if PA holds of a1, . . . , an for some a1 in b1, . . . , and an in bn . 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. If PA(a1, . . . , an ) holds, then by definition of PB, PB( j(a1), . . . , j(an )) holds; while if PB( j(a1), . . . , j(an )) holds, then again by definition of PB, PA(a1, . . . , an ) holds for some ai , where each ai belongs to the same equivalence class j(ai ) = j(ai ) as ai . The truth of CP in A guarantees that in that case PA(a1, . . . , an ) holds. Trivially, a1 ≡A a2 holds if and only if j(a1) = j(a2), which is to say, if and only if j(a1) ≡B j(a2) 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. So we have a model B of S as required.

Propositions 19.12 and 19.13 can both be stated more generally. If is any set of sentences and ± the set of all S± for S in , together with all functionality axioms, then is satisfiable if and only if ± is. If is any set of sentences not involving function symbols, and * is the set of all S* for S in together with the equivalence axiom and all congruence axioms, then is satisfiable if and only if * 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

19.1Find equivalents

(a)in negation-normal form

(b)in disjunctive normal form

(c)in full disjunctive normal form

for (( A & B) ( B & C)) ( A C).

19.2Find equivalents in prenex form for

(a)x(P(x) → x P(x))

(b)x( x P(x) → P(x)).

19.3Find an equivalent in prenex form for the following, and write out its Skolem form:

x(Qx → y(P y & Ryx)) ↔ x(P x & y(Qy → Rx y).

19.4Let T be a set of finite sequences of 0s and 1s such that any initial segment (e0, . . . , em−1), m < n, of any element (e0, . . . , en−1) in T is in T . Let T * be

the subset of T consisting of all finite sequences s such that there are infinitely many finite sequences t in T with s is an initial segment of t. Show that if T is infinite, then there is an infinite seqeunce e1, e2, . . . of 0s and 1s such that every initial segment (e0, . . . , em−1) is in T *.

19.5 State and prove a compactness theorem for truth-functional valuations.