An entirely non-self-referential Yabloesque paradox
Graham Priest (Analysis 57:236–242, 1997) has argued that Yablo’s paradox (Analysis 53:251–252, 1993) involves a kind of ‘hidden’ circularity, since it involves a predicate whose satisfaction conditions can only be given in terms of that very predicate. Even if we accept Priest’s claim that Yablo’s paradox is self-referential in this sense—that the satisfaction conditions for the sentences making up the paradox involve a circular predicate—it turns out that there are paradoxical variations of Yablo’s paradox that are not circular in this sense, since they involve satisfaction conditions that are not recursively specifiable, and hence not recognizable in the sense required for Priest’s argument. In this paper I provide a general recipe for constructing infinitely many (in fact, continuum-many) such noncircular Yabloesque paradoxes, and conclude by drawing some more general lessons regarding our ability to identify conditions that are necessary and sufficient for paradoxically more generally.
KeywordsYablo paradox Circularity Recursive function Self-reference
An early version of this paper was presented by Jesse M. Butler at the 2009 APA Central Division Meeting in Chicago, Illinois, where Roy T. Cook was the commentator. Sadly, Jesse passed away in 2013 before publishing this work, and Professor Cook graciously volunteered to oversee the refereeing and revision process for the paper. Jesse’s widow—Ivana Simic—would like to gratefully acknowledge all the work Professor Cook has put in to help prepare the paper for publication, despite the fact that it was written more than 10 years ago and that a lot has happened in the discourse on Yablo’s paradox since. Thanks are also owed to two anonymous referees for exceptionally helpful comments.