Assertional Logics and the Frege Hierarchy

In this paper we continue the investigation carried out in Albuquerque et al. (2018) on assertional logics and their relation with the Frege hierarchy, through the notions of relative point-regularity and relative congruence orderability. We provide new characterizations for the classes of logics within the Frege hierarchy under the underlying assumption of assertionality. In particular, an assertional logic \({\mathcal {S}}\) is fully Fregean if and only if the class \({\textsf{Alg}}{\mathcal {S}}\) is congruence orderable. Moreover, an assertional logic \({\mathcal {S}}\) is protoalgebraic if and only the class \({\textsf{Alg}}{\mathcal {S}}\) is point-regular. Finally, we introduce a new notion of relative strong congruence orderability and prove that the class \({\textsf{Alg}}{\mathcal {S}}\) satisfies this property if and only if the intrinsic variety \(\mathbb {V}({\mathcal {S}})\) is congruence orderable. As a consequence, we prove a sufficient condition for the variety problem in AAL.