There are numerous strengthenings of this notion in the literature, which are often confused, the usually mistakes being that finitary of L implies that L is an elementary class2 (a counterexample is given in [2]) or vice versa (a counterexample is given in [4]). Moreover, the relation of these two notions with the finiteness of ρ (called finite algebraizability) is another usual source of confusions. The goal of this talk is to clarify these confusions by considering the overlooked condition of finiteness of τ . We show that by combining these four properties we obtain 7 distinct classes of logics (the smallest class coinciding with that of B–P algebraizable logics [1]). Then we add two more well-studied properties: regularity of algebraization (a special requirement for τ ) and algebraic implicativeness of L (a special requirement for ρ). We eventually obtain a hierarchy of 17 classes logics of algebraizable logics and show their separation examples.

Peer reviewed