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 PSPACE-complete and compact.