TY - JOUR

T1 - Two Fallacies in Proofs of the Liar Paradox

AU - Eldridge-Smith, Peter

N1 - Publisher Copyright:
© 2020, Springer Nature B.V.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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.

AB - 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.

KW - Knower paradox

KW - Liar paradox

KW - Montague’s theorem

KW - Substitution of identicals

KW - Tarski’s indefinability theorem

KW - Truth

UR - http://www.scopus.com/inward/record.url?scp=85078046388&partnerID=8YFLogxK

U2 - 10.1007/s11406-019-00158-5

DO - 10.1007/s11406-019-00158-5

M3 - Article

SN - 0048-3893

VL - 48

SP - 947

EP - 966

JO - Philosophia (United States)

JF - Philosophia (United States)

IS - 3

ER -