The design of ontologies for new commonsense domains continues to pose challenges, particularly in cases where multiple potential axiomatizations satisfy the requirements for the ontology. One approach is to specify the requirements with respect to the intended semantics of the terminology; from a mathematical perspective the requirements may be characterized by the class of structures(referred to as the required models) which capturethe intended semantics. This approach leads to a natural notion of the correctness as a relationship between the models of the axiomatization of the ontology and the required models for the ontology. In this paper, we consider three possible generalizations of the notion of the correctness of an ontology in the case in which the ontology and the required models have different signatures.We show that these notions of correctness lead to different approaches for ontology evaluation and discuss the benefits and drawbacks of each approach.