Abstract In this work we present two new classes of logics determined by specific properties of the Leibniz operator that expand the Leibniz hierarchy. By doing so, we present a sufficient condition for the commutativity of the Leibniz operator with inverse substitutions/homomorphisms, which gives us a negative answer to the possibility of establishing a precise connection between Leibniz classes and the behavior of the Leibniz operator.

Abstract Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.