On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

Abstract

An important question in modal nonmonotonic logics concerns the limits of propositional 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 themodal axiom p5: ◊p⊃ (¬sp⊃s¬sp). This axiomis a naturalweakening of classical negative introspection restricting its scope to possible facts. It closely resembles the axiom w5: p⊃ (¬sp⊃s¬sp) 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.