# Bi-Intuitionistic Logics: a New Instance of an Old Problem

## Rajeev Goré, Australian National University

As anyone who reads the literature on bi-intuitionistic logic will know, the numerous papers by Cecylia Rauszer are foundational but confusing. For example: these papers claim and retract various versions of the deduction theorem for bi-intuitionistic logic; they erroneously claim that the calculus is complete with respect to rooted canonical models; and they erroneously claim the admissibility of cut in her sequent calculus for this logic. Worse, authors such as Crolard, have based some of their own foundational work on these confused and confusing results and proofs. We trace this confusion to the axiomatic formalism of RBiInt in which Rauszer first characterized bi-intuitionistic logic and show that, as in modal logic, RBiInt can be interpreted as two different consequence relations. We remove this ambiguity by using generalized Hilbert calculi, which are tailored to capture consequence relations. We show that RBiInt leads to two logics, wBIL and sBIL, with different extensional and meta-level properties, and that they are, respectively, sound and strongly complete with respect to the Kripkean local and global semantic consequence relations of bi-intuitionistic logic. Finally, we explain where they were conflated by Rauszer.