Liars with Curry: Dialetheism and the Prospects for a Uniform Solution

Abstract

Some dialetheists have claimed that one of the central advantages of their approach to the Paradoxes of Self-Reference is that they are able to offer a unified solution to structurally similar paradoxes that arise in the semantic and set-theoretic realms (Priest in Mind 103:25–34, [12] and Beyond the limits of thought. Clarendon Press, Oxford, [16]). They argue that since the structures of all of these paradoxes conform with the Inclosure Schema (IS), the Principle of Uniform Solution (PUS) dictates that we should solve them all the same way. But the dialetheist’s approach to PUS collapses when it comes to the Curry Paradox, to which any solution based on dialetheism seems inapplicable. We show that a particular version of a ‘paracomplete’ theory (inspired by Field in Saving truth from paradox. Oxford University Press, Oxford, [8]) has available to it a way of avoiding these problems that the dialetheist cannot mirror without losing Modus Ponens. JC Beall has suggested a way of minimizing this loss, but Beall’s strategy runs up against a further difficulty about the PUS. This one involves the Irrationalist’s Paradox. We conclude with a dilemma: The dialetheist can either reject the principles that are used to accommodate ordinary reasoning when Modus Ponens or Disjunctive Syllogism fail, and thus live without the ability to mimic these inferences, or they can sacrifice the Principle of Uniform Solution by solving an extremely Liar-like semantic paradox in a way that has nothing to do with their solution to the Liar and Curry’s Paradox. In either case, the prospects for a plausible and truly uniform dialetheic solution to the paradoxes of self-reference are grim.