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Pinocchio against the Semantic Hierarchies

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Abstract

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.

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Notes

  1. Cook gives the following succinct proof that the intersubstitutability of truth, which he labels ‘T-substitutivity’, is not a logical necessity.

    [L]et Φ4 be a logical truth named by t4, and Φ5 a logical falsehood named by t5. Since Φ4 is a logical truth, by T-substitutivity, T(t4) is a logical truth, but then by Logical substitutivity [that is, the substitutivity of non-logical constants in a logical truth], T(t5) is a logical truth, but then by T-substitutivity Φ5 is a logical truth. Contradiction. (Cook2012:236)

  2. Quine (1995) shows how such an identity can be derived using a syntactic function, in particular a self-predication function.

  3. Adequate definitions of truth will entail each instance of this schema, or each instance of a more general schema using the truth relation or the converse satisfaction relation. My presentation assumes for convenience that the meta-language contains (and extends) the object-language. This will make proofs in section 3 more perspicuous. The same results can readily be obtained without the use of canonical naming. Nevertheless, as I stated previously, canonical naming, being more formal, gives the T-schema its best chance of being necessary.

  4. While the Pinocchio paradox and the Liar pose obstacles for a theory of truth, whether these paradoxes need to use a principle of truth depends on one’s theory of truth, truth bearers and logical systems. For example, Prior (1958) did not use a truth predicate, but still found it necessary to devise some way of avoiding a version of the Epimenides paradox. In the case of the Pinocchio paradox, an alternative principle would be ‘Pinocchio’s nose grows iff what he is saying is not the case’, but this would only avoid having to use the T-schema if one’s theory and logical system was such that ‘is not the case’ was not just a synonym for the truth predicate. Most people think it is; in any case, my concern here is to pose a challenge for theories of truth, not to avoid the T-schema or similar principles of truth.

  5. D’Agostini and Ficara (2016) credit the idea for the Blushing Liar paradox to Victoria Schneider.

  6. Instead, Eldridge-Smith and Eldridge-Smith (2010) use the Pinocchio paradox to criticise part of Tarski’s analysis of the Liar.

    The Pinocchio paradox is, in a way, a counter-example to solutions to the Liar that would exclude semantic predicates from an object-language, because ‘is growing’ is not a semantic predicate. Tarski’s analysis of the source of pathology of which the Liar is symptomatic led him to conclude it arose from free use of semantic predicates in the object-language. Tarski’s solution was to restrict such predicates strictly to the metalanguage. Intuitively, predicates like ‘is growing’ are typical of just the sorts of predicates one wants in a useful object-language. If empirical predicates like ‘is growing’ need to be restricted in the object-language to avoid versions of the Liar, the intuitive bounds on which predicates need to be restricted in the object-language to avoid Liar-like paradoxes have been breached. [Eldridge-Smith and Eldridge-Smith 2010: 213]

  7. Personally, I think this only slightly complicates the argument against physicalism. The complication is that either the instance of the Pinocchio principle or the relevant instance of the T-schema is false by reductio, but either way I think Luna still has an argument against physicalism.

  8. The aporia of deductive logic not being ampliative is ECQ or Explosion.

  9. There are other principles of truth. There are compositional principles of interpretation, which if they are analytic, assure rules such as:

    ‘A & B’ is true iff ‘A’ is true and ‘B’ is true.

    ‘~A’ is true iff ‘A’ is not true

    On these principles as axioms, see. Volker Halbach (2014). Note, nevertheless, that these principles of interpretation do not assure the T-schema.

  10. This derivation of the Liar may usefully be compared with Heck 2012; although Heck’s Liar is premised on . ~(A & T < ~A>) and ~(~A & ~T < ~A>). These premises might be justified by the thoughts that nothing is the case and is false, and nothing is not the case but is not false. The advantage of Heck’s Liar is that it does not require double negation.

  11. However, I think paraconsistent logics are weak logics. One wants a strong non-ampliative alethic logic that supports maximal expressiveness. That is why one wants a strong deductive logic that can use its own truth predicate. But whether my semantics works with a stronger logic than dialetheism is a question that depends on further argument addressing other inferential aspects of the Liar paradox, and a question that is beyond the scope of this article.

  12. This can be thought of along the lines of precedence of operators, so that it is conceivably formally specifiable. Essentially, the semantic value of any statement made by Pinocchio in Twi-nerf is determined by the Pinocchio principle and not the T-schema.

  13. Let us say that the hierarchical T-schema makes standard assumptions about the range and ordering of the index, i.

  14. D’Agostini and Ficara (2016) are not committed to truth maker maximalism, the view that every truth has a truth maker.

  15. Non-truth functional variations of Curry’s paradox do not seem to have related hypodoxes.

  16. Nevertheless, to the best of my knowledge there is no known counterexample or exception to Eldridge-Smith’s remaining conjecture that each hypodoxical conundrum has a related paradox.

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  1. School of Philosophy, Research School of Social Sciences, Australian National University, Canberra, Australia

    Peter Eldridge-Smith

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Appendix: Pinocchio’s search for a Truth-teller: the missing Pinocchio hypodox

Appendix: Pinocchio’s search for a Truth-teller: the missing Pinocchio hypodox

Eldridge-Smith has conjectured that every paradox of a large class, broader than just the paradoxes of self-reference, has a related hypodox (Eldridge-Smith 2007; Eldridge-Smith, 2012a, 2015: Appendix). A hypodox being a generalization of the Truth-teller phenomenon. As a hypodox, the Truth-teller is either true or not, but there is lack of a principle determining which. The class of paradoxes with dual hypodoxes includes the Liar, Russell’s, Grelling’s, and their truth-functional variations; it also includes time travel paradoxes.Footnote 15 As a variation of the Liar paradox, one might expect the Pinocchio paradox to have a related Pinocchio hypodox. Then a point against considering the Pinocchio paradox as a Liar paradox is that it does not have a corresponding hypodox. Had Pinocchio said “my nose is not growing,” it might either of been the case that his nose is growing or it isn’t. In either case there is no conundrum (given the T-schema). A similar criticism, if it is a criticism, applies to close relatives of the Pinocchio paradox, such as the Blushing liar. If Victoria had said “I am not blushing,” there would be no conundrum. Either she would be blushing or not.Footnote 16

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Eldridge-Smith, P. Pinocchio against the Semantic Hierarchies. Philosophia 46, 817–830 (2018). https://doi.org/10.1007/s11406-018-9948-y

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