Studia Logica

, Volume 45, Issue 1, pp 39–53 | Cite as

A cut-free gentzen-type system for the logic of the weak law of excluded middle

  • Branislav R. Boričić


The logic of the weak law of excluded middleKC p is obtained by adding the formula ℸA ∨ ℸ ℸA as an axiom scheme to Heyting's intuitionistic logicH p . A cut-free sequent calculus for this logic is given. As the consequences of the cut-elimination theorem, we get the decidability of the propositional part of this calculus, its separability, equality of the negationless fragments ofKC p andH p , interpolation theorems and so on. From the proof-theoretical point of view, the formulation presented in this paper makes clearer the relations betweenKC p ,H p , and the classical logic. In the end, an interpretation of classical propositional logic in the propositional part ofKC p is given.


Mathematical Logic Computational Linguistic Classical Logic Propositional Logic Sequent Calculus 


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  1. [1]
    J. G. Anderson,Superconstructive propositional calculi with extra axiom schemecontaining one variable,Zeitschrift für mathematische Logik und Grunds lagen der Mathematik, 18 (1972), pp. 113–130.Google Scholar
  2. [2]
    W. Craig,Linear reasoning. A new form of the Herbrand-Gentzen theorem,The Journal of Symbolic Logic, 22 (1957), pp. 250–268.Google Scholar
  3. [3]
    D. M. Gabbay, Semantic proof of the Craig interpolation theorem for intuitio nistic logic and extensions,Logic Colloquium '69, North-Holland, Amsterdam, 1971, pp. 391–410 andThe Journal of Symbolic Logic, 42 (1977), pp. 269–271.Google Scholar
  4. [4]
    D. M. Gabbay,Semantical Investigations in Heyting's Intuitionistic Logic, D. Reidel, Dordrecht, 1981.Google Scholar
  5. [5]
    L. Henkin,An extension of the Craig-Lyndon interpolation theorem,The Journal of Symbolic Logic, 28 (1963), pp. 201–216.Google Scholar
  6. [6]
    T. Hosoi,On intermediate logics III,Journal of Tsuda College, 6 (1974), pp. 23–38.Google Scholar
  7. [7]
    V. A. Jankov,Some superconstructive propositional calculi,Soviet Mathematics, Doklady, 4 (1963), pp. 1103–1105. (Orig.:Доклады Академии наук CCCP, 151 (1963), pp. 796–798.)Google Scholar
  8. [8]
    V. A. Jankov,The calculus of the weak “law of excluded middle”,Math. USSR Izvestija, 2 (1968), pp. 997–1004, (1970). (Orig.:Иавестия Академии наук CCCP, 32 (1968), pp. 1044–1051.)Google Scholar
  9. [9]
    E. G. K. López-Escobar,On the interpolation theorem for logic of constant do mains,The Journal of Symbolic Logic, 46 (1981), pp. 87–88 and 48 (1983), pp. 595–599.Google Scholar
  10. [10]
    R. C. Lyndon,An interpolation theorem in the predicate calculus,Pacific Journal of Mathematics, 9 (1959), pp. 129–142.Google Scholar
  11. [11]
    L. L. Maximova,craig interpolation theorem and amalgamable varieties,Soviet Mathematics, Doklady, 18 (1977), pp. 1550–1553. (Orig.:Доклады Академии наук CCCP, 237 (1977), pp. 1281–1284.)Google Scholar
  12. [12]
    L. L. Maximova,Interpolation properties of superintuitionistic logics,Studia Logica, 38 (1979), pp. 419–428.Google Scholar
  13. [13]
    Л. J1. МАКСИМОВА,Интерполяуионная теорема Луноона в модальных логиках, Математическая логика и теория алгорифмов, Академия наук CCCP —Сибирское отделение, Наука, Москва, 1982, pp. 45–55.Google Scholar
  14. [14]
    C. Rauszer,On a certain cut-free axiomatization of some intermediate logics,The Journal of Symbolic Logic, 49 (1984), p. 703.Google Scholar
  15. [15]
    O. Sonobe,A Gentzen-type formulation of some intermediate prepositional logics,Journal of Tsuda College, 7 (1975), pp. 7–14.Google Scholar
  16. [16]
    M. E. Szabo ed.,The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969.Google Scholar
  17. [17]
    M. E. Szabo,Algebra of Proofs, North-Holland, Amsterdam, 1978.Google Scholar
  18. [18]
    G. Takeuti,Proof Theory, North-Holland, Amsterdam, 1975.Google Scholar
  19. [19]
    T. Umezawa,On intermediate propositional logics,The Journal of Symbolic Logic, 24 (1959), pp. 20–36.Google Scholar
  20. [20]
    T. Umezawa,On logics intermediate between intuitionistic and classical predicate logic,The Journal of Symbolic Logic, 24 (1959), pp. 141–153.Google Scholar
  21. [21]
    S. Zachorowski,Remarks on interpolation property for intermediate logics,Reports on Mathematical Logic, 10 (1978), pp. 139–146.Google Scholar

Copyright information

© Polish Academy of Sciences 1986

Authors and Affiliations

  • Branislav R. Boričić
    • 1
  1. 1.Ekonomski FakultetUniverzitet u BeograduBeogradJugoslavija

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