Expressive Power and Intensional Operators

Abstract

AbstractIn 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 $${{\varvec{actually}}}_n$$ 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 $${\mathscr {L}}^*$$ L ∗ , is as expressive as an extensional variant of it, called $${\mathscr {L}}$$ 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 $${\mathscr {L}}$$ L and $${\mathscr {L}}^*$$ L ∗ , we show that the intensional language $${\mathscr {L}}^*$$ L ∗ is not as expressive as the extensional language $${\mathscr {L}}$$ 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 $${\mathscr {L}}^*$$ L ∗ does become as expressive as $${\mathscr {L}}$$ L when we add Cresswell’s operator of universal modality $$\square $$ □ to $${\mathscr {L}}^*$$ 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.