Possibilistic vs. Relational Semantics for Logics of Incomplete Information

This paper proposes an extension of the MEL logic to a language containing modal formulae of depth 0 or 1 only. MEL is a logic of incomplete information where an agent can express both beliefs and explicitly ignored facts, that only uses modal formulae of depth 1, and no objective ones. The extended logic, called MEL has the same axioms as, and is in some sense equivalent to, S5 with a restricted language, but with the same expressive power. The semantics is not based on Kripke models with equivalence relations, but on pairs made of an interpretation (representing the real state of facts) and a non-empty set of possible interpretations (representing an epistemic state). Soundness and completeness are established. We provide a rationale for using our approach when an agent reasons about what is known of the epistemic state of another agent and compares it with what is known about the real world. Our approach can be viewed as an alternative to the basic epistemic logic not concerned with introspection. We discuss the difference with S5 used as a logic for rough sets, and the similarity with some previous non-monotonic logics of knowledge is highlighted.