What is Inconsistency

Typically, relational databases operate under the Closed World Assumption (CWA) of Reiter (Reiter, 1987). The CWA is a meta-rule that says that given a knowledge base KB and a sentence P, if P is not a logical consequence of KB, assume ~P (the negation of P). Thus, we explicitly represent only positive facts in a knowledge base. A negative fact is implicit if its positive counterpart is not present. Under the CWA we presume that our knowledge about the world is complete i.e. there are no “gaps” in our knowledge of the real world. The Open World Assumption (OWA) is the opposite point of view. Here, we “admit” that our knowledge of the real world is incomplete. Thus we store everything we know about the world – positive and negative. Consider a database which simply contains the information “Tweety is a bird”. Assume that we want to ask this database the query “Does Tweety fly?”. Under the CWA, since we assume that there are no gaps in our knowledge, every query returns a yes/no answer. In this case we get the answer “no” because there is no information in the database stating that Tweety can fly. However, under the OWA, the answer to the query is “unknown”. i.e. the database does not know whether Tweety flies or not. We would obtain a “no” answer to this query under the OWA only if it was explicitly stated in the database that Tweety does not fly.