In this paper, we introduce a novel approach to deductive databases meant to take into account the needs of current applications in the area of data integration. To this end, we extend the formalism of standard deductive databases to the context of Four-valued logic so as to account for unknown, inconsistent, true or false information under the open world assumption. In our approach, a database is a pair (E,R) where E is the extension and R the set of rules. The extension is a set of pairs of the form (f, v) where f is a fact and v is a value that can be true, inconsistent or false - but not unknown (that is, unknown facts are not stored in the database). The rules follow the form of standard Datalog{neg} rules but, contrary to standard rules, their head may be a negative atom. Our main contributions are as follows: (i) we give an expression of first-degree entailment in terms of other connectors and exhibit a functionally complete set of basic connectors not involving first-degree entailment, (ii) we define a new operator for handling our new type of rules and show that this operator is monotonic and continuous, thus providing an effective way for defining and computing database semantics, and (iii) we argue that our framework allows for the definition of a new type of updates that can be used in most standard data integration applications.