Coalgebraic completeness-via-canonicity for distributive substructural logics

We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.Comment: 36 page

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

Corrections and Remarks

This document contains corrections to errors discovered to-date in, and also some remarks upon, both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it