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Philosophia

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Two Fallacies in Proofs of the Liar Paradox

  • Peter Eldridge-SmithEmail author
Article

Abstract

At some step in proving the Liar Paradox in natural language, a sentence is derived that seems overdetermined with respect to its semantic value. This is complemented by Tarski’s Theorem that a formal language cannot consistently contain a naive truth predicate given the laws of logic used in proving the Liar paradox. I argue that proofs of the Eubulidean Liar either use a principle of truth with non-canonical names in a fallacious way or make a fallacious use of substitution of identicals. Tarski never committed the first fallacy and may have himself considered it fallacious. Nevertheless, I clarify that it is fallacious. I then argue substitution of identicals needs to be restricted within the scope of the truth predicate. A logic for truth implementing this restriction is a monotonic extension of a classical first order logic, or indeed a formalizable fragment of natural language. Proofs of Tarski’s Indefinability of Truth theorem are invalid in this logic. This approach generalizes to invalidate proofs of Liar-like paradoxes, particularly the predicate form of the Knower paradox. Consequently, such a logic can be further extended in a way that avoids Montague’s theorem for such a system. Yet, the semantic status of a Liar sentence is not fully resolved. It is no longer overdetermined; it is now underdetermined.

Keywords

Truth Liar paradox Substitution of identicals Tarski’s indefinability theorem Knower paradox Montague’s theorem 

Notes

References

  1. Boolos, G. S., Burgess, J. P., & Jeffrey, R. C. (2007). Computability and logic. Cambridge University Press.Google Scholar
  2. Eldridge-Smith, P. ( 2008). The Liar Paradox and its relatives. PhD dissertation, Australian National University. https://digitalcollections.anu.edu.au/handle/1885/49284
  3. Eldridge-Smith, Peter. 2012a. A hypodox! A hypodox! A disingenuous hypodox! The Reasoner, 6(7): 118–119. Accessible from https://research.kent.ac.uk/reasoning/the-reasoner/the-reasoner-volume-6/
  4. Eldridge-Smith, P. (2012b). The unsatisfied paradox. The Reasoner, 6(12), 184–185 Accessible from https://research.kent.ac.uk/reasoning/the-reasoner/the-reasoner-volume-6/.Google Scholar
  5. Eldridge-Smith, P. (2015). Two paradoxes of satisfaction. Mind, 124(493), 85–119.CrossRefGoogle Scholar
  6. Eldridge-Smith, P. (2018). Pinocchio against the semantic hierarchies. Philosophia, 46(4), 817–830.CrossRefGoogle Scholar
  7. Eldridge-Smith, P. (2019). The Liar hypodox: A truth-teller’s guide to defusing proofs of the liar paradox. Open Journal of Philosophy, 9(2), 152–171.  https://doi.org/10.4236/ojpp.2019.92011.CrossRefGoogle Scholar
  8. Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
  9. Friedman, H., & Sheard, M. (1987). An axiomatic approach to self-referential truth. Annals of Pure and Applied Logic, 33, 1–21.CrossRefGoogle Scholar
  10. Halbach, V. (2011). Axiomatic theories of truth. Oxford: Oxford University Press.CrossRefGoogle Scholar
  11. Hintikka, J. (1996). The principles of mathematics revisited. Cambridge University Press.Google Scholar
  12. Kaplan, D., & Montague, R. (1960). A paradox regained. Notre Dame Journal of Formal Logic, 1, 79–90.CrossRefGoogle Scholar
  13. McGee, V. (1991). Truth, vagueness, and paradox : An essay on the logic of truth. Indianapolis: Hackett Pub. Co.Google Scholar
  14. Montague, R. (1963). Syntactical treatments of modality. Acta Philosophica Fennica, XVI, 153–167.Google Scholar
  15. Murzi, J. (2014). The inexpressibility of validity. Analysis, 74(1), 65–81.CrossRefGoogle Scholar
  16. Murzi, J., & Carrara, M. (2015). Paradox and logical revision. A short introduction. Topoi, 34, 7–14.CrossRefGoogle Scholar
  17. Priest, G. (2002). Beyond the limits of thought. Oxford: Oxford University Press.CrossRefGoogle Scholar
  18. Priest, G. (2006). In contradiction: A study of the transconsistent. Oxford: Oxford University Press.CrossRefGoogle Scholar
  19. Quine, W. V. O. (1995). Truth, paradox and Gödel’s theorem. In Selected logical papers (Enlarged ed., pp. 236–241). Cambridge: Harvard University Press.Google Scholar
  20. Skyrms, B. (1984). Intensional aspects of semantical self-reference. In R. L. Martin (Ed.), Recent essays on truth and the Liar paradox (pp. 119–131). Oxford: Oxford University Press.Google Scholar
  21. Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Sudia Philosophica, 1, 261–405 Page references are to the translation ‘The concept of truth in formalised languages’ in Tarski (1983): 152–278.Google Scholar
  22. Tarski, A. (1944). The semantic conception of truth: And the foundations of semantics. Philosophy and Phenomenological Research, 4(3), 341–376.CrossRefGoogle Scholar
  23. Tarski, A. (1983). Logic, semantics, metamathematics: Papers from 1923 to 1938 (2nd ed, J. Corcoran, Ed., & Woodger, J. H. Trans.). Indianapolis: Hackett Pub. Co.Google Scholar

Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.Philosophy, Australian National UniversityCanberraAustralia

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