This talk will look at non-classical negations and their corresponding adjustment connectives from a modal viewpoint, over complete distributive lattices, and apply a very general mechanism in order to offer adequate analytic proof systems to logics that are based on them. Defining non-classical negations within usual modal semantics automatically allows one to treat equivalent formulas as synonymous, and to have a natural justification for a global version of the contraposition rule. From that perspective, our study offers a particularly useful environment in which negative modalities and their companions may be used for dealing with inconsistency and indeterminacy. After investigating modal logics based on arbitrary frames, we extend the results to serial frames, reflexive frames, and functional frames. In each case we also investigate when and how classical negation may thereby be defined. This reports on joint work with Ori Lahav and Yoni Zohar.