Abstract
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.
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Notes
See Skyrms (1978) for remarks along these lines.
Asher and Kamp (1989) equally provide a possible world semantics for the predicate setting but their framework is slightly different and only allows for one modal notion.
A sentence is a formula with no free variable.
Here, \(sub^{\bullet}(\cdot,\cdot)\) represents the binary substitution function that takes the Gödel number of a formula ϕ with exactly one free variable and the Gödel number of some expression \(\zeta\) of the language as arguments and provides the Gödel number of the sentence that results from ϕ when the free variable is replaced by the numeral of the Gödel number of \(\zeta\) as an output.
In a way this had also been the strategy of Skyrms (1978) although Skyrms also blocks the diagonalization of the modal predicate by syntactic means.
Our presentation closely follows Gupta (1982), pp. 9–15.
‘\({Term_{\mathcal{L}_{QT}}}\)’ and ‘\({Frml_{\mathcal{L}_{QT}}}\)’ denote the set of terms and, respectively, of formulas of the language \(\mathcal{L}_{QT}\). Similarly, ‘\({Sent_{\mathcal{L}_{QT}}}\)’ stands for the set of sentences of \(\mathcal{L}_{QT}\).
σ needs to respect the interpretation T as, e.g., a sentence of the form ∀xTx is of quotation degree 0.
In possible world semantics for modal operator logic the logic S5 characterizes a frame based on an equivalence relation. It can be shown that in this case accessibility relation can be dropped and instead of quantification simpliciter over a set of possible worlds. Cf. Hughes and Cresswell (1996) and Blackburn et al. [3] for more on modal logic and possible world semantics.
Since the domains of the premodels may vary we decided to avoid complication and to consider subsets of the set of sentences of \(\mathcal{L}_{QM}\) as interpretations of the truth predicate only.
We assume \(\bigcap\) to be an operation on P(Sent QM ) and thus, in particular, \(\bigcap\emptyset=Sent_{QM}\).
To see this, note that the interpretations of the truth and modal predicates cannot be disjoint since all the tautologies and the \(\mathcal{L}\)-truths are in the interpretation of the truth predicate in every model. If there exists a sentence which is not in the interpretation of the one predicate but the other, then there exist denumerable many sentences of this kind: take all the conjunctions of tautologies with this sentence. This explains the second observation. A parallel argument establishes the third.
Note that as a consequence of the uniqueness of fixed-points for every frame F and proper premodel \(M\in{\boldsymbol{\mathcal{W}}}\) there exists only one proper model which is induced by M relative to this frame.
We use calligraphic letters to denote proper models.
The argument which appears at least implicitly in the literature can be glossed as follows. The fact that you have shown that modal logic, i.e. the principles of modal operator logic, can be upheld in a predicate setting just shows that the modal predicates under consideration are just operators in disguise. This kind of argument is maybe most explicitly stated in the writings of Reinhard (1980) and Grim (1993).
See Stern (2012) for a more in depth discussion of the distinction between modal operators and predicates, and arguments in favor of expressively rich frameworks for modality.
References
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Acknowledgments
I wish to thank Catrin Campbell-Moore, Fabrice Correia, Martin Fischer, Hannes Leitgeb and Karl-Georg Niebergall for helpful discussions of this material. The research was supported by the DFG project “Syntactical Treatments of Interacting Modalities” which is hosted by the Munich Center for Mathematical Philosophy which in turn enjoys support from the Alexander von Humboldt Foundation.
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Stern, J. Montague’s Theorem and Modal Logic. Erkenn 79, 551–570 (2014). https://doi.org/10.1007/s10670-013-9523-7
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DOI: https://doi.org/10.1007/s10670-013-9523-7