Studia Logica 101 (3):487-503 (2013)

Kai Wehmeier
University of California, Irvine
Benjamin Rin
Utrecht University
We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5
Keywords Modal logic  Actuality  Eliminability  Redundancy   Kripke models  Neighborhood models
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DOI 10.1007/s11225-012-9395-x
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References found in this work BETA

Modal Logic: An Introduction.Brian F. Chellas - 1980 - Cambridge University Press.
Semantical Analysis of Modal Logic I. Normal Propositional Calculi.Saul A. Kripke - 1963 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9 (5‐6):67-96.
Expressive Completeness in Modal Language.Allen Hazen - 1976 - Journal of Philosophical Logic 5 (1):25--46.
The Eliminability of the Actuality Operator in Propositional Modal Logic.Allen Hazen - 1978 - Notre Dame Journal of Formal Logic 19 (4):617-622.

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Two-Dimensional Tableaux.David Gilbert - 2016 - Australasian Journal of Logic 13 (7).

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