Labelled Tableau Calculi Generating Simple Models for Substructural Logics∗

In this paper we apply the methodology of Labelled Deductive Systems to the tableau method in order to obtain a deductive framework for substructural logics which incorporates the facility of model generation. For this special purpose, we propose new labelled tableau calculi TL and TLe for two substructural logics L (essentially the Lambek calculus) and Le (the multiplicative fragment of intuitionistic linear logic). The use of labels makes it possible to generate countermodels in terms of a certain very simple semantics based on monoids, which we call the simple semantics. We show that, given a formula C as input, every nonredundant tableau construction procedure for TL and TLe terminates in finitely many steps, yielding either a tableau proof of C or a finite countermodel of C in terms of the simple semantics. It shows the finite model property for L and Le with respect to the simple semantics.