Quantification and modality

Abstract

Quantifiers as modals It is a familiar observation that the universal quantifier of, say, standard first-order theory behaves in a way that is very similar to the necessity operator or box in most standard modal systems. The universal quantifier acts on predicates, open sentences, or formulas in such a way that the quantification of a conjunction implies the conjunction of the conjuncts, ∀ x ( Px ) implies Px , and several other familiar truths of quantification all combine to suggest strongly that the quantifier “∀ x ” has modal character. Similar observations have been noted for the relation of the existential quantifier “ Ex ” and the corresponding modal, the diamond. Thus, it seems natural to think that “∀ x ” corresponds to some kind of □, and “ Ex ” corresponds to its dual, ◊. The network of correspondences becomes even more compelling when it is observed that (at least classically), “ Ex ” can be defined as “¬∀ x ¬”. It seems to be no accident that the quantifiers look very much like modals. We need not try to force our understanding of quantification to make it appear as if quantification is a type of necessity (or possibility). The reason that universal and existential quantifiers count as modal operators is the same as in all the other cases that we have considered: There is a certain implication structure containing the items that are quantified, and quantification over those items is modal when the corresponding operator distributes over the implication relation, but fails to distribute over its dual.