The Law of Excluded Middle and Berry’s Paradox... Finally

Abstract

This is the culmination of a discussion on Berry's Paradox with Graham Priest, over an extended period from 1983 to 2019, the central point being whether the Paradox can be avoided or not by removal of the Law of Excluded Middle (LEM). Priest is of the view that a form of the Paradox can be derived without the LEM, whilst Brady disputes this. We start by conceptualizing negation in the logic MC of meaning containment and introduce the LEM as part of the classical recapture. We then examine the usage of the LEM in some other paradoxes and see that it is applied to cases of self-reference. In relation to Priest's [2019] paper, we go on to find a similar use of the LEM in Priest's derivation of Berry's Paradox. However, it is found to be deeper and trickier than other paradoxes, requiring a special effort to untangle the relationships between the LEM, self-reference and meta-theoretic influence. We then examine Brady's previous formalization of Berry's Paradox, considering Brady's most recent view of restricted quantification and his recursive account of the least number satisfying a property. We show that neither of these methods can be used to formalize the paradox. We also examine the inductive shapes for the Substitution of Identity Rule for intensional and extensional identities, while conceding Priest's point, made in [2019], that his shape of Substitution of Identity is not relevant to his proof of Berry's Paradox.