Pinocchio against the Semantic Hierarchies

The Liar paradox is an obstacle to a theory of truth, but a Liar sentence need not contain a semantic predicate. The Pinocchio paradox, devised by Veronique Eldridge-Smith, was the first published paradox to show this. Pinocchio’s nose grows if, and only if, what Pinocchio is saying is untrue (the Pinocchio principle). What happens if Pinocchio says that his nose is growing? Eldridge-Smith and Eldridge-Smith (Analysis, 70(2): 212-5, 2010) posed the Pinocchio paradox against the Tarskian-Kripkean solutions to the Liar paradox that use language hierarchies. Eldridge-Smith (Analysis, 71(2): 306-8, 2011) also set the Pinocchio paradox against semantic dialetheic solutions to the Liar. Beall (2011) argued the Pinocchio story was just an impossible story. Eldridge-Smith (Analysis, 72(3): 749-752, 2012b) responded that unless the T-schema is a necessary truth of some sort (logical, metaphysical or analytic), the Pinocchio principle is possible. Luna (Mind & Matter 14(1): 77–86, 2016) argues that the Pinocchio contradiction proves the principle is false. D’Agostini & Ficara (2016) discuss a more plausible physical truth-tracking trait, the Blushing Liar, and argue that the Pinocchio contradiction is not a metaphysical dialetheia. I respond to Luna, and D’Agostini & Ficara, and prove that the Pinocchio paradox is a counterexample to hierarchical solutions to the Liar.