Towards Universal Logic: Gaggle Logics

International audienceA class of non-classical logics called gaggle logics is introduced, based on a Kripke-style relational semantics and inspired by Dunn's gaggle theory. These logics deal with connectives of arbitrary arity and we show that they capture a wide range of non-classical logics. In particular, we list the 96 binary connectives and 16 unary connectives of basic gaggle logic and relate their truth conditions to the non-classical logics of the literature. We establish connections between gaggle theory and group theory. We show that Dunn's abstract law of residuation corresponds to an action of transpositions of the symmetric group on the set of connectives of gaggle logics and that Dunn's families of connectives are orbits of the same action. Other operations on connectives, such as dual and Boolean negation, are also reformulated in terms of actions of groups and their combination is defined by means of free groups and free products. We show how notions of groups arise naturally from our gaggle logics and how gaggle logics can be canonically defined from given groups. Our other main contribution deals with the proof theory of gaggle logics. We show how sound and complete calculi can be systematically computed from any basic gaggle logic with or without Boolean connectives. These calculi are display calculi and we prove that the cut rule can be systematically eliminated from proofs. This allows us to prove that basic gaggle logics are decidable

On Displaying Negative Modalities

We extend Takuro Onishi’s result on displaying substructural negations by formulating display calculi for non-normal versions of impossibility and unnecessity operators, called regular and co-regular negations, respectively, by Dimiter Vakarelov. We make a number of connections between Onishi’s work and Vakarelov’s study of negation. We also prove a decidability result for our display calculus, which can be naturally extended to obtain decidability results for a large number of display calculi for logics with negative modal operators

Display and Hilbert Calculi for Atomic and Molecular Logics

Sound and strongly complete display calculi for basic atomic and molecular logics are introduced with a Kripke-style relational semantics. These logics are based on Dunn's gaggle theory and generalize modal logics. We also provide sound and strongly complete Hilbert calculi for basic atomic logics with a Kripke-style relational semantics. All these calculi can be automatically computed from the definition of the connectives constituting a basic atomic or molecular logic, yet with some restrictions on the class of molecular logics

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

Correspondence Theory for Atomic Logics

We develop the correspondence theory for the framework of atomic and molecular logics on the basis of the work of Goranko & Vakarelov. First, we show that atomic logics and modal polyadic logics can be embedded into each other. Using this embedding, we reformulate the notion of inductive formulas introduced by Goranko & Vakarelov into our framework. This allows us to prove correspondence theorems for atomic logics by adapting their results