Researchers design ontologies as a means to accurately annotate and integrate experimental data across heterogeneous and disparate data- and knowledge bases. Formal ontologies make the semantics of terms and relations explicit such that automated reasoning can be used to verify the consistency of knowledge. However, many biomedical ontologies do not sufficiently formalize the semantics of their relations and are therefore limited with respect to automated reasoning for large scale data integration and knowledge discovery. We describe a method to improve automated reasoning over biomedical ontologies and identify several thousand contradictory class definitions. Our approach aligns terms in biomedical ontologies with foundational classes in a top-level ontology and formalizes composite relations as class expressions. We describe the semi-automated repair of contradictions and demonstrate expressive queries over interoperable ontologies. Our work forms an important cornerstone for data integration, automatic inference and knowledge discovery based on formal representations of knowledge. Our results and analysis software are available at http://bioonto.de/pmwiki.php/Main/ReasonableOntologies.