Classical BI: Its Semantics and Proof Theory

TL;DR

Classical BI (CBI) extends bunched logic by incorporating classical behavior into multiplicative connectives, with a unique resource model structure and a display calculus proof system, establishing foundational semantics and proof-theoretic results.

Contribution

This paper introduces CBI, a novel bunched logic with classically behaving multiplicative connectives and a display calculus, expanding the theoretical framework of resource-sensitive logics.

Findings

CBI's semantics require resources to have unique duals.

A display calculus for CBI is formulated and shown to be complete.

Fundamental properties of CBI, including completeness, are established.

Abstract

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.