Computability and Logic

174 PROOFS AND COMPLETENESS
14.13 Example. Symmetry of identity
(1) d = d ⇒ d = d (R0)
(2) d = d, c = d ⇒ d = c (R8a), (1)
(3) c = d ⇒ d = c (R7), (2)
(4) ⇒∼c = d, d = c (R2a), (3)
(5) ⇒∼c = d ∨ d = c (R3), (4)
(6) ⇒ c = d → d = c (5)
(7) ∼(c = d → d = c) ⇒ (R2b), (6)
(8) ∃y ∼(c = y → y = c) ⇒ (R6), (7)
(9) ⇒∼∃y ∼(c = y → y = c) (R2a), (8)
(10) ⇒∀y(c = y → y = c) (9)
(11) ∼∀y(c = y → y = c) ⇒ (R2b), (10)
(12) ∃x ∼∀y(x = y → y = x) ⇒ (R6), (11)
(13) ⇒∼∃x ∼∀y(x = y → y = x) (R2a), (12)
(14) ⇒∀x∀y(x = y → y = x) (13)
The formula A(x) to which (R8a) has been applied at line (2) is d = x.
14.2 Soundness and Completeness
Let us now begin the proof of soundness, Theorem 14.1, according to which every
derivable sequent is secure. We start with the observation that every (R0) sequent
{A}⇒{A} is clearly secure. It will then suffice to show that each rule (R1)–(R9) is
sound in the sense that when applied to secure premisses it yields secure conclusions.
Consider, for instance, an application of (R1). Suppose Ɣ ⇒ is secure, where Ɣ
is a subset of Ɣ ′ and is a subset of ′ , and consider any interpretation that makes
all the sentences in Ɣ ′ true. What (R1) requires is that it should make some sentence
in ′ true, and we show that it does as follows. Since Ɣ is a subset of Ɣ ′ , it makes all
the sentences in Ɣ true, and so by the security of Ɣ ⇒ it makes some sentence in
true and, since is a subset of ′ , thereby makes some sentence of ′ true.
Each of the rules (R2)–(R9) must now be checked in a similar way. Since this proof
is perhaps the most tedious in our whole subject, it may be well to remark in advance
that it does have one interesting feature. The feature is this: that as we argue for the
soundness of the formal rules, we are going to find ourselves using something like the
unformalized counterparts of those very rules in our argumentation. This means that
a mathematical heretic who rejected one of another of the usual patterns of argument
as employed in unformalized proofs in orthodox mathematics—and there have been
benighted souls who have rejected the informal counterpart of (R9), for example—
would not accept our proof of the soundness theorem. The point of the proof is not to
convince such dissenters, but merely to check that, in putting everything into symbols,
we have not made some slip and allowed some inference that, stated in unformalized
terms, we ourselves would recognize as fallacious. (This is a kind of mistake that it
is not hard to make, especially over the side conditions in the quantifier rule, and it is
one that has been made in the past in some textbooks.) This noted, let us now return
to the proof.