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1

On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

Koutras C.D., Zikos Y.

Fundamenta Informaticae

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2009

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Vol. 96, nr 1/2

111-125

EN

An important question in modal nonmonotonic logics concerns the limits of prepositional definability for logics of the McDermott-Doyle family. Inspired by this technical question we define a variant of autoepistemic logic which provably corresponds to the logic of the McDermott-Doyle family that is based on the modal axiom p5 :[formula]. This axiomis a natural weakening of classical negative introspection restricting its scope to possible facts. It closely resembles the axiom w5 :[formula] which restricts the effect of negative introspection to true facts. We examine p5 in the context of classical possible-worlds Kripke models, providing results for correspondence, completeness and the finite model property. We also identify the corresponding condition for p5 in the context of neighbourhood semantics. Although rather natural epistemically, this axiom has not been investigated in classical modal epistemic reasoning, probably because its addition to S4 gives the well-known strong modal system S5.

2

Weaker Axioms, More Ranges

Koutras C.D., Peppas P.

Fundamenta Informaticae

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2002

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Vol. 51, nr 3

297-310

EN

In the family of many-valued modal languages proposed by M. Fitting in 1992, every modal language is based on an underlying Heyting algebra which provides the space of truth values. The lattice of truth values is explicitly represented in the language by a set of special constants and this allows for forming weak, generalized, many-valued analogs of all classical modal axioms. Weak axioms of this kind have been recently investigated from the canonicity, completeness and correspondence perspective. In this paper, we provide some results on the effect of adopting weak versions of the axioms D, T, 4, 5 and w5 in the family of many-valued modal non-monotonic logics, a` la McDermott and Doyle. For many-valued modal languages built on finite chains, we extend the results by proving two quite general range theorems. We then hint on the relation between the modal non-monotonic logics obtained: we prove that there exist ranges which selectively pick out some of the expansions produced by the many-valued autoepistemic logics, actually the ones with a confidence-bounded set of beliefs. However, an exact characterization of the relation between the various ranges created by the weak many-valued modal axioms still remains to be explored.

3

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

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

Fundamenta Informaticae

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1999

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Vol. 38, Nr 3

281-324

EN

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.