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First-order non-monotonic modal logics

Kaminski M., Rey G.

Fundamenta Informaticae

2000

Vol. 42, nr 3,4

303-333

In this paper we introduce first-order non-monotonic modal logic and study its properties. Our definition is semantical and is based on the intuition similar to that lying behind the definition of first-order default logic. Thus, our definition of first-order non-monotonic modal logic well complies with that of first-order default logic and circumscription, providing a substantial evidence for its acceptance. In addition, our definition implies all properties of propositional non-monotonic modal logic, which supplies an additional support for its adequacy.

Many-valued modal non-monotonic reasoning: sequential stable sets and logics with linear truth spaces

Koutras C.D., Koletsos G., Zachos S.

Fundamenta Informaticae

1999

Vol. 38, Nr 3

281-324

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. Non-monotonic extensions of these logics are introduced with a fixpoint construction a la McDermott & Doyle and employ sequential belief sets as epistemic states. 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.