A Simple Modal Logic for Reasoning about Revealed Beliefs

Abstract

Even though in Artificial Intelligence, a set of classical logical formulae is often called a belief base, reasoning about beliefs requires more than the language of classical logic. This paper proposes a simple logic whose atoms are beliefs and formulae are conjunctions, disjunctions and negations of beliefs. It enables an agent to reason about some beliefs of another agent as revealed by the latter. This logic, called MEL, borrows its axioms from the modal logic KD, but it is an encapsulation of propositional logic rather than an extension thereof. Its semantics is given in terms of subsets of interpretations, and the models of a formula in MEL is a family of such non-empty subsets. It captures the idea that while the consistent epistemic state of an agent about the world is represented by a non-empty subset of possible worlds, the meta-epistemic state of another agent about the former’s epistemic state is a family of such subsets. We prove that any family of non-empty subsets of interpretations can be expressed as a single formula in MEL. This formula is a symbolic counterpart of the Möbius transform in the theory of belief functions.