The Principle of Uniform Solution (of the Paradoxes of Self-Reference)

Schema: there is a property φ and function δ such that (1) w = {x|φ(x)} exists, and (2) if y is a subset of w, then (i) δ(y) ∉ y, and (ii) δ(y) ∈ w. Thus, for example, in the case of Burali-Forti's Paradox, φ(x) is the property of being an ordinal, w is the set of all ordinals, and δ(y) is the least ordinal greater than every member of y. Priest spends the main part of his paper showing that the other paradoxes mentioned above also fit Russell's Schema. I have nothing to say against this part of Priest's paper; the part with which I wish to take issue comes near the end, where Priest introduces the Principle of Uniform Solution (PUS): if two paradoxes are of the same kind, then they should have the same kind of solution. What is it, according to Priest, for two paradoxes to be of the same kind? It is for there to be a certain structure that produces contradiction, and for the two paradoxes to share that structure. Russell's Schema is a contradiction-producing structure, and all the paradoxes mentioned above share this structure; hence these paradoxes are all of the same kind. Hence, by the PUS, these paradoxes should all have the same kind of solution. " Any solution that can handle only some members of the family is bound to appear somewhat one-eyed, and as not having got to grips with the fundamental issue " (Priest 1994, p. 32). Generally speaking, however, logicians have adopted two different kinds of solution: one for the set-theoretic paradoxes (Russell's, Burali-Forti's and Miri-manoff's), which is of the Zermelo-Fraenkel sort, and involves denying the existence of the totality w of clause (1) of Russell's Schema; and one for the semantic paradoxes, which involves denying the T-schema (or rel