Abstract
ABSTRACTIn 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 BCI and stronger systems, 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 logic is a fragment of a corresponding sorted, residuated 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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DOI 10.1080/11663081.2019.1703469
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References found in this work BETA

Collected Works.Kurt Gödel - 1986 - Oxford University Press.
The Algebra of Topology.J. C. C. Mckinsey & Alfred Tarski - 1944 - Annals of Mathematics, Second Series 45:141-191.

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Game-Theoretic Semantics for Non-Distributive Logics.Chrysafis Hartonas - 2019 - Logic Journal of the IGPL 27 (5):718-742.

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