Logical Foundations of Evidential Reasoning with Contradictory Information

Abstract

Inconsistent or contradictory information is quite common in modern information technology such as the Web or unstructured databases. In this paper, we employ two levels of epistemic logics to provide logical foundations for evidential reasoning with this kind of information. The first-level logic is the well-known Belnap–Dunn four-valued logic. This logic provides a formalism for reasoning about both incomplete and contradictory information. In addition to the two standard Boolean truth values T and F, there are two new values: N and B. They are used to designate incomplete and contradictory information, respectively. The four-valued logic is externally epistemic in the sense that the truth values are intended to reflect what the agents may have been informed about and are passed over to the agents from the external environment. By using the semantics for this logic, we enrich Carnap’s universe for consistent information by replacing standard possible worlds with states, set-ups or situations where a proposition may be both true and false. We shall call such a universe a Belnap–Dunn universe. The second-level logic is epistemic logic S5. When the information is uncertain and imprecise, it usually fails to provide probability values for every subset of the Belnap–Dunn universe. Probabilities are defined only on those subsets which are known with certainty. We employ epistemic logic S5 to distinguish those known subsets and to characterize the notion that such known part of the information improves our knowledge by reducing the scope of possible valid states. S5 is internally epistemic in the sense that the knowledge is determined by the agents. Probabilistic reasoning with the combination of the four-valued logic and epistemic logic S5 is nothing but evidential reasoning over bilattices or de Morgan lattices.