Equivalential and algebraizable logics (Q2563460)

The author investigates the process of algebraization of the so-called equivalential and finitely equivalential logics. His approach is based on matrix semantics. In the paper, a logic need not be finitary (i.e., have only finitary rules). As to algebraizability, the author distinguishes between finitely algebraizable logics (i.e. those admitting a finite set of equivalence formulas; finitary finitely algebraizable logics are precisely the logics algebraizable in the sense of \textit{W. J. Blok} and \textit{D. Pigozzi} [Algebraizable logics, Mem. Am. Math. Soc. 396 (1989; Zbl 0664.03042)]) and possibly infinitely algebraizable, or p.i.-algebraizable, ones. The main result of the paper states that a logic is finitely algebraizable (p.i.-algebraizable) iff it is finitely equivalential (resp., equivalential) and the truth predicate in the reduced matrix models is equivalentially definable. The paper contains the necessary background on equivalential and algebraizable logics. A natural example of an infinitary logic that is p.i.-algebraizable but not finitary algebraizable is presented, and known examples of nonfinitary finitely algebraizable logics are reminded.