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Studia Logica

, Volume 53, Issue 3, pp 433–457 | Cite as

Cut-free sequent and tableau systems for propositional Diodorean modal logics

  • Rajeev Goré
Article

Abstract

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality □ is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.

Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the “superformulae” involved are always bounded by a finite set of formulaeX*L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive.

Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems. The techniques are due to Hintikka and Rautenberg.

Keywords

Mathematical Logic Modal Logic Decision Procedure Computational Linguistic Completeness Proof 
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.

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References

  1. [1]
    R. A. Bull,An algebraic study of Diodorean modal systems Journal of Symbolic Logic 30(1), 1965, pp. 58–64.Google Scholar
  2. [2]
    R. A. Bull,Review of ‘Melvin Fitting, Proof Methods for Modal and Intuitionistic Logics, Synthese Library, Vol. 169, Reidel, 1983’ Journal of Symbolic Logic 50, 1985, pp. 855–856.Google Scholar
  3. [3]
    R. A. Bull andK. Segerberg,Basic modal logic, In D. Gabbay and F. Guenthner, editors,Handbook of Philosophical Logic, Volume II: Extensions of Classical Logic, pp. 1–88. D. Reidel 1984.Google Scholar
  4. [4]
    M. Fitting,Proof Methods for Modal and Intuitionistic Logics, volume 169of Synthese Library, D. Reidel, Dordrecht, Holland 1983.Google Scholar
  5. [5]
    M. Fitting,First order modal tableaux Journal of Automated Reasoning 4, 1988, pp. 191–213.Google Scholar
  6. [6]
    R. I. Goldblatt,Logics of Time and Computation, CSLI Lecture Notes Number 7, CSLI Stanford 1987.Google Scholar
  7. [7]
    G. E. Hughes andM. J. Cresswell,Introduction to Modal Logic Methuen, London 1968.Google Scholar
  8. [8]
    G. E. Hughes andM. J. Cresswell,A Companion to Modal Logic Methuen, London 1984.Google Scholar
  9. [9]
    E. J. Lemmon andD. Scott,An Introduction To Modal Logic American Philosophical Quarterly, Monograph Series, Basil Blackwell, Oxford 1977.Google Scholar
  10. [10]
    W. Rautenberg,Modal tableau calculi and interpolation Journal of Philosophical Logic 12, 1983, pp. 403–423.Google Scholar
  11. [11]
    W. Rautenberg,Corrections for modal tableau calculi and interpolation by W. Rautenberg, JPL 12 (1983) Journal of Philosophical Logic 14, 1985, p. 229.Google Scholar
  12. [12]
    W. Rautenberg, Personal communication, December 5th, 1990.Google Scholar
  13. [13]
    K. Segerberg,An essay in classical modal logic (3 vols.), Technical Report Filosofiska Studier, nr 13, Uppsala Universitet, Uppsala 1971.Google Scholar
  14. [14]
    T. Shimura,Cut-free systems for the modal logic S4.3 and S4.3GRZ Reports on Mathematical Logic 25, 1991, pp. 57–73.Google Scholar
  15. [15]
    S. Valentini,A syntactic proof of cut elimination for GL lin Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32, 1986, pp. 137–144.Google Scholar
  16. [16]
    J. F. A. K. van Benthem andW. Blok,Transitivity follows from Dummett's axiom Theoria 44 1978, pp. 117–118.Google Scholar
  17. [17]
    P. Wolper,Temporal logic can be more expressive Information and Control 56, 1983, pp. 72–99.Google Scholar
  18. [18]
    J. J. Zeman,Modal Logic: The Lewis-Modal Systems, Oxford University Press 1973.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Rajeev Goré
    • 1
  1. 1.Department of Computer ScienceUniversity of ManchesterManchesterEngland

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