  Non-classical negations may fail to be contradictory-forming operators in
more than one way, and they often fail also to respect fundamental meta-logical
properties such as the replacement property. Such drawbacks are witnessed by
intricate semantics and proof systems, whose philosophical interpretations and
computational properties are found wanting. In this paper we investigate
congruential non-classical negations that live inside very natural systems of
normal modal logics over complete distributive lattices; these logics are
further enriched by adjustment connectives that may be used for handling
reasoning under uncertainty caused by inconsistency or undeterminedness. Using
such straightforward semantics, we study the classes of frames characterized by
seriality, reflexivity, functionality, symmetry, transitivity, and some
combinations thereof, and discuss what they reveal about sub-classical
properties of negation. To the logics thereby characterized we apply a general
mechanism that allows one to endow them with analytic ordinary sequent systems,
most of which are even cut-free. We also investigate the exact circumstances
that allow for classical negation to be explicitly defined inside our logics.
