Calculi for Basic Atomic Logics
Résumé
Atomic logics are based on Dunn’s gaggle theory and generalize modal logic and the
Lambek calculus. Sound and complete Hilbert, display and sequent calculi for basic atomic logics with a Kripke-style relational semantics are introduced. These calculi can be automatically computed from the definition of the connectives defining a basic atomic logic. Sufficient conditions for the cut admissibility of these calculi are given based on the shape of the connectives considered. Also, a novel Hilbert axiomatization of modal logic is found out by applying our general results to modal logic. Finally, we prove that basic atomic logics are all in PSPACE and compact.
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