Tutorials

Abstract.Logics can be interpreted in various ways. Relational structures are quite versatile in providing semantics for non-classical logics. First, I will briefly overview S. A. Kripke's semantics for normal modal logic. Then, I outline the ternary relational semantics for relevance logics created by R. K. Meyer and R. Routley. A general theory of relational semantics based on the notion of residuation (called gaggle theory) was developed by J. M. Dunn. Time permitting, I will mention general frames and topological duality theory, correspondence theory and canonicity too.

Abstract.

Abstract. After a gentle introduction to proof systems, we will show local versions of ordinary sequent rules. This allows to modularly define calculi for a number of logics, including (classical) simply dependent multimodal logics, and (classical) non-normal modal logics.

We then show how to obtain a local system for linear logic, and propose a general framework for modularly describing sub-structural systems combining, coherently, the sub-structural behaviours inherited from LL with simply dependent multimodalities. This class of systems includes linear, elementary, affine, bounded and subexponential linear logics and extensions of multiplicative additive linear logic (MALL) with normal modalities, as well as general combinations of them.

From the theoretical point of view, we give a uniform presentation of modal and sub-structural logics featuring different axioms for their modal operators. From the practical point of view, the results presented in this tutorial lead to a generic way of constructing theorem provers for different logics, all of them based on the same grounds. This opens the possibility of using the same logical framework for reasoning about all such logical systems.