Labelled Tableau Calculi Generating Simple Models for Substructural Logics
Kazushige Terui
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.