On a Russellian Paradox about Propositions and Truth

We deal with a paradox involving the relations between propositions and sets (Appendix B of Principles of Mathematics), and the problem of its formalization. We first propose two (mutually incompatible) abstract theories of propositions and truth. The systems are predicatively inspired and are shown consistent by constructing suitable inductive models. We then consider a reconstruction of a theory of truth in the context of (a consistent fragment of) Quine’s set theory NF. The theory is motivated by an alternative route to the solution of the Russellian difficulty and yields an impredicative semantical system, where there exists a high degree of self-reference and yet paradoxes are blocked by restrictions to the diagonalization mechanism.