7 May 2019: Giorgio Sbardolini

Aboutness paradox

Abstract: On a plausible reconstruction, the Russell-Myhill paradox (also called Russell’s paradox of propositions, or Appendix B paradox) trades on aboutness distinctions. Interestingly, this paradox is neither due to set-theoretic naivete, nor to semantic ascent. In fact, the Russell-Myhill contradiction follows from an hitherto unrecognized abstraction principle for propositions, and so it is formally very closely related to the Russell’s more famous contradiction from Basic Law V. Appreciation of this insight undermines a recent interpretive line, according to which the Russell-Myhill paradox is a reductio of theories of structured propositions: in fact, Russell’s reasoning is a threat for any hyperintensional theory on which propositional identity implies sameness of subject matter. Since certain forms of predicativism offer a principled approach to the dangers of abstraction, we may to some extent vindicate Russell’s own conclusion, which was to restrict the quantifiers. An added benefit of the variety of higher-order predicativism presented here is that some puzzles about propositions discussed by Arthur Prior are seen to disappear.