Two Fallacies in Proofs of the Liar Paradox


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.

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  1. 1.

    In first-order logic substitution of identicals is the Identity Elimination rule. The corresponding metaphysical principle is the indiscernibility of identicals.

  2. 2.

    Furthermore, Eldridge-Smith (2015) argues that Grelling’s and Russell’s paradoxes are of a different kind than those relying on substitution of identicals, and so one should not expect a restriction on substitution of identicals to avoid them. He explicitly distinguishes his own Unsatisfied paradox, as a paradox that does rely on substitution of identicals, from Grelling’s. If Eldridge-Smith (2015) is correct, there are two kinds of pathology tracked by the two kinds of paradox he formally distinguishes, and therefore each kind may well need a different solution. Correspondingly, the solution proposed here will resolve Eldridge-Smith’s own Unsatisfied paradox but not Grelling’s paradox.

  3. 3.

    See Tarski (1944: 344) for the origin of the name ‘T-schema’. ‘The T-schema’ is also used in the literature to refer to a more sophisticated schema based on Tarski’s Convention T (1935/1983: 187–8). That schema requires an object- and meta-language distinction.

  4. 4.

    An anonymous referee is certain of this. In which case, I am merely arguing for a restriction that Tarski himself intended.

  5. 5.

    My thanks to an anonymous referee for encouraging me to make this schema explicit and clarify its justification.

  6. 6.

    I have presented similar grammatical justifications in various forms in a number of conference papers, such as the 1982 Australasian Association for Logic conference and the 1983 Australasian Association of Philosophy conference, as part of a more complex semantic evaluation of the Liar, but have not previously published them.

  7. 7.

    For a good account of the structural rules usually suspected in derivations of the Liar, see Murzi & Carrara (2015).

  8. 8.

    In K3 with three values {T, N, F} where only T is designated, ‘A xor B’ can be evaluated as T just in the cases where A and B take values (T,N), (N,T), (T,F) or (F,T). In LP with three values {T,B,F} where both T and B are designated, ‘A xor B’ can be evaluated as taking a designated value just in the cases where A and B take values (T,F), (F,T), (B,F), or (F,B).

  9. 9.

    As well as truth schemata, the literature contains discussion of rules for truth introduction and elimination, and a principle about the intersubstitutivity of a sentence for a sentence saying it is true. A similar restriction would affect rules for truth introduction and elimination, i.e. those rules should be restricted to predicating truth of canonical names. Then derivations using such rules will also require restricted use of substitution of identicals. Moreover, any rule about the intersubstitutability of a sentential expression and a sentence predicating truth to that expression needs to be restricted so as to only substitute sentences predicating truth of a canonical name of that sentence (cf. Field 2008). Derivations using such intersubstitutivity will also require formally restricted use of substitution of identicals.

  10. 10.

    An abbreviation is different to a name. An abbreviation is a convenience. One can always eliminate use of an abbreviation by using the abbreviated expression. Thus, Liar proofs have no dependency on use of abbreviations.

  11. 11.

    Of course, arithmetization can be used to much the same effect, for example see Hintikka (1996: 142). For another example, see the first part of the proof of Gödel’s diagonal lemma in Priest 2006 (Ch. 3).

  12. 12.

    Self-predication can be used with n-ary predicates φn(x1, …, xn), and a general version of Gödel’s diagonal lemma derived in this way (cf. Eldridge-Smith 2015). Then other paradoxes can use that lemma, such as the Unsatisfied paradox (Eldridge-Smith 2012b) and validity paradoxes (Murzi 2014). Thus, these paradoxes have the same dependency on substitution of identicals as above, and their paradoxical proofs are also resolved by managing the invalid use of substitution of identicals.

  13. 13.

    Line (20) is a valid introduction of truth with a non-canonical name because it uses only sentential logic.

  14. 14.

    This solution applies as well to proofs of “chain Liars,” also known as “circular liars”, as well as Yablo’s paradox. One can see how these proofs use Identity Elimination (=E), given schemata restricted to canonical names, in Eldridge-Smith (2015). Eldridge-Smith’s (2015) various proofs of his Unsatisfied paradox use a canonical schema for the truth relation, but one can easily map these to proofs of the Liar using the CT-schema.


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Eldridge-Smith, P. Two Fallacies in Proofs of the Liar Paradox. Philosophia (2020) doi:10.1007/s11406-019-00158-5

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