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


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.

This is a preview of subscription content, log in to check access.


  1. 1.

    Ariola, Z. M., H. Herbelin, and A. Sabry, A proof-theoretic foundation of abortive continuations, Higher-Order and Symbolic Computation 20(4):403–429, 2007.

    Article  Google Scholar 

  2. 2.

    Brown, C. E., and C. Rizkallah, Glivenko and Kuroda for Simple Type Theory, The Journal of Symbolic Logic 79(2):485–495, 2014.

    Article  Google Scholar 

  3. 3.

    Ertola, R., and M. Sagastume, Subminimal logic and weak algebras, Reports on Mathematical Logic 44:153–166, 2009.

    Google Scholar 

  4. 4.

    Espíndola, C., A short proof of Glivenko theorems for intermediate predicate logics, Archive for Mathematical Logic 52(7):823–826, 2013.

    Article  Google Scholar 

  5. 5.

    Farahani, H., and H. Ono, Glivenko theorems and negative translations in substructural predicate logics, Archive for Mathematical Logic 51(7):695–707, 2012.

    Article  Google Scholar 

  6. 6.

    Ferreira, G., and P. Oliva, On the relation between various negative translations, in U. Berger, H. Diener, P. Schuster, and M. Seisenberger (eds.), Logic, Construction, Computation, vol. 3 of Ontos-Verlag Mathematical Logic Series. De Gruyter, 2012, pp. 227–258.

  7. 7.

    Galatos, N., and H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71:1353–1384, 2006.

    Article  Google Scholar 

  8. 8.

    Glivenko, V., Sur quelques points de la logique de M. Brouwer, Bulletins de la Classe des Sciences 15(5):183–188, 1929.

    Google Scholar 

  9. 9.

    Guerrieri, G., and A. Naibo, Postponement of raa and Glivenko’s Theorem, Revisited (long version), ArXiv e-prints,, 2017.

  10. 10.

    Kleene, S. C., Introduction to metamathematics, North-Holland, 1952.

  11. 11.

    Kuroda, S., Intuitionistische untersuchungen der formalistischen logik, Nagoya Mathematical Journal 2:35–47, 1951.

    Article  Google Scholar 

  12. 12.

    Murthy, C., Extracting Constructive Content From Classical Proofs, Ph.D. thesis, Cornell University, 1990.

  13. 13.

    Nadathur, G., Uniform provability in classical logic, Journal of Logic and Computation 8(2):209–229, 1998.

    Article  Google Scholar 

  14. 14.

    Ono, H., Glivenko theorems revisited, Annals of Pure and Applied Logic 161(2):246–250, 2009.

    Article  Google Scholar 

  15. 15.

    Pereira, L. C., Translations and normalization procedures (abstract), in WoLLIC’2000—7th Workshop on Logic, Language, Information and Computation, pp. 21–24., 2000.

  16. 16.

    Pereira, L. C., and E. H. Haeusler, On constructive fragments of classical logic, in H. Wansing, (ed.), Dag Prawitz on Proofs and Meaning, vol. 7 of Outstanding Contributions to Logic, Springer, Berlin, 2015, pp. 281–292.

    Google Scholar 

  17. 17.

    von Plato, J., Elements of Logical Reasoning, Cambridge University Press, 2013.

  18. 18.

    von Plato, J., and A. Siders, Normal derivability in classical natural deduction, The Review of Symbolic Logic 5:205–211, 2012.

    Article  Google Scholar 

  19. 19.

    Prawitz, D., Natural Deduction: A proof-theoretical study, Almqvist & Wiksell, Stockholm, 1965.

    Google Scholar 

  20. 20.

    Seldin, J. P., On the proof theory of the intermediate logic MH, The Journal of Symbolic Logic 51(3):626–647, 1986.

    Article  Google Scholar 

  21. 21.

    Seldin, J. P., Normalization and excluded middle. I, Studia Logica 48(2):193–217, 1989.

    Article  Google Scholar 

  22. 22.

    Stålmarck, G., Normalization theorems for full first order classical natural deduction, The Journal of Symbolic Logic 56(1):129–149 1991.

    Article  Google Scholar 

  23. 23.

    Statman, R., Structural Complexity of Proofs, Ph.D. thesis, Stanford University, 1974.

  24. 24.

    Tennant, N., Anti-Realism and Logic: Truth as eternal, Oxford University Press, Oxford, 1987.

    Google Scholar 

  25. 25.

    Zdanowski, K., On second order intuitionistic propositional logic without a universal quantifier, The Journal of Symbolic Logic 74(1):157–167, 2009.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Giulio Guerrieri.

Additional information

Special Issue: General Proof Thery.

Edited by Thomas Piecha and Peter Schroeder-Heister

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guerrieri, G., Naibo, A. Postponement of \(\mathsf {raa}\) and Glivenko’s Theorem, Revisited. Stud Logica 107, 109–144 (2019).

Download citation


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