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

TL;DR

This paper proposes a predicative approach to resolving the Russell-Myhill paradox within Church's intensional logic, providing a consistency proof and exploring implications for set theory and related paradoxes.

Contribution

It introduces a predicative restriction on comprehension in Church's intensional logic and demonstrates its consistency, addressing longstanding paradoxes and criticisms.

Findings

Provides a consistency proof for the predicative response

Models key axioms of Church's intensional logic with criticized features

Offers a predicative set conception resolving the Wehmeier problem

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.