Affiliations: Department of Electrical and Computer Engineering, CS Division, National Technical University of Athens, 157 73 Zographou, Athens, Greece. [email protected] | General Department, Division of Mathematics, National Technical University of Athens, 157 73 Zographou, Athens, Greece. [email protected] | CIS Department, Brooklyn College, City University of New York, 2900 Bedford Avenue, NY 11210, USA. [email protected]
Note: [] Address for correspondence: Graduate Program in Logic, Algorithms & Computation, Department of Mathematics, University of Athens, Panepistimioupolis 157 73 Zographou, Athens, Greece
Note: [] Address for correspondence: General Department, Division of Mathematics, National Technical University of Athens, 157 73 Zographou, Athens, Greece
Note: [] Address for correspondence: CIS Department, Brooklyn College, City University of New York, 2900 Bedford Avenue, NY 11210, USA
Abstract: A family of many-valued modal logics which correspond to possible-worlds models with many-valued accessibility relations, has been recently proposed by M. Fitting [7, 8]. Non-monotonic extensions of these logics are introduced with a fixpoint construction à la McDermott & Doyle and employ sequential belief sets as epistemic states [9]. In this paper we take a logical investigation of many-valued modal non-monotonic reasoning in Fitting's formal framework. We examine the notion of MV-stable sets which emerges as a sequential many-valued analog of Stalnaker-Moore stable sets and prove that several attractive epistemic properties are essentially retained in the many-valued setting, esp. when focusing on a syntactically simple epistemic fragment of MV-stable sets. We show that MV-stable sets are always closed under S4 consequence and identify three sufficient conditions for capturing axioms of negative introspection. Also, the relation of MV-stable sets to many-valued analogs of classical S5 models and to many-valued extensions of universal models is discussed. Finally, we pay special attention to the subclass of logics built on linear Heyting algebras and show that inside this subclass, the situation is very similar - in many respects - to the machinery devised by W. Marek, G. Schwarz and M. Truszczyński. In particular, the normal fragments of the two important classical ranges of modal non-monotonic logics remain intact: many-valued autoepistemic logic is captured by any non-monotonic logic in K5 – KD45 and many-valued reflexive autoepistemic logic corresponds to KTw5 – Sw5.
