunction introduction,
together with (R1), allows us to pass from Ɣ ⇒{A}∪ or Ɣ ⇒{B}∪ by way of
Ɣ ⇒{A, B}∪ to Ɣ ⇒{(A ∨ B)}∪, which corresponds to inferring a disjunction
from one disjunct. (R4) or left disjunction introduction corresponds to ‘proof by
cases’, where something that has been shown to follow from each disjunct is concluded
to follow from a disjunction. (R5) or right existential quantifier introduction
corresponds to inferring an existential generalization from a particular instance. (R6)
or left existential-quantifier introduction is a bit subtler: it corresponds to a common
procedure in mathematical proof where, assuming there is something for which a