Curry, Yablo and duality

Analysis 69 (4):612-620 (2009)
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Abstract

The Liar paradox is the directly self-referential Liar statement: This statement is false.or : " Λ: ∼ T 1" The argument that proceeds from the Liar statement and the relevant instance of the T-schema: " T ↔ Λ" to a contradiction is familiar. In recent years, a number of variations on the Liar paradox have arisen in the literature on semantic paradox. The two that will concern us here are the Curry paradox, 2 and the Yablo paradox. 3The Curry paradox demonstrates that neither negation nor a falsity predicate is required in order to generate semantic paradoxes. Given any statement Φ whatsoever, we need merely consider the statement: If this statement is true, then Φor: " Ξ: T → Φ" Here, via familiar reasoning, one can ‘prove’ Φ merely through consideration of statement Ξ and the Ξ-instance of the T-schema.Interestingly, the Liar paradox can be viewed as nothing more than a special case of the Curry paradox. If we define negation in terms of the conditional and a primitive absurdity constant ‘⊥’: 4" ∼ Ψ = df Ψ → ⊥" then the Liar paradox is simply the instance of the Curry paradox obtained by substituting ‘⊥’ for Φ.The Yablo paradox demonstrates that circularity is also not required in order to generate semantic paradox. 5 The paradox proceeds by considering an infinite ω-sequence of statements of the form: S 1: ) S 2: ) S 3: ) : : : : : S i: ) : : : : : :– that is, the set of …

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Roy T. Cook
University of St. Andrews

Citations of this work

Truth, Revenge, and Internalizability.Kevin Scharp - 2014 - Erkenntnis 79 (3):597-645.

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References found in this work

Paradox without Self-Reference.Stephen Yablo - 1993 - Analysis 53 (4):251-252.
The inconsistency of certain formal logic.Haskell B. Curry - 1942 - Journal of Symbolic Logic 7 (3):115-117.
Truth and reflection.Stephen Yablo - 1985 - Journal of Philosophical Logic 14 (3):297 - 349.
The Inconsistency of Certain Formal Logics.Alonzo Church & Haskell B. Curry - 1942 - Journal of Symbolic Logic 7 (4):170.

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