Archive for Mathematical Logic

, Volume 45, Issue 5, pp 581–599 | Cite as

On the rules of intermediate logics

  • Rosalie Iemhoff


If the Visser rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of the Visser rules are admissible is not known. In this paper we give a brief overview of results on admissible rules in the context of intermediate logics. We apply these results to some well-known intermediate logics. We provide natural examples of logics for which the Visser rule are derivable, admissible but nonderivable, or not admissible.

Mathematics Subject Classification (2000)

03B20 03F50 03B55 03F03 03F55 

Keywords or phrases

Intermediate logics admissible rules realizability Rieger-Nishimura formulas Medvedev logic Independence of Premise 


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  1. 1.
    Baaz, M., Ciabattoni, A., Fermüller, C.F.: Hypersequent Calculi for Gödel Logics-a Survey. J. Logic Comput. To appearGoogle Scholar
  2. 2.
    Blackburn, P., Rijke, de M., Venema, Y.: Modal Logic. Cambridge University Press, 2001Google Scholar
  3. 3.
    Chagrov, A., Zakharyaschev, M., Modal Logic. Oxford University Press, 1998Google Scholar
  4. 4.
    Dummett, M.: A propositional logic with denumerable matrix. J. Symbolic Logic 24, 96–107 (1959)MathSciNetGoogle Scholar
  5. 5.
    Fiorentini, C.: Kripke Completeness for Intermediate Logics. PhD-thesis, University of Milan, 2000Google Scholar
  6. 6.
    Gabbay, D., de Jongh, D.: A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property. J. Symbolic Logic 39, 67–78 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ghilardi, S.: Unification in intuitionistic logic. J. Symbolic Logic 64, 859–880 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ghilardi, S.: A resolution/tableaux algorithm for projective approximations in IPC. Logic J. IGPL. 10, 229–243 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Gödel, K.: Über unabhängigkeitsbeweise im Aussagenkalkül. Ergebnisse eines mathematischen Kolloquiums 4, 9–10 (1933)zbMATHGoogle Scholar
  10. 10.
    Harrop, R.: Concerning formulas of the types Open image in new window in intuitionistic formal systems. J. Symbolic Logic 25, 27–32 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Iemhoff, R.: Provability logic and admissible rules. PhD thesis, University of Amsterdam, 2001Google Scholar
  12. 12.
    Iemhoff, R.: A(nother) characterization of Intuitionistic Propositional Logic. Ann. Pure Appl. Logic 113, 161–173 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Iemhoff, R.: Towards a proof system for admissibility. Computer Science Logic '03. LNCS vol. 2803, Springer, 2003, pp. 255–270Google Scholar
  14. 14.
    Iemhoff, R.: Intermediate logics and Visser's rules. Notre Dame J. Formal Logic 46 (1), (2005)Google Scholar
  15. 15.
    Kreisel, G., Putnam, H.: Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül. Archiv für mathematische Logic und Grundlagenforschung 3, 74–78 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Medvedev, Ju.T.: Finite problems. Soc. Math. Dokl. 3, 227–230 (1962)zbMATHGoogle Scholar
  17. 17.
    Minari, P., Wronski, A.: The property (HD) in intermediate logics. Rep. Math. Logic 22, 21–25 (1988)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Nishinura, I.: On formulas of one propositional variable in intuitionistic propositional calculus. J. Symbolic Logic 25 (4), 327–331 (1960)Google Scholar
  19. 19.
    van Oosten, J.: Realizability and independence of premise. Technical report, University Utrecht, 2004Google Scholar
  20. 20.
    Prucnal, H.: On two problems of Harvey Friedman. Studia Logica 38, 257–262 (1979)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rose, G.F.: Propositional calculus and realizability. Trans. Am. Math. Soc. 75, 1–19 (1953)zbMATHCrossRefGoogle Scholar
  22. 22.
    Roziere, P.: Regles Admissibles en calcul propositionnel intuitionniste. PhD-thesis (in french), Université Paris VII, 1992Google Scholar
  23. 23.
    Rybakov, V.V.: A Criterion for Admissibility of Rules in the Modal System S4 and the Intuitionistic Logic. Algebra and logic 23 (5), 369–384 (1984)CrossRefGoogle Scholar
  24. 24.
    Rybakov, V.V.: Admissibility of Logical Inference Rules. Elsevier, 1997Google Scholar
  25. 25.
    Smoryński, C.: Applications of Kripke Models. Mathematical Investigations of Intuitionistic Arithmetic and Analysis, Springer, 1973Google Scholar
  26. 26.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, vol. I, North-Holland, 1988Google Scholar
  27. 27.
    Visser, A.: Substitutions of Σ-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic. Ann. Pure Appl. Logic 114 (1–3), 227–271 (2002)Google Scholar
  28. 28.
    Visser, A.: Rules and Arithmetics. Notre Dame J. Formal Logic 40 (1), 116–140 (1999)MathSciNetGoogle Scholar
  29. 29.
    Wronski, A.: Remarks on intermediate logics with axiomatizations containing only one variable. Reports on Mathematical Logic 2, 63–75 (1974)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Institute for Discrete Mathematics and Geometry E104Vienna University of TechnologyViennaAustria

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