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