In an article dating back in 1992, Kosta Do\v{ s}en initiated a project of modal translations in substructural logics, aiming at generalizing the well-known G\"{ o}del-McKinsey-Tarski translation of intuitionistic logic into { \bf S4}. Do\v{ s}en's translation worked well for (variants of) { \bf BCI} and stronger systems ({ \bf BCW}, { \bf BCK}), but not for systems below { \bf 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, L\"{ o}wenheim-Skolem property and decidability.
Categories: math.LO