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

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.

FRAGMENTS OF FREGE’SGRUNDGESETZEAND GÖDEL’S CONSTRUCTIBLE UNIVERSE

“…The extra information about the injection χ n : λ → θ −1 n ({0, 1}) in Theorem 5.2 will be primarily useful for our companion paper [41], where we use constructible sets to build models of an intensional type theory (cf. §5 of [41]). The reason for the focus on {0, 1} is that in the tradition of type-theory these are used as ersatzes for the truth-values {F, T }.…”

“…This paper is the first in a series of three papers -the other two being [42], [41]-which collectively constitute a sequel to the Basic Law V components of our paper [40]. In that earlier paper [40], we gave a proof of the consistency of Frege's Grundgesetze system with limited amounts of comprehension using tools from hyperarithmetic theory.…”

Impredicativity and Paradox

“…Given impredicative comprehension, we find that two operators O and Q may apply to different propositions and nevertheless be such that the proposition ∀ p ( Op → p ) is one and the same as the proposition ∀ p ( Qp → p ). Walsh () provides predicativists with the resources to reject the relevant generalization of Cantor's theorem.…”

“…Parsons () proved the consistency of a schematic version of (V) against the backdrop of first‐order logic; Heck () extended the result to ramified second‐order logic; and Wehmeier () and Ferreira and Wehmeier () did it for ‐comprehension . More recently, Sean Walsh has isolated models of (V) and other abstraction principles in fragments of Gödel's constructible universe in Walsh ()) and Walsh ()), and Walsh () has extended the construction to model predicative fragments of type theory.…”

“…Poincaré appears to have in mind the fact that some definitions are not invariant under expansions of the domain of quantification. Sean Walsh, who has recently revived the predicativist stance in Walsh (), Walsh (), and Walsh (), motivates a predicative restriction of comprehension in terms of semantic stability.…”