Notre Dame Journal of Formal Logic

Ekman’s Paradox

Peter Schroeder-Heister and Luca Tranchini

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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.

Article information

Notre Dame J. Formal Logic, Volume 58, Number 4 (2017), 567-581.

Received: 26 February 2014
Accepted: 6 July 2015
First available in Project Euclid: 18 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F05: Cut-elimination and normal-form theorems 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 00A30: Philosophy of mathematics [See also 03A05]

paradoxes Russell’s paradox normalization general proof theory identity of proofs


Schroeder-Heister, Peter; Tranchini, Luca. Ekman’s Paradox. Notre Dame J. Formal Logic 58 (2017), no. 4, 567--581. doi:10.1215/00294527-2017-0017.

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