Ontologies define classes of entities and their interrelations. They are used to organize data according to a theory of the domain. Towards that end, ontologies provide class definitions (i.e., the necessary and sufficient conditions for defining class membership). In medical ontologies, it is often difficult to establish such definitions for diseases. We use three examples (anemia, leukemia and schizophrenia) to illustrate the limitations of ontologies as classification resources. We show that eligibility criteria are often more useful than the Aristotelian definitions traditionally used in ontologies. Examples of eligibility criteria for diseases include complex predicates such as ' x is an instance of the class C when at least n criteria among m are verified' and 'symptoms must last at least one month if not treated, but less than one month, if effectively treated'. References to normality and abnormality are often found in disease definitions, but the operational definition of these references (i.e., the statistical and contextual information necessary to define them) is rarely provided. We conclude that knowledge bases that include probabilistic and statistical knowledge as well as rule-based criteria are more useful than Aristotelian definitions for representing the predicates defined by necessary and sufficient conditions. Rich knowledge bases are needed to clarify the relations between individuals and classes in various studies and applications. However, as ontologies represent relations among classes, they can play a supporting role in disease classification services built primarily on knowledge bases.