Béziau developed the paraconsistent logic $\mathbb{Z}$, which is definitionally equivalent to the modal logic $\mathbb{S}5$, and gave an axiomatization of the logic $\mathbb{Z}$: the system HZ. Omori and Waragai proved that some axioms of HZ are not independent and then proposed another axiomatization for $\mathbb{Z}$ that includes two inference rules and helps to understand the relation between $\mathbb{S}5$ and classical propositional logic. In the present paper, we analyze logic $\mathbb{Z}$ in detail; in the process we also construct a family of paraconsistent logics that are characterized by different properties that are relevant in the study of logics.