Paraconsistency, paracompleteness, Gentzen systems, and trivalent semantics

A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form, or of the form, and all the active formulas of its premises belong to the set. In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion for such systems, and show that a quasi-canonical system of the type we investigate is coherent iff it is strongly paraconsistent or strongly paracomplete (in a sense defined in the paper), iff it has a trivalent, non-deterministic semantics of a special type (also defined in the paper) for which it is sound and complete. Finally, we determine when a system of this sort admits cut-elimination, and provide a simple procedure for transforming one which does not into one which does.