Labelled Calculi for Lukasiewicz Logics

In this paper, we define new decision procedures for Łukasiewicz logics. They are based on particular integer-labelled hypersequents and of logical proof rules for such hypersequents. These rules being proved strongly invertible our procedures naturally allow one to generate countermodels. From these results we define a “merge”-free calculus for the infinite version of Łukasiewicz logic and prove that it satisfies the sub-formula property. Finally we also propose for this logic a new terminating calculus by using a focusing technique.