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Studia Logica

, Volume 107, Issue 1, pp 109–144 | Cite as

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

  • Giulio GuerrieriEmail author
  • Alberto Naibo
Article

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.

Keywords

Proof theory Natural deduction Negative translation Reductio ad absurdum. 

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Informatica–Scienza e Ingegneria (DISI)Alma Mater Studiorum–Università di BolognaBolognaItaly
  2. 2.IHPST (UMR 8590)Université Paris 1 Panthéon–Sorbonne, CNRS, ENSParisFrance

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