We propose a new sequent calculus for bi-intuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cut-elimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proof-search strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for bi-intuitionistic logic. As far as we know, our new calculus is the first sequent calculus for bi-intuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cut-elimination proof, and which can be used naturally for backwards proof-search.

Abstract. Bi-intuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cut-free labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, following S. Negri’s general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search. To describe a proof search procedure, we develop a more algorithmic version that also allows for counter-model extraction from a failed proof attempt. 1

Abstract. Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusion-left inferences to be singleconclusion resp. single-assumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Goré et al., where a complete class of cuts is encapsulated into special “unnest ” rules and (3) a cut-free labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standard-style sequent calculus. 1