Formal ontologies model classes and their properties and relationships. This paper considers various choices for modelling of classes and properties, and the interrelationship of these within a formal logical framework. Unlike predicate logical usage with quantification over individuals only, in the applied metalogic classes and properties appear as first class non-extensional objects. Using this framework miscellaneous classification structures are examined ranging from mere partial orders to distributive lattices. Moreover, we seek to capture notions such as intensionality of classes and properties ascribed to individuals and classes in an coherent ontological framework. In this analytic framework we further present generative ontologies in which novel classes can be produced systematically by means of given classes and properties.