# Modal Translation of Substructural Logics

## TL;DR

This paper extends modal translation techniques to all substructural logics, including non-distributive systems, enabling classical interpretations and transfer of key logical properties.

## Contribution

It demonstrates that every substructural logic can be embedded into a sorted, residuated modal logic, generalizing previous translations and covering non-distributive cases.

## Key findings

- Every non-distributive logic is a fragment of a sorted, residuated modal logic.
- The translation provides a classical interpretation of non-distributive propositional calculi.
- Results like compactness and decidability transfer seamlessly via the translation.

## Abstract

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.