A Simple Solution to Friedman's Fourth Problem

It is shown that Friedman's problem, whether there exists a proper extension of first order logic satisfying the compactness and interpolation theorems, has extremely simple positive solutions if one considers extensions by generalized (finitary) propositional connectives. This does not solve, however, the problem of whether such extensions exist which are also closed under relativization of formulas. It is well known that the classical propositional connectives form a complete set in the sense that any Boolean function is definable from them. However, if one considers connectives as generalized quantifiers (cf. [4]), then the possibilities multiply since a given connective may vary its meaning according to the size of the universe. Consider, for example, the connective Q defined by The classical connectives are thus the "constant" generalized connectives. It turns out that many of the logics obtained by adding these generalized connectives to first order logic satisfy both the compactness and interpolation theorem, providing an extremely simple solution to Friedman's problem 4 in [2]. In fact, for these logics countable compactness and interpolation are equivalent properties. We characterize them and give examples. Since generalized connectives may be considered generalized quantifiers of monadic type, and we have shown elsewhere [I] that proper extensions of L,, by quantifiers of monadic type do not satisfy many-sorted interpolation, and if they enjoy relativizations they do not satisfy (plain) interpolation, we conclude that compactness + interpolation + many-sorted interpolation fi closure under relativizations. To our knowledge, all attempts to construct a (countably) compact proper extension of L,, satisfying interpolation and closed under relativizations have failed so far. Generalized connectives are implicit in Lindstrom's general definition of quantifier 141. They correspond to classes of structures of type (0,. . . ,0) (0-ary relations identified with truth values) closed under isomorphism. However, nowhere in the literature is there a treatment of these specific quantifiers and their properties. Received June 17,1985; revised October 23, 1985. 01986, Association for Symbolic Logic 0022-4812/86/5lO3-0022/$01.70 Notice that new infinitary propositional connectives may be added to infinitary logic, L,,,, preserving interpolation and other pleasant properties (cf. [3]). DEFINITION. Let n E o.A generalized n-ary propositional connective is a function c: Cardinals -+ (0, l ) (O, l )n (this allows for 0-ary connectives c:Card -+ (0,l)) . L,,(c) will be the logic obtained by allowing formulas of the form c(cpl, . ..,cpn) with the semantics given by where cpa[s] = 1 if 'U t = , cp, 0 otherwise. Analogously, one may define L,,(ci I i E I) for any family {ci I i E I) of generalized connectives. DEFINITION. If c is an n-ary generalized connective, define for each Boolean function f:{O,l)" -+ (0 , l ) the class Ef(c) = {KE Card ( C(K)= f ). Note that we may always write a sentence af(c) in L,,(c) such that For example, if c = 0 ,as described before, and f = A , then E, ( 0 ) = o and we may choose as a, ( o ) the sentence where T is any valid formula. THEOREM . . ,ck) satisjies the interpolation theorem if and only if for any 1. L,,(cl,. f,,. . . ,fk (of the appropriate arities) the class Ef,(ci) is jinite or has infinite cardinals PROOF. ''exFor each formula cp E L,,(cl, . . . ,ck) and Boolean functions f,, . . . ,fk of the corresponding arities, let c ~ s ~ . . . , ~ be the result of replacing every E L,, occurrence of ci in cp by a classical propositional schema representing the function f;:. Then we have and, if cp is valid, If Ef,(ci) has infinite cardinals, then by the Lowenheim-Skolem theorem in L,,, (1) implies (2) 'U + cp,,...fk for all infinite 'U. By compactness of L,,, (2) implies that there is n( fl. . . f,)E o such that