The equational definability of truth predicates

By a ‘logic’ we mean here a substitution-invariant consequence relation on formulas over an algebraic signature. Propositional logics are obvious examples, but even first order logic can be re-formulated in this way. The notion of an ‘algebraizable’ logic was made precise in the 1980s, mainly by Blok and Pigozzi, who provided intrinsic characterizations of the logics that are indeed algebraizable. One of their characterizations yields a practical strategy for showing that a logic is inherently non-algebraizable. The meaning of ‘algebraizable’ decomposes into two parts. In a slogan,