Computability and Logic

258 NORMAL FORMS
if P A holds of a 1 , ..., a n for some a 1 in b 1 , ..., and a n in b n . We also need to specify
what the denotation ≡ B of the new sign is to be. We take it to be the genuine identity
relation.
Let now j be the function from |A| to |B| whose value for argument a is the equivalence
class of a. IfP A (a 1 , ..., a n ) holds, then by definition of P B , P B ( j(a 1 ), ...,
j(a n )) holds; while if P B ( j(a 1 ), ..., j(a n )) holds, then again by definition of P B ,
P A (a
1 ′ , ..., a′ n ) holds for some a′ i , where each a′ i
belongs to the same equivalence class
j(a
i ′) = j(a i)asa i . The truth of C P in A guarantees that in that case P A (a 1 ,...,a n )
holds. Trivially, a 1 ≡ A a 2 holds if and only if j(a 1 ) = j(a 2 ), which is to say, if and
only if j(a 1 ) ≡ B j(a 2 ) holds. Thus the function j has all the properties of an isomorphism
except for not being one-to-one. If we look at the proof of the isomorphism
lemma, according to which exactly the same sentences are true in isomorphic interpretations,
we see that the property of being one-to-one was used only in connection
with identity. Hence, so far as sentences not involving identity are concerned, by the
same proof as that of the isomorphism lemma, the same ones are true in B as in A.
(See Proposition 12.5 and its proof.) In particular S* is true in B. But since ≡ B is
the genuine identity relation, it follows that the result of replacing ≡ by = in S* will
also be true in B—and the result of this substitution is precisely the original S.Sowe
have a model B of S as required.
Propositions 19.12 and 19.13 can both be stated more generally. If Ɣ is any set
of sentences and Ɣ ± the set of all S ± for S in Ɣ, together with all functionality
axioms, then Ɣ is satisfiable if and only if Ɣ ± is. If Ɣ is any set of sentences not
involving function symbols, and Ɣ* is the set of all S* for S in Ɣ together with the
equivalence axiom and all congruence axioms, then Ɣ is satisfiable if and only if Ɣ*
is satisfiable. Applications of the function-free and identity-free normal forms of the
present section will be indicated in the next two chapters.
Problems
19.1 Find equivalents
(a) in negation-normal form
(b) in disjunctive normal form
(c) in full disjunctive normal form
for ∼((∼A & B) ∨ (∼B & C)) ∨∼(∼A ∨ C).
19.2 Find equivalents in prenex form for
(a) ∃x(P(x) →∀xP(x))
(b) ∃x(∃x P(x) → P(x)).
19.3 Find an equivalent in prenex form for the following, and write out its Skolem
form:
∀x(Qx →∃y(Py& Ryx)) ↔∃x(Px & ∀y(Qy → Rxy).
19.4 Let T be a set of finite sequences of 0s and 1s such that any initial segment
(e 0 , ..., e m−1 ), m < n, of any element (e 0 , ..., e n−1 )inT is in T . Let T *be