David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Thought: A Journal of Philosophy 1 (1):36-40 (2012)
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.
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References found in this work BETA
Hartry H. Field (2008). Saving Truth From Paradox. Oxford University Press.
Harvey Friedman & Michael Sheard (1987). An Axiomatic Approach to Self-Referential Truth. Annals of Pure and Applied Logic 33 (1):1--21.
Harvey Friedman & Michael Sheard (1988). The Disjunction and Existence Properties for Axiomatic Systems of Truth. Annals of Pure and Applied Logic 40 (1):1--10.
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Citations of this work BETA
Richard Heck (2012). More on 'A Liar Paradox'. Thought: A Journal of Philosophy 1 (4):270-280.
Julien Murzi (2012). On Heck's New Liar. Thought: A Journal of Philosophy 1 (2):258-269.
Elia Zardini (2012). It Is Not the Case That [P and 'It Is Not the Case That P' Is True] nor Is It the Case That [P and 'P' Is Not True]. Thought: A Journal of Philosophy 1 (4):309-319.
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