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Dialetheism and distributed sorites

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Abstract

Noniterative approaches to the sorites paradox accept single steps of soritical reasoning, but deny that these can be combined into valid chains of soritical reasoning. The distributed sorites is a puzzle designed to undermine noniterative approaches to the sorites paradox, by deriving an inconsistent conclusion using only single steps, but not chains, of soritical reasoning. This paper shows how a dialetheist version of the noniterative approach, the strict-tolerant approach, also solves the distributed sorites paradox, at no further cost, by accepting the inconsistent conclusion.

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Notes

  1. See, for example, Egré (2015) and the references therein.

  2. See, for example, Hyde and Diana (2018) for an overview of the sorites paradox.

  3. See, for example, Zardini (2008) and Gaifman (2010). See also Barnett (2019, p. 1073) for further references as well as the name “noniteration”.

  4. See Barnett (2019, p. 1073). See also Barnett (2018, pp. 8–9) and Carlson et al. (2021) for a closely related puzzle in normative ethics.

  5. Barnett (2019, p. 1075) concedes that Gaifman’s contextualist development of noniteration avoids the distributed sorites paradox, but at the cost of qualifying tolerance.

  6. The strict-tolerant approach was proposed by Van Rooij (2011) in response to Zardini (2008), and is elaborated on at length by Cobreros et al. (2012).

  7. See, for example, Priest (2006, p. 4) for this characterisation of dialetheism. For a helpful overview of dialetheism about vagueness see Hyde and Colyvan (2008). See Cobreros et al. (2012, p. 356) for the connection of their approach with dialetheism.

  8. Dominic Hyde, for example, uses this symmetry to argue that supervaluationism is no better than subvaluationism, its dialetheic dual (Hyde, 1997, 2010). Cobreros (2011), on the other hand, argues subvaluationism is better than supervaluationism.

  9. See Beall and Restall (2006, pp. 27–28) for example, on vagueness and logical pluralism.

  10. In Cobreros et al. (2012, p. 376), their Version 1 of the sorites is our implicit sorites, and their Version 2 of the sorites is our explicit sorites.

  11. Essentially due to the “cut rule”, for which see Ripley (2013a) and references therein.

  12. See, for example, Sorensen (1988, p. 239).

  13. See van Rooij (2011, p. 213) and Cobreros et al. (2012, p. 353) for this mutually recursive definition of strict and tolerant truth.

  14. This is lemma 1 in Cobreros et al. (2012, p. 357).

  15. See van Rooij (2011, p. 213) and Cobreros et al. (2012, p. 354) for this point.

  16. \(K_3\) is named for Kleene (1952, p. 334). Tye (1990, p. 544) applies \(K_3\) to vagueness.

  17. Asenjo (1966) and Priest (1979) introduced LP. Colyvan (2008), Priest (2010, p. 73) and Ripley (2013b) apply LP to vagueness. Weber (2010, pp. 1043–1045) applies the relevant logic DK, which extends LP with a relevant conditional. Beall (2014) criticises applying LP to vagueness; Weber et al. (2014) reply.

  18. See van Rooij (2011, p. 213) and Cobreros et al. (2012, p. 373) for the invalidity of modus ponens in tt.

  19. Priest (2010, pp. 73–74), Ripley (2013b, pp. 346–347), and Weber (2010, p. 1040) all allow that LP may be extended with conditionals which do validate modus ponens. They concede sorites arguments restated with these conditionals are valid, but deny they are sound.

  20. We also have sc, ct, cs, tc and ts, for “strict-classical”, “classical-tolerant”, “classical-strict”, “tolerant-classical” and “tolerant-strict” respectively for the five remaining logics (Cobreros et al., 2012, p. 366).

  21. See Cobreros et al. (2012, p. 376). The logics ct and sc also validate single-step sorites, but invalidate sorites arguments of two-steps or more. Cobreros et al. (2012, p. 373) prefer st to sc because tolerance is not a tautology of the latter. Although van Rooij (2011, p. 213) initially singled-out ct as an appropriate logic for vagueness, Cobreros et al. (2012, p. 373) prefer st because it satisfies the deduction theorem. But most of the points in this paper would go through with only superficial changes if ct were adopted instead of st.

  22. As Cobreros et al. (2012, p. 376) admit.

  23. Restall (1997), for example, helpfully distinguishes paraconsistency from dialetheism.

  24. See Cobreros et al. (2012, p. 357) for their definition of satisfiability.

  25. The idea for this paper arose from a discussion with my colleagues Zach Barnett and Lavinia Picollo; I’m especially grateful to Zach for much discussion. I’m also grateful to the students in my non-classical logic class, Eugene Ho, Jethro Peck, Ashish Stephen Peter, and Thomas Tan for working through the material with me. I thank my colleagues Zach Barnett, Ethan Jerzak, Michael Pelczar, Hsueh Qu, Mattias Skipper, Lavinia Picollo, Weng-Hong Tang and Daniel Waxman for discussing the paper in our reading group. Finally, I thank Nathaniel Gan for reading two recent drafts of the paper, and anonymous referees for their comments. This paper was part of our Metaphysics of Humanity project, supported by the Ministry of Education, Singapore, grant number A-0003057-00-00.

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Acknowledgements

The idea for this paper arose from a discussion with my colleagues Zach Barnett and Lavinia Picollo; I’m especially grateful to Zach for much help with the paper. I’m also grateful to the students in my non-classical logic class, Eugene Ho, Jethro Peck, Ashish Stephen Peter, and Thomas Tan for working through the material with me. I thank my colleagues Zach Barnett, Ethan Jerzak, Michael Pelczar, Hsueh Qu, Mattias Skipper, Lavinia Picollo, Weng-Hong Tang and Dan Waxman for discussing the paper in our work-in-progress reading group. Finally, I thank Nathaniel Gan for reading two recent drafts of the paper, and anonymous referees for their comments. This paper was part of our Metaphysics of Humanity project, supported by a grant from the Ministry of Education, Singapore, Grant Number A-0003057-00-00.

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  1. Department of Philosophy, National University of Singapore, Block AS3 Level 5, 3 Arts Link, Singapore, 117570, Singapore

    Ben Blumson

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Blumson, B. Dialetheism and distributed sorites. Synthese 202, 116 (2023). https://doi.org/10.1007/s11229-023-04346-5

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