In Nicola Olivetti & Rineke Verbrugge (eds.), Advances in Modal Logic, Vol. 13. London: College Publications (2020)

Yifeng Ding
University of California, Berkeley
Wesley H. Holliday
University of California, Berkeley
In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics—the atomicity of the algebra of propositions—but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic.
Keywords modal logic  Kripke incompleteness  Kripke inconsistency  atomic inconsistency  possibility semantics  algebraic semantics  Kaplan's paradox
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References found in this work BETA

On the Plurality of Worlds.David Lewis - 1986 - Wiley-Blackwell.
A Problem in Possible Worlds Semantics.David Kaplan - 1995 - In Walter Sinnott-Armstrong, Diana Raffman & Nicholas Asher (eds.), Modality, Morality and Belief: Essays in Honor of Ruth Barcan Marcus. Cambridge University Press. pp. 41-52.
Lectures on Boolean Algebras.Paul R. Halmos - 1966 - Journal of Symbolic Logic 31 (2):253-254.
Intensional Logics Without Interative Axioms.David K. Lewis - 1974 - Journal of Philosophical Logic 3 (4):457-466.
Post Completeness in Congruential Modal Logics.Peter Fritz - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. College Publications. pp. 288-301.

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Elusive Propositions.Gabriel Uzquiano - 2021 - Journal of Philosophical Logic 50 (4):705-725.

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