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Unlike in classical modal logic, in non-classical modal logics the box anddiamond operators frequently fail to be interdefinable. Instead, these logicsimpose some compatibility conditions which tie the box and diamond operatorstogether and ensure that in terms of Kripke semantics they arise from the sameaccessibility relation. This is, for instance, the case in the intuitionisticmodal logic of Fischer Servi and the positive modal logic of Dunn. In theselogics, however, such compatibility conditions also impose further conditionson the accessibility relation. In this paper, we identify the basiccompatibility conditions which ensure that modal operators in distributivemodal logics arise from a single accessibility relation without imposing anyrestrictions on the relation. As in the distributive logic of Gehrke,Nagahashi, and Venema, we allow for negative box and diamond operators here inaddition to the usual positive ones. Intuitionistic modal logic and positivemodal logic, or more precisely the corresponding classes of algebras, are thenobtained in a modular way by adding certain canonical axioms which we calllocality conditions on top of these basic compatibility conditions.

Compatibility between modal operators in distributive modal logic