Property Theories

Abstract

We begin with a truism.1 A property theory is a theory that deals with properties. More precisely, it is a theory that formulates general, non-contingent laws that deal with properties. There are two salient ways of talking of properties. First, they can be talked about as predicables (i.e., as instantiables). Accordingly, one sort of property theory would be a theory that provides general non-contingent laws of the behaviour of the predication relation (instantiation relation). Nothing prevents the logical framework of such a theory from being extensional; that is, it could be formulated in a logical framework in which equivalent formulas are intersubstitutable salva veritate. For example, this sort of property theory could be formulated in a first-order extensional language with identity and a distinguished binary logical predicate for the predication relation. The major challenge facing this sort of property theory is to resolve various paradoxes that result from naive predication principles such as the following analogue of Russell’s paradox: (∃x)(∀y) (x is a predicable of y ↔ y is not predicable of itself). The second salient way of talking about properties is by means of property abstracts such as ‘the property of being a man’. Property abstracts belong to a family of complex singular terms known as intensional abstracts. These include gerundive phrases, infinitive phrases and ‘that’-clauses.