Modal translation of substructural logics

Abstract

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalizing the wellknown 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, Löwenheim-Skolem property and decidability.intuitionistic formulas. Grzegorczyk [32] showed that adding to S4 the axiom (G) ◻(◻(p→◻p)→p)→p, the (now known as) Grzegorczyk system Grz = S4+G is strictly stronger than S4, it is not contained in S5 and the same result about the GMT-translation of IL into Grz is provable. This was also shown to be true for McKinsey's system S4.1 = S4+(◻ ϕ → ◻ ϕ). This quickly led to the investigation of logics, variably called super-intuitionistic, or intermediate logics between IL and CPL and their modal companions, i.e. modal logics containing S4, cf. [5,12,21,47,48], in some cases logics between S4 and S5 [16], which culminated in the Blok-Esakia theorem, independently proven by Blok [8] and Esakia [21]. The topological interpretation of S4 [49-51], interpreting the box operator as an interior operator, provided furthermore the background for investigating dualities between modal companions of superintuitionistic logics and topological spaces [5]. In all of the above cases distribution is assumed and topological dualities investigated are based on the Priestley [59] duality for distributive lattices, or on the Esakia [20-22] duality for Heyting algebras.For logics without distribution, Goldblatt [29] provided a translation of Orthologic O into the B system of modal logic. This quickly led to modal approaches to quantum logic [4,58], as an immediate application. Goldblatt's result is specific to Orthologic and it was Kosta Došen [15] who embarked in a project of extending the GMT translation to the case of substructural logics. Došen's main result is an embedding of BCI (to which he refers as BC), BCK, BCW with intuitionistic negation into S4-type modal extensions of versions of BCI, BCK, BCW, respectively, all equipped with a classical-type negation (satisfying double negation and De Morgan laws). The result is established by proof-theoretic means, but the approach does not apply uniformly to substructural logics at large. Indeed, Došen points out that difficulties arise when dealing with weak systems such as the non-commutative and/or non-associative full Lambek Calculus (N)FL (see [15], Introduction, as well as end of Section 4).A paralle...