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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. [ABSTRACT FROM AUTHOR]