Postponement of \(\mathsf {raa}\) and Glivenko’s Theorem, Revisited

Abstract

We study how to postpone the application of the reductio ad absurdum rule (\(\mathsf {raa}\)) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of \(\mathsf {raa}\), which induces a negative translation (a variant of Kuroda’s one) from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

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Correspondence to Giulio Guerrieri.

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Special Issue: General Proof Thery.

Edited by Thomas Piecha and Peter Schroeder-Heister

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Guerrieri, G., Naibo, A. Postponement of \(\mathsf {raa}\) and Glivenko’s Theorem, Revisited. Stud Logica 107, 109–144 (2019). https://doi.org/10.1007/s11225-017-9781-5

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Keywords

  • Proof theory
  • Natural deduction
  • Negative translation
  • Reductio ad absurdum.