In this paper, we introduce a novel approach for dealing with databases containing inconsistent information. Considering four-valued logics in the context of OWA (Open World Assumption), a database ∆ is a pair (E, R) where E is the extension and R the set of rules. In our formalism, the set E is a set of pairs of the form ϕ, v where ϕ is a fact and v is either t, or b, or f (meaning respectively true, inconsistent or false), given that unknown facts are not stored. Moreover the rules extend Datalog neg rules allowing their heads to be a negative atom. We then define the notion of model of such a database, we show how to compute one particular model called semantics, and we investigate properties of this model. We also show how our approach applies to data integration and we review examples from the literature.