Notre Dame Journal of Formal Logic 58 (4):567-581 (2017)

Authors
Luca Tranchini
Universität Tübingen
Abstract
Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to the standard ones. This shows that the notion of reduction is in need of clarification. Referring to the notion of identity of proofs in general proof theory, we argue that reduction steps should not merely remove redundancies, but must respect the identity of proofs. Consequentially, we propose to modify Tennant’s paradoxicality test by basing it on this refined notion of reduction.
Keywords Russell’s paradox   general proof theory   identity of proofs   normalization  paradoxes
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DOI 10.1215/00294527-2017-0017
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References found in this work BETA

Ideas and Results in Proof Theory.Dag Prawitz & J. E. Fenstad - 1971 - Journal of Symbolic Logic 40 (2):232-234.
Identity of Proofs Based on Normalization and Generality.Kosta Došen - 2003 - Bulletin of Symbolic Logic 9 (4):477-503.
Proof and Paradox.Neil Tennant - 1982 - Dialectica 36 (2‐3):265-296.

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