Expressive Power and Intensional Operators

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2024年4月24日

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Pablo Cubides Kovacsics,
David Rey

摘要

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 ${{\varvec{actually}}}_n$ ${actually}_{n}$ 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.