Journal of Logic, Language and Information

, Volume 12, Issue 2, pp 213–225

A Cut-Free Gentzen Formulation of Basic Propositional Calculus

  • Kentaro Kikuchi
  • Katsumi Sasaki


We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. The cut-elimination theorem isproved in a syntactic way by modifying Gentzen's method. Thisdual-context style system exemplifies the effectiveness of dual-contextformulation in formalizing various non-classical logics.

Basic Propositional Calculus cut-elimination dual-context system Gentzen system Kripke model 


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  1. Aghaei, M. and Ardeshir, M., 2000, “A bounded translation of intuitionistic propositional logic into basic propositional logic,” Mathematical Logic Quarterly 46, 199–206.CrossRefGoogle Scholar
  2. Aghaei,M. and Ardeshir, M., 2001, “Gentzen-style axiomatizations for some conservative extensions of basic propositional logic,” Studia Logica 68, 263–285.CrossRefGoogle Scholar
  3. Ardeshir, M., 1995, “Aspects of basic logic,” Ph.D. Thesis, Marquette University, Milwaukee.Google Scholar
  4. Ardeshir, M. and Ruitenburg, W., 1998, “Basic propositional calculus I,” Mathematical Logic Quarterly 44, 317–343.Google Scholar
  5. Ardeshir, M. and Ruitenburg, W., 2001, “Basic propositional calculus II. Interpolation,” Archive for Mathematical Logic 40, 349–364.Google Scholar
  6. Barber, A., 1997, “Linear type theories, semantics and action calculi,” Ph.D. Thesis, LFCS, University of Edinburgh.Google Scholar
  7. Bierman, G.M. and de Paiva, V.C.V., 2000, “On an intuitionistic modal logic,” Studia Logica 65, 383–416.Google Scholar
  8. Davies, R. and Pfenning, F., 1996, “A modal analysis of staged computation,” pp. 258–270 in Proceedings of the 23rd Annual Symposium on Principles of Programming Languages, St. Petersburg Beach, FL, G. Steele, Jr., ed., New York: ACM Press.Google Scholar
  9. Gentzen, G., 1935, “Untersuchungen über das logische Schliessen,” Mathematische Zeitschrift 39, 176–210, 405–431. English translation: pp. 68–131 in The Collected Papers of Gerhard Gentzen, M.E. Szabo, ed., Amsterdam: North-Holland.Google Scholar
  10. Girard, J.-Y., 1993, “On the unity of logic,” Annals of Pure and Applied Logic 59, 201–217.Google Scholar
  11. Heuerding, A., Seyfried, M., and Zimmermann, H., 1996, “Efficient loop-check for backward proof search in some non-classical propositional logics,” pp. 210–225 in Proceedings of Theorem Proving with Analytic Tableaux and Related Methods, 5th International Workshop, TABLEAUX’ 96, Terrasini, Palermo, Italy, P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, eds., Lecture Notes in Artificial Intelligence, Vol. 1071, Berlin: Springer-Verlag.Google Scholar
  12. Hodas, J.S. and Miller, D., 1994, “Logic programming in a fragment of intuitionistic linear logic,” Information and Computation 110, 327–365.Google Scholar
  13. Ishii, K., Kashima, R., and Kikuchi, K., 2000, “Sequent calculi for Visser's propositional logics,” Notre Dame Journal of Formal Logic, to appear.Google Scholar
  14. Kikuchi, K., 2002, “Dual-context sequent calculus and strict implication,” Mathematical Logic Quarterly 48, 87–92.Google Scholar
  15. Masini, A., 1992, “2-Sequent calculus: A proof theory of modalities,” Annals of Pure and Applied Logic 58, 229–246.Google Scholar
  16. Ruitenburg, W., 1999, “Basic logic, K4, and persistence,” Studia Logica 63, 343–352.Google Scholar
  17. Sasaki, K., 1998, “A Gentzen-style formulation for Visser's propositional logic,” Nanzan Management Review 12, 343–351.Google Scholar
  18. Sasaki, K., 1999, “Formalizations for the consequence relation of Visser's propositional logic,” Reports on Mathematical Logic 33, 65–78.Google Scholar
  19. Suzuki, Y., Wolter, F., and Zakharyaschev, M., 1998, “Speaking about transitive frames in propositional languages,” Journal of Logic, Language and Information 7, 317–339.Google Scholar
  20. Szabo, M.E., ed., 1969, The Collected Papers of Gerhard Gentzen, Amsterdam: North-Holland.Google Scholar
  21. Visser, A., 1981, “A propositional logic with explicit fixed points,” Studia Logica 40, 155–175.Google Scholar
  22. Wansing, H., 1997, “Displaying as temporalizing, sequent systems for subintuitionistic logics,” pp. 159–178 in Logic, Language and Computation, S. Akama, ed., Dordrecht: Kluwer Academic Publishers.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Kentaro Kikuchi
    • 1
  • Katsumi Sasaki
    • 2
  1. 1.Department of Mathematics and InformaticsChiba UniversityInage-ku, ChibaJapan
  2. 2.Department of Mathematical SciencesNanzan UniversitySeto-si, AichiJapan

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