Review: Richard Montague, Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability

In the present piece we defend predicate approaches to modality, that is approaches that conceive of modal notions as predicates applicable to names of sentences or propositions, against the challenges raised by Montague’s theorem. Montague’s theorem is often taken to show that the most intuitive modal principles lead to paradox if we conceive of the modal notion as a predicate. Following Schweizer (J Philos Logic 21:1–31, 1992) and others we show this interpretation of Montague’s theorem to be unwarranted unless a further non trivial assumption is made—an assumption which should not be taken as a given. We then move on to showing, elaborating on work of Gupta (J Philos Logic 11:1–60, 1982), Asher and Kamp (Properties, types, and meaning. Vol. I: foundational issues, Kluwer, Dordrecht, pp 85−158, 1989), and Schweizer (J Philos Logic 21:1–31, 1992), that the unrestricted modal principles can be upheld within the predicate approach and that the predicate approach is an adequate approach to modality from the perspective of modal operator logic. To this end we develop a possible world semantics for multiple modal predicates and show that for a wide class of multimodal operator logics we may find a suitable class of models of the predicate approach which satisfies, modulo translation, precisely the theorems of the modal operator logic at stake.

Montague's Theorem and Modal Logic

1. Our interest here is in the relations between the syntactical predicate fi_is true' and the syntactical modal predicates such as '_is necessary', *_is possible', '_is contingent', and *_entails...' That there should be similarities between these predicates is not surprising since the modal predicates are so-called because they deal with what used to be known as 'modes' of truth. '_is necessary', for example, is synonymous with '_is necessarily true'. The alleged pre-analytic view of the correct use of the predicate *_ is true' is that we may accept all instances of the schema

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Gunnar Niemi

Papers

On the existence of a modal antinomy