In this talk we address the curious phenomenon by which some pairs of different logics defined over the same signature (e.g. the paraconsistent logic $P_1$ and the paracomplete logic $I_1$) admit a Blok-Pigozzi algebraization within the same quasi-variety. An apparent paradox then arises because $I_1$ and $P_1$ admit identical algebraizators, and so they cannot be algebraizable within the same quasi-variety. We will show that the paradox vanishes if we take into account that different presentations of the given logics are used in each case. Finally, we present an isomorphism between the category of algebraizable logics and the category of deductivizable quasi-varieties, which sheds light into the above situation. Our categories are based upon natural notions of morphisms, in contrast to the unusual and restricted notion of morphisms used by Jánossy, Kurucz, and Eiben, and so our result can be considered an improvement of the latter.