Studia Logica

, Volume 53, Issue 2, pp 299–324 | Cite as

Refutation systems in modal logic

  • Valentin Goranko


Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.


Mathematical Logic Modal Logic Decision Procedure Computational Linguistic Deductive System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. F. A. K. van Benthem,Modal logic and classical logic, Bibliopolis, Napoli, 1986.Google Scholar
  2. [2]
    G. Bryll andM. Maduch,Rejected axioms for Łukasiewicz's many-valued logics, (in Polish), Zeszyty Naukowe Wyższej Szkoly Pedagogicznej w Opolu,Mathematyka VI, Logika i algebra (1969), pp. 3 – 19.Google Scholar
  3. [3]
    R. Dutkiewicz,The method of axiomatic rejection for the intuitionistic propositional logic,Studia Logica,XLVIII,4, (1989), pp. 449–460.Google Scholar
  4. [4]
    G. G. Hughes, M. J. Cresswell,A companion to modal logic, Methuen, London, 1984.Google Scholar
  5. [5]
    T. Inoue,On Ishimoto's theorem in axiomatic rejection — the philosophy of unprovability (in Japanese),Philosophy of Science, 22 (Waseda University Press, Tokyo, 1989), pp. 77–93.Google Scholar
  6. [6]
    T. Inoue, A. Ishimoto andM. Kobayashi,Axiomatic rejection for the propositional fragment of Leśniewski's ontology, Soviet-Japan Symposium “Leśniewski's ontology and its applications”, Moscow 1990.Google Scholar
  7. [7]
    G. Kreisel andH. Putnam,Unableitbarkeitbeweismethode für den intuitionistischen Aussagenkalkul,Archiv Math. Logik Grundlagenforsch, 3, (1957), pp. 74–78.Google Scholar
  8. [8]
    J. Łukasiewicz,On the intuitionistic theory of deduction,Indag. Math., 14, (1952), pp. 202–212.Google Scholar
  9. [9]
    J. Łukasiewicz,Aristotle's syllogistic from the standpoint of modern formal logic, Oxford 1957.Google Scholar
  10. [10]
    D. Scott,Completeness proofs for the intuitionistic sentential calculus, Summaries of talks presented at the Summer Institute for Symbolic Logic, Inst. for defense Analyses, Cornell University 1957, 2nd. ed., Communications Research division, Princeton 1960, pp. 231–241.Google Scholar
  11. [11]
    K. Segerberg,An essay in classical modal logic Filosofiska Studier 13, Uppsala, 1971.Google Scholar
  12. [12]
    T. Skura,A complete syntactical characterization of the intuitionistic logic,Reports on mathematical logic,23 (1989), pp. 75–80.Google Scholar
  13. [13]
    T. Skura,On pure refutation formulations of sentential logics,Bull. of Sect. of Logic, PAS, vol. 19, 3 (1990), pp. 102–107.Google Scholar
  14. [14]
    T. Skura,On decision procedures for sentential logics,Studia Logica, L 2 (1991), pp. 173–179.Google Scholar
  15. [15]
    T. Skura,Refutation rules for three modal logics,Bull. of Sect. of Logic, vol.21 1 (1992), pp. 31–32.Google Scholar
  16. [16]
    J. Słupecki andG. Bryll,Proof of Ł-decidability of Lewis system S5,Studia Logica,XXXII (1973), pp. 99–105.Google Scholar
  17. [17]
    J. Słupecki, G. Bryll andU. Wybraniec-Skardowska,Theory of rejected propositions, I,Studia Logica XXIX (1971), pp. 75–115;II,Studia Logica, XXX (1972), pp. 97 – 115.Google Scholar
  18. [18]
    W. Staszek,On proofs of rejection,Studia Logica,XXIX (1971), pp. 17–23.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Valentin Goranko
    • 1
  1. 1.Department of MathematicsUniversity of the North, Qwaqwa BranchPhuthaditjhabaRepublic of South Africa

Personalised recommendations