8 A Structural Property on Modal Frames Characterizing Default

We show that modal logics characterized by a class of frames satisfying the insertion property are suitable for Reiter's default logic. We reene the canonical x point construction deened by Marek, Schwarz and Truszczy nski for Reiter's default logic and thus we address a new paradigm for nonmonotonic logic. In fact, diierently from the construction deened by these authors, we show that suitable modal logics for such a construction must indeed contain KD4. When reeexivity is added to the modal logic used for the x point construction then we come to the Marek Schwarz and Truszczy nski framework for Reiter's default logic. Our framework, in fact, is appropriate also to the family of modal logics in between S4 and S4f. If, instead, reeexivity is dropped, then we show that a new family of modal logics is gained, namely the modal logics in between KD4 and KD4Z. The upper bound can be extended to the modal logic KD4LZ whenever the propositional language taken into account is nite.