A neglected resolution of Russell's paradox of propositions

Abstract

Bertrand Russell offered an influential paradox of propositions in Appendix B of "The Principles of Mathematics", but there is little agreement as to what to conclude from it. We suggest that Russell’s paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the uses to which modern descendants of Russell’s paradox of propositions have been put in recent literature.