Traditionally, a property theory is a theory of abstract entities that can be predicated of things. A theory of properties in this sense is a theory of predication – just as a theory of classes or sets is a theory of membership. In a formal theory of predication, properties are taken to correspond to some (or all) one-place predicate expressions. In addition to properties, it is usually assumed that there are n-ary relations that correspond to some (or all) n-place predicate expressions (for n≥ 2). A theory of properties is then also a theory of relations.
In this entry we shall use the traditional labels ‘realism’ and ‘conceptualism’ as a convenient way to classify theories. In natural realism, where properties and relations are the physical, or natural, causal structures involved in the laws of nature, properties and relations correspond to only some predicate expressions, whereas in logical realism properties and relations are generally assumed to correspond to all predicate expressions.
Not all theories of predication take properties and relations to be the universals that predicates stand for in their role as predicates. The universals of conceptualism, for example, are unsaturated concepts in the sense of cognitive capacities that are exercised (saturated) in thought and speech. Properties and relations in the sense of intensional Platonic objects may still correspond to predicate expressions, as they do in conceptual intensional realism, but only indirectly as the intensional contents of the concepts that predicates stand for in their role as predicates. In that case, instead of properties and relations being what predicates stand for directly, they are what nominalized predicates denote as abstract singular terms. It is in this way that concepts – such as those that the predicate phrases ‘is wise’, ‘is triangular’ and ‘is identical with’ stand for – are distinguished from the properties and relations that are their intensional contents – such as those that are denoted by the abstract singular terms ‘wisdom’, ‘triangularity’ and ‘identity’, respectively. Once properties are represented by abstract singular terms, concepts can be predicated of them, and, in particular, a concept can be predicated of the property that is its intensional content. For example, the concept represented by ‘is a property’ can be predicated of the property denoted by the abstract noun phrase ‘being a property’, so that ‘Being a property is a property’ (or, ‘The property of being a property is a property’) becomes well-formed. In this way, however, we are confronted with Russell’s paradox of (the property of) being a non-self-predicable property, which is the intensional content of the concept represented by ‘is a non-self-predicable property’. That is, the property of being a non-self-predicable property both falls and does not fall under the concept of being a non-self-predicable property (and therefore both falls and does not fall under the concept of being self-predicable).
VII.—Natural Realism and Present Tendencies in Philosophy
