Sequent Calculi with Context Restrictions and Applications to Conditional Logic

By Bjorn Lellmann,
Imperial College London
October 2013


In this thesis we consider generic tools and techniques for the proof-theoretic investigation of not necessarily normal modal logics based on minimal, intuitionistic or classical propositional logic. The underlying framework is that of ordinary symmetric or asymmetric two-sided sequent calculi without additional structural connectives, and the point of interest are the logical rules in such a system. We introduce the format of a sequent rule with context restrictions and the slightly weaker format of a shallow rule. The format of a rule with context restrictions is expressive enough to capture most normal modal logics in the S5 cube, standard systems for minimal, intuitionistic and classical propositional logic and a wide variety of non-normal modal logics. For systems given by such rules we provide sufficient criteria for cut elimination and decidability together with generic complexity results. We also explore the expressivity of such systems with the cut rule in terms of axioms in a Hilbert-style system by exhibiting a corresponding syntactically defined class of axioms along with automatic translations between axioms and rules. This enables us to show a number of limitative results concerning amongst others the modal logic S5. As a step towards a generic construction of cut free and tractable sequent calculi we then introduce the notion of cut trees as representations of rules constructed by absorbing cuts. With certain limitations this allows the automatic construction of a cut free and tractable sequent system from a finite number of rules. For cases where such a system is to be constructed by hand we introduce a graphical representation of rules with context restrictions which simplifies this process. Finally, we apply the developed tools and techniques and construct new cut free sequent systems for a number of Lewis’ conditional logics extending the logic V. The systems yield purely syntactic decision procedures of optimal complexity and proofs of the Craig interpolation property for the logics at hand.