Operators Solve the Many Categories Problem with Universals

By the Many Categories problem, I mean the prima facie violation of Ockham's Razor by realists about universals: there is, it might seem, just too much variety. Thus, David Armstrong posits both properties and relations. He also theorises about determinates of determinables. Another influential realist, E. J. Lowe distinguishes non-substantial from substantial universals. Yet again, both Armstrong and Lowe include in their ontology abstract particulars in addition to universals. My aim in this paper is to offer a unification of these categories using an operator-theory. First, I show how operators provide a theory of abstract particulars of relations and of impure relational properties. Then, using impure relational properties I derive the Fregean Abstraction Principle. After some metaphysics of mathematics, I provide a theory of determinables. Finally, I consider kinds and substances.