Expressive Power and Intensional Operators

In Entities and Indices , M. J. Cresswell argued that a first-order modal language can reach the expressive power of natural-language modal discourse only if we give to the formal language a semantics with indices containing infinite possible worlds and we add to it an infinite collection of operators actuallyₙ actually n and Ref ₙ R e f n which store and retrieve worlds. In the fourth chapter of the book, Cresswell gave a proof that the resulting intensional language, which he called L^* L ∗ , is as expressive as an extensional variant of it, called L L , which has full quantification over worlds. In both linguistics and philosophy, Cresswell’s book has been viewed as offering a compelling argument for preferring extensional systems in the study of natural language. In this paper, after providing a model-theoretic definition of the relation being as expressive as that can be applied to Cresswell’s languages L L and L^* L ∗ , we show that the intensional language L^* L ∗ is not as expressive as the extensional language L L . This result, we claim, undermines Cresswell’s argument to the effect that English modal discourse has the power of explicit quantification over worlds. Additionally, we show that L^* L ∗ does become as expressive as L L when we add Cresswell’s operator of universal modality □ to L^* L ∗ , which provides an extra amount of expressive power. Recently, I. Yanovich has advocated a view that is similar to ours in important respects. At the end of the paper we offer a short discussion of his formalism.