And the Problems of Predication and Impredicativity I. Self-predication and Paradox

Russell's paradox and the resultant distinction of logical types have been central topics of philosophical discussion for almost a century. In this essay I claim that a more fundamental distinction, that which distinguishes properties and relations as monadic, dyadic, etc., provides a basis for blocking Russell's paradox as applied to properties, not sets, without distinctions of type (or equivalent distinctions). The distinction also points to fundamental features of predication that bear on the nature of relations, the extension of relations, the problem of the analysis of relational order, and questions about symmetrical relations (identity) and purported monadic relational properties (self-identity). In particular, an early unpublished analysis of relational order that Russell proposed is examined in detail and contrasted with the standard set-theoretical analysis of Wiener and Kuratowski. I argue that the analysis of Kuratowski presupposes a basic ordering relation, and hence does not provide an analysis of order, whereas a modification of Russell's analysis provides a viable alternative analysis. The discussion of Russell's paradox and relational predication brings out a connection with Bradley's lesser-known paradox of predication, which, like the Russell paradox, is found to stem from a mistaken conception of the exemplification relation. Though I argue that a Principia-sty\Q schema without type distinctions avoids the familiar paradox involving self-predication, I defend Russell's rejection of impredicative properties, and his use of a ramified theory of orders, from Ramsey's arguments attacking Russell's claim that a "vicious circle" is involved in the use of impredicative predicates. This discussion involves a consideration of Ramsey's view of quantification, which he derived from Wittgenstein. As both the ramified theory, involving Russell's axiom of reducibility, and the acceptance of impredicative predicates are found to be problematic, I conclude that complex predicate abstracts involving predicate quantifiers should not be taken to represent properties.