In memoriam José Arrazola Ramírez (1962–2018) The logic $\textbf{G}^{\prime}_3$ was introduced by Osorio et al. in 2008; it is a three-valued logic, closely related to the paraconsistent logic $\textbf{CG}^{\prime}_3$ introduced by Osorio et al. in 2014. The logic $\textbf{CG}^{\prime}_3$ is defined in terms of a multi-valued semantics and has the property that each theorem in $\textbf{G}^{\prime}_3$ is a theorem in $\textbf{CG}^{\prime}_3$. Kripke-type semantics has been given to $\textbf{CG}^{\prime}_3$ in two different ways by Borja et al. in 2016. In this work, we continue the study of $\textbf{CG}^{\prime}_3$, obtaining a Hilbert-type axiomatic system and proving a soundness and completeness theorem for this logic.
In 2016, Béziau introduces a restricted notion of paraconsistency, the so-called genuine paraconsistency. A logic is genuine paraconsistent if it rejects the laws $\varphi ,\neg \varphi \vdash \psi $ and $\vdash \neg (\varphi \land \neg \varphi )$. 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 $ \vdash \varphi , \neg \varphi $ and $\neg (\psi \vee \neg \psi ) \vdash $. 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.
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