Proof theory for modal logic Sara Negri Department of Philosophy 00014 University of Helsinki, Finland e-mail: sara.negri@helsinki.fi Abstract The axiomatic presentation of modal systems and the standard formula- tions of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generaliza- tions of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective. 1 Introduction In the literature on modal logic, an overall skepticism on the possibility of developing a satisfactory proof theory was widespread until recently. This attitude was often accompanied by a belief in the superiority of model-theoretic methods over proof- theoretic ones. Standard proof systems have been shown insufficient for modal logic: Natural deduction presentations of even the most basic modal logics present difficulties that have been resolved only partially, and the same has happened with sequent calculus. These traditional proof system have failed to meet in a satisfactory way such basic requirements as analyticity and normalizability of formal derivations. Therefore alternative proof systems have been developed in recent years, with a lot of emphasis on their relative merits and on applications. We review first the axiomatic presentations of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic, and the difficulties that emerge with these approaches. We then move to the generalizations of standard proof systems. These include, among others, display calculi, hyperse- quent, and labelled systems. The last ones are presented from a closer perspective; we discuss their methodological, computational, and metatheoretical properties: the way they answer the challenge of extending proof-theoretic semantics to non- classical logics, their expressive power, analyticity, applicability to proof search, the possibility to obtain direct completeness proofs without artificial Henkin-set con- structions, and their use in the solution of problems that usually involve complex model-theoretic constructions such as negative results in correspondence theory and modal embeddings among different logics. All these results can be obtained in an elementary way through methods of proof analysis of labelled sequent calculi. These calculi cover most modal and non-classical logics that permit a relational semantics. 2 Axiom systems The language of propositional modal logic is obtained by adding to the language of propositional logic the two modal operators ✷ and ✸, to form from any given formula 1