Modal Nonmonotonic Logic with Restricted Application of the Negation as Failure to Prove Rule 1

Abstract

In the paper we study a family of modal nonmonotonic logics closely related to the family of modal nonmonotonic logics proposed by McDermott and Doyle. For a modal logic $\mathcal{S}$ and a fixed collection of formulas X we introduce the notion of an ( $\mathcal{S}$ , X)-expansion. We restrict to modal logics which have a complete Kripke semantics. We study the properties of ( $\mathcal{S}$ , X)-expansions and show that in many respects they are analogous to the properties of $\mathcal{S}$ -expansions in nonmonotonic modal logics of McDermott and Doyle.