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We look at non-classical negations and their corresponding adjustmentconnectives from a modal viewpoint, over complete distributive lattices, andapply a very general mechanism in order to offer adequate analytic proofsystems to logics that are based on them. Defining non-classical negationswithin usual modal semantics automatically allows one to treat equivalentformulas as synonymous, and to have a natural justification for a globalversion of the contraposition rule. From that perspective, our study offers aparticularly useful environment in which negative modalities and theircompanions may be used for dealing with inconsistency and indeterminacy. Afterinvestigating modal logics based on arbitrary frames, we extend the results toserial frames, reflexive frames, functional frames, and symmetric frames. Ineach case we also investigate when and how classical negation may thereby bedefined.

It ain't necessarily so: Basic sequent systems for negative modalities