We propose a logical language for representing networks with nonmonotonic multiple inheritance. The language is based on a variant of annotated logic studied in [5, 6, 17–21]. The use of annotated logic provides a rich setting that allows to disambiguate networks whose topology does not provide enough information to decide how properties are to be inherited. The proposed formalism handles inheritance via strict as well as defeasible links. We provide a formal account of the language, describe its semantics, and show how a unique intended model can be associated with every inheritance specification written in the language. Finally, we present an algorithm that correctly propagates inherited properties according to the given semantics. The algorithm is also complete in the sense that it computes the set of all properties that must be inherited by any given individual object, and then terminates.