Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified (...) only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages. (shrink)

While standard first-order modal logic is quite powerful, it cannot express even very simple sentences like “I could have been taller than I actually am” or “Everyone could have been smarter than they actually are”. These are examples of cross-world predication, whereby objects in one world are related to objects in another world. Extending first-order modal logic to allow for cross-world predication in a motivated way has proven to be notoriously difficult. In this paper, I argue that the standard accounts (...) of cross-world predication all leave something to be desired. I then propose an account of cross-world predication based on quantified hybrid logic and show how it overcomes the limitations of these previous accounts. I will conclude by discussing various philosophical consequences and applications of such an account. (shrink)

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. (shrink)

Besides their ordinary cardinal and proportional meanings, many and few have been argued to allow for a ‘reverse proportional’ reading. This reading has later been characterised in two opposite directions: Cohen’s reading where the proportion \ matters and Herburger’s where it does not. We develop a compositional analysis that derives the correct truth conditions for both characterisations of Westerståhl-style sentences while maintaining conservativity, assuming a standard syntax/semantics mapping and reducing their context-dependence to mechanisms independently needed for degree constructions in general. (...) In a nutshell, mirroring the decomposition of other degree expressions like tall, many is decomposed into the parametrized determiner many and the operator POS, where POS combines with a contextually salient comparison class C matching the alternatives triggered by some element X\ in the sentence. Non-reverse readings obtain when X\ is external to the original host NP and reverse readings when X\ is internal to the host NP. Cohen’s truth conditions for Westerståhl-style sentences are derived as a reverse proportional reading and Herburger’s interpretation as a sub-case of the non-reverse cardinal reading. (shrink)

ABSTRACT Backwards-looking operators Saarinen, E. [1979. “Backwards-Looking Operators in Tense Logic and in Natural Language.” In Essays on Mathematical and Philosophical Logic, edited by J. Hintikka, I. Niiniluoto, and E. Saarinen, 341–367. Dordrecht: Reidel] that have the material in their scope depend on higher intensional operators, are known to increase the expressivity of some intensional languages and have thus played a central role in debates about approaches to intensionality in terms of implicit parameters vs. variables explicitly quantifying over them. The (...) current contribution takes a look at these operators from a type-logical perspective. It is shown that extending Gallin's translation from intensional type logic to two-sorted type theory so as to include a version of Yanovich's backwards-looking operators, does not increase the expressive power of formulae with exclusively intensional parameters. The result, which makes use of a theorem by Zimmermann : 65–77]), is illustrated by pertinent examples from the literature. The paper closes by indicating alternative strategies of incorporating backwards-looking operators. (shrink)

Reformulation strategies for solving Fitch’s paradox of knowability date back to Edgington. Their core assumption is that the formula \, from which the paradox originates, does not correctly express the intended meaning of the verification thesis, which should concern possible knowledge of actual truths, and therefore the contradiction does not represent a logical refutation of verificationism. Supporters of these solutions claim that can be reformulated in a way that blocks the derivation of the paradox. Unfortunately, these reformulation proposals come with (...) other problems, on both the logical and the philosophical side. We claim that in order to make the reformulation idea consistent and adequate one should analyze the paradox from the point of view of a quantified modal language. An approach in this line was proposed by, among others, Kvanvig but was not fully developed in its technical details. Here we approach the paradox by means of a first order hybrid modal logic, a tool that strikes us as more adequate to express transworld reference and the rigidification needed to consistently express this idea. The outcome of our analysis is ambivalent. Given a first order formula we are able to express the fact that it is knowable in a way which is both consistent and adequate. However, one must give up the possibility of formulating as a substitution free schema of the kind \. We propose that one may instead formulate by means of a recursive translation of the initial formula, being aware that many alternative translations are possible. (shrink)