Metalogic of Intuitionistic Propositional Calculus
Alex Citkin
Source: Notre Dame J. Formal Logic Volume 51, Number 4
(2010), 485-502.
Abstract
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.
First Page:
Show
Hide
Keywords: intuitionistic logic; modal logic; admissible rule; Heyting algebra; monadic algebra; intermediate logic
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1285765801
Digital Object Identifier: doi:10.1215/00294527-2010-031
Zentralblatt MATH identifier: 05822359
Mathematical Reviews number (MathSciNet): MR2741839
References
[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.
Mathematical Reviews (MathSciNet): MR0447074
Zentralblatt MATH: 0356.08006
[2] Bezhanishvili, G., "Varieties of monadic Heyting algebras. I", Studia Logica, vol. 61 (1998), pp. 367--402.
Mathematical Reviews (MathSciNet): MR1657117
Zentralblatt MATH: 0964.06008
Digital Object Identifier: doi:10.1023/A:1005073905902
[3] Bezhanishvili, G., "Glivenko type theorems for intuitionistic modal logics", Studia Logica, vol. 67 (2001), pp. 89--109.
Mathematical Reviews (MathSciNet): MR1833655
Zentralblatt MATH: 1045.03020
Digital Object Identifier: doi:10.1023/A:1010577628486
[4] Citkin, A., "On admissible rules of intuitionistic propositional logic", Mathematics of the USSR, Sbornik, vol. 31 (1977), pp. 279--88.
Zentralblatt MATH: 0386.03011
Mathematical Reviews (MathSciNet): MR465801
[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.
Mathematical Reviews (MathSciNet): MR0373838
Zentralblatt MATH: 0289.02032
Digital Object Identifier: doi:10.2307/2272344
JSTOR: links.jstor.org
[8] Ghilardi, S., "Unification in intuitionistic logic", The Journal of Symbolic Logic, vol. 64 (1999), pp. 859--80.
Mathematical Reviews (MathSciNet): MR1777792
Zentralblatt MATH: 0930.03009
Digital Object Identifier: doi:10.2307/2586506
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet): MR0130832
Zentralblatt MATH: 0098.24201
Digital Object Identifier: doi:10.2307/2964334
JSTOR: links.jstor.org
[10] Iemhoff, R., Provability Logic and Admissible Rules, ILLC Publications, Amsterdam, 2001.
Mathematical Reviews (MathSciNet): MR1838256
Zentralblatt MATH: 0995.03045
[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.
Mathematical Reviews (MathSciNet): MR1875741
Zentralblatt MATH: 0988.03045
Digital Object Identifier: doi:10.1016/S0168-0072(01)00056-2
[12] Iemhoff, R., and G. Metcalfe, "Proof theory for admissible rules", Annals of Pure and Applied Logic, vol. 159 (2009), pp. 171--86.
Mathematical Reviews (MathSciNet): MR2523716
Zentralblatt MATH: 1174.03024
Digital Object Identifier: doi:10.1016/j.apal.2008.10.011
[13] Jeřábek, E., "Complexity of admissible rules", Archive for Mathematical Logic, vol. 46 (2007), pp. 73--92.
Mathematical Reviews (MathSciNet): MR2298605
Zentralblatt MATH: 1115.03010
Digital Object Identifier: doi:10.1007/s00153-006-0028-9
[14] Jeřábek, E., "Canonical rules", The Journal of Symbolic Logic, vol. 74 (2009), pp. 1171--1205.
Mathematical Reviews (MathSciNet): MR2583815
Zentralblatt MATH: 1186.03045
Digital Object Identifier: doi:10.2178/jsl/1254748686
Project Euclid: euclid.jsl/1254748686
[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).
Mathematical Reviews (MathSciNet): MR0363810
Zentralblatt MATH: 0315.02027
[16] Ono, H., "On some intuitionistic modal logics", Research Institute for Mathematical Sciences. Kyoto University, vol. 13 (1977/78), pp. 687--722.
Mathematical Reviews (MathSciNet): MR0476407
Zentralblatt MATH: 0373.02026
Digital Object Identifier: doi:10.2977/prims/1195189604
[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).
Mathematical Reviews (MathSciNet): MR816572
Zentralblatt MATH: 0598.03014
[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.
Mathematical Reviews (MathSciNet): MR1454360
Zentralblatt MATH: 0872.03002
[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.
Mathematical Reviews (MathSciNet): MR0480273
Zentralblatt MATH: 0375.02044
Notre Dame Journal of Formal Logic
