## Notre Dame Journal of Formal Logic

### Intermediate Logics and Visser's Rules

Rosalie Iemhoff

#### Abstract

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

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

Dates
First available in Project Euclid: 31 January 2005

https://projecteuclid.org/euclid.ndjfl/1107220674

Digital Object Identifier
doi:10.1305/ndjfl/1107220674

Mathematical Reviews number (MathSciNet)
MR2131547

Zentralblatt MATH identifier
1102.03032

Subjects
Primary: 03B55: Intermediate logics

#### Citation

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

#### References

• 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.