Notre Dame Journal of Formal Logic

Metalogic of Intuitionistic Propositional Calculus

Alex Citkin


With each superintuitionistic propositional logic L with a disjunction property we associate a set of modal logics the assertoric fragment of which is L. Each formula of these modal logics is interdeducible with a formula representing a set of rules admissible in L. The smallest of these logics contains only formulas representing derivable in L rules while the greatest one contains formulas corresponding to all admissible in L rules. The algebraic semantic for these logics is described.

Article information

Notre Dame J. Formal Logic, Volume 51, Number 4 (2010), 485-502.

First available in Project Euclid: 29 September 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B55: Intermediate logics 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25]
Secondary: 06D20: Heyting algebras [See also 03G25]

intuitionistic logic modal logic admissible rule Heyting algebra monadic algebra intermediate logic


Citkin, Alex. Metalogic of Intuitionistic Propositional Calculus. Notre Dame J. Formal Logic 51 (2010), no. 4, 485--502. doi:10.1215/00294527-2010-031.

Export citation


  • [1] Baker, K. A., "Finite equational bases for finite algebras in a congruence-distributive equational class", Advances in Mathematics, vol. 24 (1977), pp. 207--43.
  • [2] Bezhanishvili, G., "Varieties of monadic Heyting algebras. I", Studia Logica, vol. 61 (1998), pp. 367--402.
  • [3] Bezhanishvili, G., "Glivenko type theorems for intuitionistic modal logics", Studia Logica, vol. 67 (2001), pp. 89--109.
  • [4] Citkin, A., "On admissible rules of intuitionistic propositional logic", Mathematics of the USSR, Sbornik, vol. 31 (1977), pp. 279--88.
  • [5] Citkin, A., On Modal Logics for Reviewing Admissible Rule of Intuitionistic Logic, VINITI, Moscow, 1978. Preprint, (In Russian).
  • [6] Citkin, A., "On modal logic of intutionistic admissibility", pp. 105--107 in Modal and Tense Logic, Second Soviet-Finnish Colloquium in Logic, Vilnus, 1979. (In Russian).
  • [7] 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.
  • [8] Ghilardi, S., "Unification in intuitionistic logic", The Journal of Symbolic Logic, vol. 64 (1999), pp. 859--80.
  • [9] Harrop, R., "Concerning formulas of the types $A\rightarrow B\bigvee C,\,A\rightarrow (Ex)B(x)$" in intuitionistic formal systems, The Journal of Symbolic Logic, vol. 25 (1960), pp. 27--32.
  • [10] Iemhoff, R., Provability Logic and Admissible Rules, ILLC Publications, Amsterdam, 2001.
  • [11] Iemhoff, R., "A(nother) characterization of intuitionistic propositional logic", First St. Petersburg Conference on Days of Logic and Computability (1999), Annals of Pure and Applied Logic, vol. 113 (2002), pp. 161--73.
  • [12] Iemhoff, R., and G. Metcalfe, "Proof theory for admissible rules", Annals of Pure and Applied Logic, vol. 159 (2009), pp. 171--86.
  • [13] Jeřábek, E., "Complexity of admissible rules", Archive for Mathematical Logic, vol. 46 (2007), pp. 73--92.
  • [14] Jeřábek, E., "Canonical rules", The Journal of Symbolic Logic, vol. 74 (2009), pp. 1171--1205.
  • [15] Maksimova, L. L., and V. V. Rybakov, "The lattice of normal modal logics", Algebra and Logic, vol. 13 (1976), pp. 105--22. (Algebra i Logika, vol. 13 (1974), pp. 188--216, 235).
  • [16] Ono, H., "On some intuitionistic modal logics", Research Institute for Mathematical Sciences. Kyoto University, vol. 13 (1977/78), pp. 687--722.
  • [17] Rosière, P., Règles Admissibles en Calcul Propositionnel Intuitionniste, Ph.D. thesis, Université Paris VII, Paris, 1992.
  • [18] Rybakov, V. V., "Bases of admissible rules of the logics $\rm S4$" and $\rm Int$, Algebra and Logic, vol. 24 (1985), pp. 55--68. (Algebra i Logika, vol. 24 (1985), pp. 87--107, 123).
  • [19] 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.
  • [20] Werner, H., "Varieties generated by quasi-primal algebras have decidable theories", pp. 555--75 in Colloquia Mathematica Societatis János Bolyai. Vol. 17. Contributions to Universal Algebra (József Attila University, Szeged, 1975), North-Holland, Amsterdam, 1977.