On Building Abstract Terms in Type Systems *

This paper offers some historical and conceptual remarks on the philosophical and logical procedures of abstraction, based on an account of the notions of concept and function. In order to provide a complete analyis, one should start by considering Plato’s theory of Ideas, which provides the first interpretation of “abstract terms” in the history of philosophy1. The nature of the most general Forms, the related problem of the knowledge thereof, their connection to existing (concrete) objects, are the essential features of the Platonic theory of knowledge and of his metaphysics. The Platonic approach is grounded on the principle of conceptual priority of Ideas over their partecipations, the Forms existing separeted from all the particulars: the former are interpreted as standard particulars to which other particulars conform. Nonetheless, my investigation will start rather by Aristotle, who held first the relation of predication to be the basis for defining abstraction: from this I will try to consider some important ideas for the notion of abstraction in Type Systems. By rejecting the Platonic understanding of general Forms, Aristotle maintains that the logical relation of predication is the starting point for any account of the Categories: these are intended as the forms of both what there is, and of what can be said2. Consequently, to speak about the existence of a certain category means properly to make a predication about a certain substance (as the first category, to which the others refer). To analyse predicative expressions (in order to explain the related predicables) will thus amount to a proper abstraction procedure. The Aristotelian understanding of abstract terms is in turn given by two related aspects, expressing the underlying distinction between abstract methods and abstract terms: