From Axioms to Structural Rules, Then Add Quantifiers

We survey recent developments in the program of generating proof calculi for large classes of axiomatic extensions of a non-classical logics by translating each axiom into a set of structural rules, starting from a base calculus. We will introduce three proof formalisms: the sequent calculus, the hypersequent calculus and the display calculus. The calculi that are obtained derive exactly the theorems of the logic and satisfy a subformula property which ensures that a proof of a theorem only contains statements that are ‘related to the conclusion’. These calculi can be used as a starting point for developing automated reasoning systems and to facilitate a proof-theoretic investigation of the logic (e.g. to prove interpolation, consistency, decidability, complexity). In the final section we discuss how first-order quantifiers may be added to these propositional calculi. Much of the content here can be found in an extended form in the survey paper [10]. The main purpose of this abstract is to provide a concise ‘hands-on’ description of the methods. We present a subjective selection of problems while directing the reader to the references for a more exhaustive exposition.