The design of formal ontologies is an interdisciplinary area of research that draws on logic, philosophy, cognitive science, linguistics, as well as computer science, with major applications in the Semantic Web. The main motivation for the construction of ontologies is to provide a shared conceptualisation of a domain for knowledge representation, reasoning and information sharing.

As the scope and relevance of ontologies grows, both for supporting Semantic Web applications and for knowledge-rich processing in general, the issue of re-using/importing developed ontological components takes on an ever more critical role. The current solutions being pursued within OWL-oriented Semantic Web approaches exhibit some severe limitations in this respect. For the next generation of ontology-based systems it will be essential to move beyond this.

In this course, we present major methodologies and techniques that allow ontology developers to correctly construct, modify, and relate ontologies - understood in a broad sense as logical theories formulated in various formal languages - with an emphasis on heterogeneity, structuring and modularity, and foundations of ontology design.

As illustrative examples, we take prominent ontologies from the spatial, philosophical, and linguistic domains. These will be discussed, analysed and shown 'at work' using the Common Algebraic Specification Language (CASL) and the tool HeTS.

CASL is a many sorted first-oder logic with logic independent structuring mechanisms, while HeTS is a tool for the analysis and manipulation of ontologies specified in CASL (possibly using different languages) and supporting theorem provers ranging from second-order theorem provers such as Isabelle, to first-order provers such as Vampire, to OWL-DL reasoners such as Pellet. An essential property of the HeTS/CASL approach is that it supports heterogeneity: ontologies can be build up from various parts that can be specified in different formal languages, corresponding to different flavours of CASL. These languages for ontology design range from relational schemes, to OWL-DL, full first-order logic, modal logic, and higher-order logic.

Taking these powerful tools together provides new possibilities for structuring and reusing ontological components, as well as significant insights for ontology developers and ontology design methodology.

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