Semantical Analysis of the Logic of Bunched Implications

TL;DR

This paper introduces a novel, model-independent approach to proving soundness and completeness for the logic of Bunched Implications by reasoning directly with validity through eigenworlds and a sequent calculus.

Contribution

It presents a new method that bypasses traditional truth-in-model semantics, using reductive logic and eigenworlds to establish semantic properties of BI and related logics.

Findings

Develops a sequent calculus for a meta-logic capturing BI semantics

Proves soundness and completeness of BI using the new approach

Applies the method to IPL and MILL without negation

Abstract

We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a meta-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the object-logic. The method proceeds through the perspective of reductive logic (as opposed to the more traditional paradigm of deductive logic), using the space of reductions as a medium for showing the behavioural equivalence of reduction in the sequent calculus for the object-logic and in the validity calculus. Rather than study the technique in general, we illustrate it for the logic of Bunched Implications (BI), thus IPL and MILL (without negation) are also treated. Intuitively, BI is the free combination of intuitionistic propositional logic and multiplicative intuitionistic linear logic, which renders its meta-theory is quite complex. The literature on BI contains many similar, but ultimately different, algebraic structures and satisfaction relations that either capture only fragments of the logic (albeit large ones) or have complex clauses for certain connectives (e.g., Beth's clause for disjunction instead of Kripke's). It is this complexity that motivates us to use BI as a case-study for this approach to semantics.