Subnormal Modal Logics for Knowledge Representation

Grigori Schwarz, Miroslaw Truszczynski

Several widely accepted modal nonmonotonic logics for reasoning about knowledge and beliefs of rational agents with introspection powers are based on strong modal logics such as KD45, S4.4, S4F and S5. In this paper we argue that weak modal logics, without even the axiom K and, therefore, below the range of normal modal logics, also give rise to useful nonmonotonic systems. We study two such logics: the logic N, containing propositional calculus and necessitation but no axiom schemata for manipulating the modality, and the logic NT - the extension of N by the schema T. For the nonmonotonic logics N and NT we develop minimal model semantics. We use it to show that the nonmonotonic logics N and NT are at least as expressive as autoepistemic logic, reflexive autoepistemic logic and default logic. In fact, each can be regarded as a common generalization of these classic nonmonotonic systems. We also show that the nonmonotonic logics N and NT have the property of being conservative with respect to adding new definitions, and prove that computationally they are equivalent to autoepistemic and default logics.

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