Actuality, Tableaux, and Two-Dimensional Modal Logics

Erkenntnis 83 (3):403-443 (2018)
Abstract
In this paper we present tableau methods for two-dimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain first-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux modulate over different notions of validity such as local, general, and diagonal, besides being general enough for several two-dimensional logics proposed in the literature. We also motivate the introduction of a new operator into two-dimensional languages and explore some of the philosophical questions raised by it concerning the relations there are between actuality, necessity, and the a priori, that seem to undermine traditional intuitive interpretations of two-dimensional operators.
Keywords modal logic  two-dimensional semantics  a priori  actuality  tableaux  necessity  multidimensional modal logics
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DOI 10.1007/s10670-017-9896-0
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References found in this work BETA

Demonstratives.David Kaplan - 1989 - In Joseph Almog, John Perry & Howard Wettstein (eds.), Themes From Kaplan. Oxford University Press. pp. 481-563.
Epistemic Two-Dimensional Semantics.David J. Chalmers - 2004 - Philosophical Studies 118 (1-2):153-226.
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