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Knights, Knaves, Truth, Truthfulness, Grounding, Tethering, Aboutness, and Paradox

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Raymond Smullyan on Self Reference

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 14))

Abstract

Knights always tell the truth; Knaves always lie. Knaves for familiar reasons cannot coherently describe themselves as liars. That would be like Epimenides the Cretan accusing all Cretans of lying. Knights do not *intuitively* run into the same problem. What could prevent a Knight from truly reporting that s/he always tells the truth? Standard theories of truth DO prevent this, however, for such a report is self-referentially ungrounded. Standard theories have a problem, then! We try to fix it.

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Notes

  1. 1.

    Self-identified knights are the group Smullyan admires the most. If they were talking nonsense, he would have noticed it.

  2. 2.

    Kripke does allow ungrounded sentences to be intrinsically true: true in a fixed point none of whose assignments are reversed in other fixed points. But the Truthfulness-Teller cannot claim that lesser status either, for there are fixed points in which it is uniquely false.

  3. 3.

    Kripke cites Herzberger (1970). See also Davis (1979), Hazen (1981), Yablo (1982), and Yablo (1993). For the relation to grounding in set theory, see Mirimanoff (1917), Yuting (1953), Boolos (1971), Barwise and Etchemendy (1989), McLarty (1993), and Yablo (2006).

  4. 4.

    There could be an “unwinding” of K that does not depend on itself, yet is equally untethered. Kripke notes the possibility of “an infinite sequence of sentences P \(_{i}\), where P \(_{i}\) says that P \(_{i+1}\) is true” (Kripke 1975, 693). For unwindings more generally see Schlenker (2007) and Cook (2014).

  5. 5.

    Compressed for readability.

  6. 6.

    Taken from (3) above.

  7. 7.

    The underlying model M is a model, possibly partial, of the T-free part of the language.

  8. 8.

    Yablo (1982).

  9. 9.

    No consistent fixed point that is; but we have defined fixed points so that all of them are consistent.

  10. 10.

    If \(\varphi \) is TRUE, then \(|\varphi |^\mathbf{t}\) has a consistent tethered dependence tree and \(|\varphi |^\mathbf{f}\) doesn’t. By the Lemma, \(\varphi \) is true in a fact-dependent fixed point but not false in any fact-dependent fixed points. The converse is similar.

  11. 11.

    Lewis (1988).

  12. 12.

    Lewis (1988).

  13. 13.

    We will be interested only in fact-dependent fixed points, more carefully, fixed points that are fact-dependent relative to some choice \(\mathcal {A}\) of non-semantic atomic facts.

References

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Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Stephen Yablo

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  1. Stephen Yablo
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Correspondence to Stephen Yablo .

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Editors and Affiliations

  1. City University of New York (CUNY) System, City University of New York, New York City, New York, USA

    Melvin Fitting

  2. NYC Department of Education, Brooklyn, New York, USA

    Brian Rayman

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Yablo, S. (2017). Knights, Knaves, Truth, Truthfulness, Grounding, Tethering, Aboutness, and Paradox. In: Fitting, M., Rayman, B. (eds) Raymond Smullyan on Self Reference. Outstanding Contributions to Logic, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-68732-2_7

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