A Liar Paradox

Authors
Richard Kimberly Heck
Brown University
Abstract
The purpose of this note is to present a strong form of the liar paradox. It is strong because the logical resources needed to generate the paradox are weak, in each of two senses. First, few expressive resources required: conjunction, negation, and identity. In particular, this form of the liar does not need to make any use of the conditional. Second, few inferential resources are required. These are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬(p ∧ p), infer ¬ p. It is, interestingly enough, also essential to the argument that the ‘strong’ form of the diagonal lemma be used: the one that delivers a term λ such that we can prove: λ = ¬ T(⌈λ⌉); rather than just a sentence Λ for which we can prove: Λ ≡ ¬T(⌈Λ⌉). The truth-theoretic principles used to generate the paradox are these: ¬(S ∧ T(⌈¬S⌉); and ¬(¬S ∧ ¬T(⌈¬S⌉). These are classically equivalent to the two directions of the T-scheme, but they are intuitively weaker. The lesson I would like to draw is: There can be no consistent solution to the Liar paradox that does not involve abandoning truth-theoretic principles that should be every bit as dear to our hearts as the T-scheme. So we shall have to learn to live with the Liar, one way or another.
Keywords liar  truth
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DOI 10.1002/tht3.5
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References found in this work BETA

Saving Truth From Paradox.Hartry H. Field - 2008 - Oxford University Press.
Outline of a Theory of Truth.Saul A. Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Self-Reference and the Languages of Arithmetic.Richard Heck - 2007 - Philosophia Mathematica 15 (1):1-29.
Truth and Disquotation.Richard Heck - 2005 - Synthese 142 (3):317--352.
An Axiomatic Approach to Self-Referential Truth.Harvey Friedman & Michael Sheard - 1987 - Annals of Pure and Applied Logic 33 (1):1--21.

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Citations of this work BETA

More on 'A Liar Paradox'.Richard Heck - 2012 - Thought: A Journal of Philosophy 1 (4):270-280.
On Heck's New Liar.Julien Murzi - 2012 - Thought: A Journal of Philosophy 1 (2):258-269.
Response to Heck.David Ripley - 2012 - Thought: A Journal of Philosophy 1 (4):254-257.

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