Axiomatisations of the Genuine Three-Valued Paraconsistent Logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$

Abstract

Genuine Paraconsistent logics $$\mathbf {L3A}$$ and $$\mathbf {L3B}$$ were defined in 2016 by Beziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernandez-Tello et al, provide implications for both logics and define the logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ . In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ satisfy a restricted version of the Substitution Theorem, and that both of them are maximal with respect to Classical Propositional Logic. To conclude we make some comparisons between $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ and among other logics, for instance $${\mathbf {Int}}$$ and some $${\mathbf {LFI}}$$ s.