Modal translation of substructural logics

Journal of Applied Non-Classical Logics 30 (1):16-49 (2020)
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Abstract

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems.

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Citations of this work

Game-theoretic semantics for non-distributive logics.Chrysafis Hartonas - 2019 - Logic Journal of the IGPL 27 (5):718-742.

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References found in this work

Collected works.Kurt Gödel - 1986 - New York: Oxford University Press. Edited by Solomon Feferman.
Relevance Logic.Michael Dunn & Greg Restall - 1983 - In Dov M. Gabbay & Franz Guenthner (eds.), Handbook of Philosophical Logic. Dordrecht, Netherland: Kluwer Academic Publishers.
The Algebra of Topology.J. C. C. Mckinsey & Alfred Tarski - 1944 - Annals of Mathematics, Second Series 45:141-191.

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