Axiomatizing Universal Properties of Quantifiers

Abstract

We axiomatize all quantifier properties which can be expressed by a universal condition on the class of algebras of sets. If A = (A,...) is a model with universe A, a (say, binary) quantifier on A is just a (binary) relation on 9Y(A). Using such a quantifier q, we can conceive of sentences Qx[p(x); +(x)] (using first-order means plus the new quantifier-symbol Q) as expressing the fact that ({a E A I A k p[a]}, {a E A I A l= #f[a]}) E q. Several properties of such quantifiers have been investigated, for instance monotonicity (cf. for instance Makowsky and Tulipani [1976] and Barwise [1978]): q is monotone iff from (X, Y) E q and Y c Z it always follows that (X, Z) e q. Another such property (for a unary quantifier) is to be an (ultra-) filter (cf. Bruce [1978]). Restricting the interpretations q of the quantifier-symbol Q to quantifiers satisfying such a property, one may ask whether the resulting notion of logical truth is axiomatizable (cf. Westerstahl [1989]). Specific axiomatizability results are contained in the references. The appendix of van der Does [199?] contains a case study. The result below was motivated by the appendix of Westerstahl [1989] and covers van der Does' cases as instances. It roughly says that, for a property to be axiomatizable, it suffices that it be expressible as a first-order universal condition in the language of set-algebras. To be more specific: Consider the (first-order) language of Boolean algebras plus the extra binary relation symbol Q. If P is a sentence in this language, we say that the quantifier q on A := (A,. . .) has (the property expressed by) P iff the model (9(A),', A, u, 0, A, c:, q) (the expansion resulting from the addition of q as a relation to the algebra of subsets over A) satisfies P (where the relation q now interprets the symbol Q). For instance, if P is the universal sentence (mon): VuVvVv'(Q(u, v) A v v'-+ Q(u, v')), then, clearly, q c 9Y(A) x 9Y(A) has P iff q is monotone. Received June 1, 1990; revised August 13, 1990. ?) 1991, Association for Symbolic Logic 0022-4812/91./5603-0014/$01 .50