To effectively use and exchange information among AI systems, a formal specification of the representation of their shared domain of discourse—called an ontology—is indispensable. In this article we introduce a special kind of knowledge representation based on a dual view of the universe of discourse and show how it can be used in human activities such as searching, in-depth exploration, and browsing. After a formal definition of dualistic ontologies, we exemplify this definition with three different (well-known) kinds of ontologies, based on the vector model, on formal concept analysis, and on fuzzy logic, respectively. The vector model leads to concepts derived by latent semantic indexing using the singular value decomposition. Both the set model as the fuzzy set model led to Formal Concept Analysis, in which the fuzzy set model is equipped with a parameter that controls the fine-graining of the resulting concepts. We discuss the relation between the resulting systems of concepts. Finally, we demonstrate the use of this theory by introducing the dual search engine. We show how this search engine can be employed to support the human activities addressed above.