Our purpose is to show a new way of algebraizing logics, called possible-translations algebraic semantics. This new method is able to algebraize some logics known not to be algebraizable (neither in the classical Lindenbaum-Tarski sense, nor even in the extended Blok- Pigozzi sense), as it is the case of certain paraconsistent logics. The method is based on the idea of extending the finitary character of Blok-Pigozzi algebraization by means of possible-translations semantics. We show some concrete examples and discuss related problems.