In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press (2007)

Greg Restall
University of Melbourne
The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes.
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Logical Consequence and the Paradoxes.Edwin Mares & Francesco Paoli - 2014 - Journal of Philosophical Logic 43 (2-3):439-469.
Pluralism and Paradox.Aaron J. Cotnoir - 2013 - In Nikolaj J. L. L. Pedersen & Cory D. Wright (eds.), Truth and Pluralism: Current Debates. Oxford University Press. pp. 339.
Model-Theoretic Semantics and Revenge Paradoxes.Lorenzo Rossi - 2019 - Philosophical Studies 176 (4):1035-1054.
Dialetheism and the Graphic Liar.Greg Littmann - 2012 - Canadian Journal of Philosophy 42 (1):15-27.

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