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)

Starting from an arbitrary MTL-chain, two constructions are considered: the MacNeille completion of the underlying order with suitable extensions of the other operations, and a finite embeddability construction. The preservation of properties via these constructions is investigated, that is, if a property that holds in the initial MTL-chain also holds in the constructed MTL-chain. General syntactic descriptions are given of terms s and t for which the inequality s ≤ t is preserved.

Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic (MALL) and for affine logic (LLW), i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL (which we call IMALL), and intuitionistic LLW (which we call ILLW). In addition, we shall show the finite model property for contractive linear logic (LLC), i.e., linear logic with contraction, and for its intuitionistic version (ILLC). (...) The finite model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL - -systems except FL c and GL - c of Ono [11], that will settle the open problems stated in Ono [12]. (shrink)

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)