In many data-centric semantic web applications, it is desirable to use OWL to encode the Integrity Constraints (IC) that must be satisfied by instance data. However, challenges arise due to the Open World Assumption (OWA) and the lack of a Unique Name Assumption (UNA) in OWL’s standard semantics. In particular, conditions that trigger constraint violations in systems using the ClosedWorld Assumption (CWA), will generate new inferences in standard OWL-based reasoning applications. In this paper, we present an alternative IC semantics for OWL that allows applications to work with the CWA and the weak UNA. Ontology modelers can choose which OWL axioms to be interpreted with our IC semantics. Thus application developers are able to combine open world reasoning with closed world constraint validation in a flexible way. We also show that IC validation can be reduced to query answering under certain conditions. Finally, we describe our prototype implementation based on the OWL reasoner Pellet.