Notre Dame Journal of Formal Logic

Intermediate Logics and Visser's Rules

Rosalie Iemhoff


Visser's rules form a basis for the admissible rules of ${\sf IPC}$. Here we show that this result can be generalized to arbitrary intermediate logics: Visser's rules form a basis for the admissible rules of any intermediate logic ${\sf L}$ for which they are admissible. This implies that if Visser's rules are derivable for ${\sf L}$ then ${\sf L}$ has no nonderivable admissible rules. We also provide a necessary and sufficient condition for the admissibility of Visser's rules. We apply these results to some specific intermediate logics and obtain that Visser's rules form a basis for the admissible rules of, for example, De Morgan logic, and that Dummett's logic and the propositional Gödel logics do not have nonderivable admissible rules.

Article information

Notre Dame J. Formal Logic, Volume 46, Number 1 (2005), 65-81.

First available in Project Euclid: 31 January 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B55: Intermediate logics
Secondary: 03B35: Mechanization of proofs and logical operations [See also 68T15]

intermediate logics intuitionistic logic admissible rules projective formulas


Iemhoff, Rosalie. Intermediate Logics and Visser's Rules. Notre Dame J. Formal Logic 46 (2005), no. 1, 65--81. doi:10.1305/ndjfl/1107220674.

Export citation


  • Baaz, M., A. Ciabattoni, and C. G. Fermüller, "Hypersequent calculi for G"ödel logics–-A survey, Journal of Logic and Computation, vol. 13 (2003), pp. 835–61.
  • Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2001.
  • Chagrov, A., and M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, The Clarendon Press, New York, 1997.
  • Dummett, M., "A propositional calculus with denumerable matrix", The Journal of Symbolic Logic, vol. 24 (1959), pp. 97–106.
  • Gabbay, D. M., and D. H. J. De Jongh, "A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property", The Journal of Symbolic Logic, vol. 39 (1974), pp. 67–78.
  • Ghilardi, S., "Unification in intuitionistic logic", The Journal of Symbolic Logic, vol. 64 (1999), pp. 859–80.
  • Ghilardi, S., "A resolution/tableaux algorithm for projective approximations in I"PC, Logic Journal of the IGPL, vol. 10 (2002), pp. 229–43.
  • Gödel, K., "Über U"nabhängigkeitsbeweise im Aussagenkalkül, Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1933), pp. 9–10.
  • Iemhoff, R., "On the admissible rules of intuitionistic propositional logic", The Journal of Symbolic Logic, vol. 66 (2001), pp. 281–94.
  • Iemhoff, R., "A(nother) characterization of intuitionistic propositional logic", Annals of Pure and Applied Logic, vol. 113 (2002), pp. 161–73. First St. Petersburg Conference on Days of Logic and Computability (1999).
  • Iemhoff, R., "Towards a proof system for admissibility", pp. 255–70 in Computer Science Logic, vol. 2803 of Lecture Notes in Computer Science, Springer, Berlin, 2003.
  • Kreisel, G., and H. Putnam, "Eine U"nableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkul, Archiv für mathematische Logic und Grundlagenforschung, vol. 3 (1957), pp. 74–78.
  • Roziere, P., Regles Admissibles en Calcul Propositionnel Intuitionniste, Ph.D. thesis, Université Paris VII, 1992.
  • Rybakov, V. V., Admissibility of Logical Inference Rules, vol. 136 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1997.