Classical BI

Abstract

We show how to extend O'Hearn and Pym's logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a non-conservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cut-eliminating proof system for CBI, based on Belnap's display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.