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Reply to Priest on Berry's Paradox

Graham Priest, in his paper, "The Logical Paradoxes and the Law of Excluded Middle", [5], argues that a uniform solution' to the logical and semantic paradoxes,2 based on a logic without the law of excluded middle, such as T W,3 fails because the definability paradoxes, in particular Berry's Paradox, can be derived using such a logic. Such a failure of a uniform solution would dash any hopes of extending the simple consistency result for naive set theory,4 based on the sentential logic T W, to include a solution to the semantic and remaining logical paradoxes. The aim of this reply is to revive the program for obtaining a uniform solution to the logical and semantic paradoxes,5 by showing that Priest needs effectively to assume the law of excluded middle to establish his derivation of Berry's Paradox, and hence that his derivation fails to go through with a logic such as T W. It will be shown that the same applies to the derivation of the definabiity paradox due to K6nig. However, for the other definability paradox mentioned by Priest, i.e. Richard's Paradox, I will argue that it is a pseudo-paradox. A plausible starting point for our treatment of Berry's Paradox is the following equivalence: