Abstract The Semantic Web (SW) can be seen as abstract representation and exchange of data and metadata. Metadata is given in terms of data mark-up and reference to shared, Web-accessible ontologies. Several interesting languages are now available for the Semantic Web. They exploit XML allowing data/metadata communication, yet are endowed with a logical semantics. Such languages allow compact descriptions by means of inheritance mechanisms that permit one to describe an object as belonging to one or more classes whose hierarchy is already described on the Web. With few exceptions, SW logical languages are designed to be monotonic, thus they cannot employ the closed-world assumption i) to make object description compact and most importantly ii) to prevent large ontologies from admitting inconsistency resulting from multiple inheritance. We address these problems by proposing a version of Local Closed-World Assumption that fits SW purposes. Its extent is itself the subject of negotiatio...