The paper by da Costa and Carnielli “On Paraconsistent deontic logic” (1986) introduced the first modal paraconsistent system, C1D, defined by combining the paraconsistent system C1 and the deontic system D. That paper aimed to show how a paraconsistent deontic system could be used as a new approach for dealing with certain kinds of deontic paradoxes. We call anodic the purely positive versions of modal systems, and cathodic any non-classical (w.r.t. negations) version of modal systems. Paradigmatic examples of the latter are modal paraconsistent systems based upon logics of formal inconsistency (LFIs), i.e., based upon paraconsistent systems where the concept of consistency is internalized within the object language, so as to control the “strength” of negation inside the systems. We discuss here some strategies for obtaining completeness to cathodic systems w.r.t possible-translations semantics, as well as some limitative results concerning incompleteness, emphasizing the role of negation in such constructions.