We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic (...) logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers ∀ new and ∃ new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels. (shrink)

A formulation of Full Lambek Calculus in the framework of natural deduction is given.

A version of strong negation is introduced into Categorial Grammar. The resulting syntactic calculi turn out to be systems of connexive logic.

We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus ( DFL ) whose algebraic semantics is the class of distributive residuated lattices ( DRL ). The problem was left open in [8, 5]. We use the method of nuclei and quasi–embedding in the style of [10, 1].

A residuated algebra is a generalization of a residuated groupoid; instead of one basic binary operation with residual operations \,/, it admits finitely many basic operations, and each n-ary basic operation is associated with n residual operations. A logical system for RAs was studied in e.g. [6, 8, 15, 16] under the name: Generalized Lambek Calculus GL. In this paper we study GL and its extensions in the form of sequent systems. We prove an interpolation property which allows to replace (...) a substructure of the antecedent structure by a single formula in a provable sequent. Together with model constructions, based on nuclei [13], interpolation leads to proofs of Finite Embeddability Property for different classes of RAs, as e.g. all RAs, distributive lattice-ordered RAs, boolean RAs, Heyting RAs and double RAs. (shrink)

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.

Models for the Lambek calculus of syntactic categories surveyed here are based on frames that are in principle of the same type as Kripke frames for intuitionistic logic. These models are extracted from the literature on models for relevant logics, in particular the ternary relationed models introduced in the early seventies. The purpose of this brief survey is to locate some open completeness problems for variants of the Lambek calculus in the context of completeness results based on various types of (...) ternary relational models. (shrink)