In the domain of ontology design as well as in Conceptual Modeling, representing universals is a challenging problem. Most approaches which have addressed this problem rely either on Description Logics (DLs) or on First Order Logic (FOL), but many difficulties remain especially about expressiveness. In mathematical logic and program checking, type theories have proved to be appealing but so far, they have not been applied in the formalization of ontologies. To bridge this gap, we present here the main capabilities of a theory for representing ontological structures in a dependently-typed framework which relies both on a constructive logic and on a functional type system. The usability of the theory is demonstrated with the Coq language which defines in a precise way what ontological primitives such as classes, relations, properties and meta-properties, are in terms of type classes.
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