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The Liar

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Abstract

The liar sentence, “This sentence is false”, presents a paradox: it is true if and only if it is false. The solution is to hold that the sentence fails to express a proposition. Our language contains implicit rules for the interpretation of sentences. These rules are inconsistent as applied to this case, for they require that the liar sentence be interpreted as expressing the proposition that holds if and only if it does not hold. No proposition can satisfy this condition, so no proposition can be the content of the liar sentence. Analogous solutions apply to the Barber Paradox, Curry’s Paradox, Grelling’s Paradox, and Russell’s Paradox.

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Notes

  1. 1.

    Why not use the more general T-schema, “┌P┐ is true if and only if P”, where the second “P” may be replaced with any declarative sentence, and “┌P┐” replaced with a name for that sentence (as in Tarski 1983)? This is usually correct; for instance, the following is correct: “‘Snow is white’ is true if and only if snow is white.” But the schema does not always work; consider: “‘This is the beginning of this sentence’ is true if and only if this is the beginning of this sentence.” The formulation given in the text avoids such complications.

  2. 2.

    Unless otherwise stated, I use “or” inclusively; thus, “A or B” means “At least one of {A, B} holds, possibly both.”

  3. 3.

    Priest 2006a, b; Beall 2009.

  4. 4.

    Cf. Quine 1986, p. 81: “[S]urely the notation ceased to be recognizable as negation when they took to regarding some conjunctions of the form [p & ~p] as true.”

  5. 5.

    Objections of this sort appear in Littman and Simmons 2004, pp. 322–4, and Sainsbury 2009, pp. 157–8.

  6. 6.

    Priest (2006b, ch. 5) and Beall (2009, pp. 48–50) take essentially this line, except that they frame the issue in terms of whether the negation operator validates explosion (the principle that (A & ~A) entails B, for any arbitrary B). My claim is not that the meaning of negation by definition validates explosion. My claim is that the meaning of negation directly rules out a sentence and its negation both being true.

    Priest and Beall construct arguments that the classical notion of negation is incoherent, which I will not discuss in detail. Priest’s main argument turns on shifting the burden of proof onto classical theorists to show that classical negation is coherent, and then arguing that they cannot do so without begging the question. I believe, though I will not argue the point here, that the burden is on Priest to show that “not!” is incoherent. Beall’s argument, in essence, uses the Liar Paradox itself to impugn the existence of the “not!” operator. In reply, I have a solution to the paradox that does not require abandoning classical logic.

  7. 7.

    This appears to be the most common view. See, e.g., Rescher 2001, pp. 196–203; Clark 2002, p. 102. Philosophers with this view disagree, however, on exactly why L fails to express a proposition.

  8. 8.

    This sort of view was taken by some medieval philosophers; see the anonymous authors discussed in Braakuis 1967 and in Spade 1975, pp. 33–4. Cf. Rescher (2001, pp. 196–7).

  9. 9.

    Geach 1950; Strawson 1950.

  10. 10.

    I thank Iskra Fileva (p.c.) for this suggestion.

  11. 11.

    Cf. Beall and Glanzberg 2011, section 2.3.1.

  12. 12.

    Tarski 1944. Russell (1908) adopts a similar but more restrictive view.

  13. 13.

    Of course, this only applies to simple sentences in subject-predicate form; however, if you know what this footnote means, then you probably already know how to extend the definition of truth1 to other sentence forms. If you don’t know what this footnote means, then you probably don’t care.

  14. 14.

    This means that the second premise in the paradoxical reasoning, “L says that L is false”, is also problematic, since it attempts to deploy a generic notion of “saying” something. If “saying” is limited to first-order saying, second-order saying, and so on, then the liar sentence does not say anything, since it does not first-order say anything, nor does it second-order say anything, and so on.

  15. 15.

    This is what everyone says about it, e.g., Rescher 2001, p. 144; Sainsbury 2009, pp. 1–2; Irvine and Deutsch 2016, section 4. The “paradox” first appears in Russell (1972, p. 101; originally published 1918), attributed to an unnamed person. Russell appears to have found it so trivial that he does not bother to state the solution.

  16. 16.

    More specifically, propositions are the primary bearers of truth, falsity, and so on; that is, anything else that is true, false, or the like, is so in virtue of its relation to a true or false, etc., proposition. A true sentence is one that expresses a true proposition, a true belief is a belief directed at a true proposition, and so on.

  17. 17.

    Herzberger (1967), for example, argues that it is impossible for a language to be inconsistent.

  18. 18.

    As Eklund (2002) maintains.

  19. 19.

    Previously mentioned in section 2.4.3 above.

  20. 20.

    Cases like this are discussed by Sloman (1971, pp. 136, 139).

  21. 21.

    Sloman (1971, pp. 142–3) takes a similar view.

  22. 22.

    This paradox derives from Curry 1942; see also Prior 1955.

  23. 23.

    This paradox derives from Russell (1902, 1903, p. 102). Grelling and Nelson (1908) restate the paradox in slightly different words, which somehow has led some to call it “Grelling’s Paradox” or “the Grelling-Nelson Paradox” (as in Clark 2002, pp. 80–81).

  24. 24.

    Russell 1902, 1903, p. 102.

  25. 25.

    Except that perhaps no sets exist; see my 2016, ch. 8.

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  1. Philosophy Department, University of Colorado Boulder, Boulder, CO, USA

    Michael Huemer

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Huemer, M. (2018). The Liar. In: Paradox Lost. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-90490-0_2

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