Axiomatisations of the Genuine Three-Valued Paraconsistent Logics 𝐋3𝐀_𝐆 and 𝐋3𝐁_𝐆

Genuine Paraconsistent logics 𝐋3𝐀 and 𝐋3𝐁 were defined in 2016 by Béziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernández-Tello et al, provide implications for both logics and define the logics 𝐋3𝐀_𝐆 and 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 𝐋3𝐀_𝐆 and 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 𝐋3𝐀_𝐆 and 𝐋3𝐁_𝐆 and among other logics, for instance 𝐈𝐧𝐭 and some 𝐋𝐅𝐈 s.