We study a modal language for negative operators — an intuitionistic-like negation and its paraconsistent dual — added to (bounded) distributive lattices. For each non-classical negation an extra operator is introduced in order to allow for standard logical inferences to be restored, whenever appropriate. We characterize the minimal normal logic and a few other basic logics with such negative modalities and their companions. The talk reports on joint work with Adriano Dodó.