A useful four-valued database logic

Recently there has been an effort to solve the problems caused by the infamous NULL in relational databases, by systematically applying Kleene's three-valued logic to SQL. The third truth-value is unknown. In this paper we show that by using a fourth truth-value inconsistent, all the advantages of the three-valued approach can be retained, and that negation can be given a constructive, intuitionistic meaning that allows negative knowledge to be specified in the logic explicitly, without having to resort to extra-logical notions of stratification or to non-monotonic reasoning. The four-valued approach also allows for a computationally efficient treatment of query answering in the presence of inconsistencies. This is in contrast to the computationally intractable repair approach to inconsistency management. From a practical view-point we show that the Cylindric Star Algebra, developed by the authors, is particularly well suited for evaluating First Order queries on four-valued databases, and that the framework of data exchange can smoothly adapted to the four truth-values.