Categorial Grammars and Substructural Logics

Substructural logics are formal logics whose Gentzen-style sequent systems abandon some/all structural rules (Weakening, Contraction, Exchange, Associativity). They have extensively been studied in current literature on nonclassical logics from different points of view: as sequent axiomatizations of relevant, multiple-valued and fuzzy logics [73, 44], as so-called resource-sensitive logics for computer science (here linear logics of Girard [42] are most representative), and as type logics for categorial grammars. In this paper we mainly focus on the latter perspective, but several basic ideas are also significant for other approaches. For example, applications of linear logics in linguistics have been proposed in the style of type logics (see Casadio [30], Lambek [61], de Groote and Lamarche [43]), and algebraic models for linear logics (phase-space models) have been constructed in a similar way as powerset models for type logics [12, 21]. In type logics formulae are interpreted as types. In semantic terms, A→ B is a type of functions (procedures) which send inputs of type A to outputs of type B, and A⊗B is a type of pairs (f, g) such that f is of type A and g is of type B. In syntactic terms, A→ B (resp. B ← A) is a type of functional expressions (functors) b which together with any argument a of type A form a complex expression ab (resp. ba) of type B.The scope of possible realizations is huge: from constructivism in mathematics to logics of computation, from combinators and lambda calculus to linear logics, from type theories of Russell and Church to theories of syntax, proposed by Leśniewski, Ajdukiewicz, Chomsky, Curry and Lambek. Connections between logic and grammar is a leitmotive of philosophy of language and many threads of philosophical logic. Categorial grammar whose main components are: (i) an assignment of logical types to lexical atoms (words), (ii) a system of type logic which provides a syntactic description of complex expressions, is a kind of formal grammar especially close to symbolic logic (much more than e.g. context-free grammar and context-sensitive grammar which are purely combinatorial devices). The general possibility of language description by means of logical types and logical relations between them was regarded by Hiż as the doctrine of grammar logicism. It seems to be important that the impact is