 Not all interesting logics are protoalgebraic. In this section we will briefly discuss four examples of non-protoalgebraic logics: the logic of conjunction and disjunction, positive modal logic, the strict implication fragment of \(\blK\) and Visser’s subintuitionistic logic. All of them are selfextensional. In the next section, we will expound the semantics of abstract logics and generalized matrices that serves to develop a really general theory of the algebraization of logic systems. As we will see, the perspective changes in an important respect from the perspective taken in logical matrix model theory.