Relational Proof Systems for Modal Logics

Abstract

The purpose of this paper is to give a survey of the relational formalization of modal logics. The paradigm ‘formulas are relations’ leads to the development of a relational logic based on algebras of relations. The logic can be viewed as a generic logic for the representation of nonclassical logics; in particular a broad class of multimodal logics can be specified within its framework. As a consequence, proof systems for the relational logic become a convenient tool for the development of a proof theory for nonclassical logics. The relational logic enables us to represent within a uniform formalism the three basic components of any propositional logical system: syntax, semantics and deduction apparatus. The essential observation, leading to a relational formalization of logical systems, is that a standard relational structure (a Boolean algebra with a monoid) constitutes a common core of a great variety of nonclassical logics. Exhibiting this common core on all the three levels of syntax, semantics and deduction, enables us to create a general framework for representation, investigation and implementation of nonclassical logics.