Studia Logica

, Volume 105, Issue 3, pp 649–664 | Cite as

Rasiowa–Harrop Disjunction Property

Article

Abstract

We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus (\(\mathbf {IPC}\)), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of \(\mathbf {IPC}\) into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of \(\mathbf {IPC}\) (presented in a recent paper co-authored with Fernando Ferreira) and answers a question then posed by Pierluigi Minari.

Keywords

Rasiowa–Harrop disjunction property Intuitionistic propositional calculus Predicative polymorphism Natural deduction Strong normalization 

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References

  1. 1.
    Dinis, B., and G. Ferreira, Instantiation overflow, Reports on Mathematical Logic 51:15–33, 2016.Google Scholar
  2. 2.
    Ferreira, F., Comments on predicative logic, Journal of Philosophical Logic 35:1–8, 2006.CrossRefGoogle Scholar
  3. 3.
    Ferreira, F., and G. Ferreira, Commuting conversions vs. the standard conversions of the “good” connectives, Studia Logica 92:63–84, 2009.CrossRefGoogle Scholar
  4. 4.
    Ferreira, F., and G. Ferreira, Atomic polymorphism, The Journal of Symbolic Logic 78:260–274, 2013.CrossRefGoogle Scholar
  5. 5.
    Ferreira, F., and G. Ferreira, The faithfulness of \({{\bf F}}_{{\bf at}}\): a proof-theoretic proof, Studia Logica 103(6):1303–1311, 2015.CrossRefGoogle Scholar
  6. 6.
    Ferreira, G., Eta-conversions of \({\bf IPC} \) implemented in atomic \({\bf F}\), To appear in Logic Jnl IGPL, published online July 1, 2016. doi:10.1093/jigpal/jzw035.
  7. 7.
    Girard, J.-Y., Y. Lafont, and P. Taylor, Proofs and Types, Cambridge University Press, 1989.Google Scholar
  8. 8.
    Minari, P., and A. Wronski, The property (HD) in intermediate logics. A partial solution of a problem of H. Ono, Reports on Mathematical Logic 22:21–25, 1988.Google Scholar
  9. 9.
    Prawitz, D., Natural Deduction, Almkvist & Wiksell, Stockholm, 1965. Reprinted, with a new preface, in Dover Publications, 2006.Google Scholar
  10. 10.
    Reynolds, J. C., Towards a theory of type structure, in B. Robinet (ed.), Lecture Notes in Computer Science, vol. 19, Colloque sur la programmation, Springer, 1974, pp. 408–425.Google Scholar
  11. 11.
    Russell, B., Principles of Mathematics, 2nd edn., George Allen and Unwin, London, 1903 (1937).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisbonPortugal

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