 The protoalgebraic logics that are truth-equational are in fact the weakly algebraizable logics studied already in Czelakowski &amp; Jansana 2000. Every algebraizable logic is weakly algebraizable. In fact, the algebraizable logics are the equivalential logics that are truth-equational. But not every weakly algebraizable logic is equivalential. An example is the  logic determined by the ortholattices, namely by the class of the matrices \(\langle \bA, \{1\} \rangle\) where \(\bA\) is an ortholattice and 1 is its greatest element (see Czelakowski &amp; Jansana 2000 and Malinowski 1990).