# Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic

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## Abstract

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.

## Keywords

Russell-Myhill Intensional logic Predicativity## Notes

### Acknowledgments

I was lucky enough to be able to present parts of this work at a number of workshops and conferences, and I would like to thank the participants and organizers of these events for these opportunities. I would like to especially thank the following people for the comments and feedback which I received on these and other occasions: Robert Black, Roy Cook, Matthew Davidson, Walter Dean, Marie Duží, Kenny Easwaran, Fernando Ferreira, Martin Fischer, Rohan French, Salvatore Florio, Kentaro Fujimoto, Jeremy Heis, Joel David Hamkins, Volker Halbach, Ole Thomassen Hjortland, Luca Incurvati, Daniel Isaacson, Jönne Kriener, Graham Leach-Krouse, Hannes Leitgeb, Øystein Linnebo, Paolo Mancosu, Richard Mendelsohn, Tony Martin, Yiannis Moschovakis, John Mumma, Pavel Pudlák, Sam Roberts, Marcus Rossberg, Tony Roy, Gil Sagi, Florian Steinberger, Iulian Toader, Gabriel Uzquiano, Albert Visser, Kai Wehmeier, Philip Welch, Trevor Wilson, and Martin Zeman. This paper has likewise been substantially bettered by the feedback and comments of the editors and referees of this journal, to whom I express my gratitude. While composing this paper, I was supported by a Kurt Gödel Society Research Prize Fellowship and by Øystein Linnebo’s European Research Council funded project “Plurals, Predicates, and Paradox.”

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