Genuine paracomplete logics

In 2016, Beziau introduces a restricted notion of paraconsistency, the so-called genuine paraconsistency. A logic is genuine paraconsistent if it rejects the laws phi, (sic)phi proves psi and proves (sic)(phi boolean AND (sic) phi In that paper, the author analyzes, among the three-valued logics, which of them satisfy this property. If we consider multiple-conclusion consequence relations, the dual properties of those above -mentioned are proves phi, (sic)phi and (sic)(psi boolean OR (sic)psi) proves. We call genuine paracomplete logics those rejecting the mentioned properties. We present here an analysis of the three-valued genuine paracomplete logics. A very natural twist structures semantics for these logics is also found in a systematic way. This semantics produces automatically a simple and elegant Hilbert-style characterization for all these logics. Finally, we introduce the logic LGP which is genuine paracomplete is not genuine paraconsistent, not even paraconsistent and cannot be characterized by a single finite logical matrix.