Graded modalities in epistemic logic

  • W. van der Hoek
  • J. -J. Ch. Meyer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 620)


We propose an epistemic logic with so-called graded modalities, in which certain types of knowledge are expressible that are less absolute than in traditional epistemic logic. Beside ‘absolute knowledge’ (which does not allow for any exception), we are also able to express ‘accepting ϕ if there at most n exceptions to ϕ’. This logic may be employed in decision support systems where there are different sources to judge the same proposition. We argue that the logic also provides a link between epistemic logic and the more quantitative (even probabilistic) methods used in AI systems. In this paper we investigate some properties of the logic as well as some applications.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AP88]
    P. Atzeni and D.S. Parker. Set containment inference and syllogisms. Theoretical Computer Science, 62:39–65, 1988.CrossRefGoogle Scholar
  2. [FC88]
    M. Fattorosi-Barnaba and C. Cerrato. Graded modalities III. Studia Logica, 47:99–110, 1988.CrossRefGoogle Scholar
  3. [FH88]
    R.F. Fagin and J.Y. Halpern. Belief, awareness, and limited reasoning. Artificial Intelligence, 34:39–76, 1988.CrossRefGoogle Scholar
  4. [Fi72]
    K. Fine. In so many possible worlds. Notre Dame Journal of Formal Logic, 13:516–520, 1972.Google Scholar
  5. [GP90]
    V. Goranko and S. Passy. Using the universal modality: Gains and questions. Preprint, Sofia University, 1990.Google Scholar
  6. [HC68]
    G.E. Hughes and M.J. Cresswell. Introduction to Modal Logic. Methuen, London, 1968.Google Scholar
  7. [HM85]
    J.Y. Halpern and Y.O. Moses. A guide to the modal logics of knowledge and belief. Proceedings IJCAI-85. Los Angeles, CA, 1985, pages 480–490.Google Scholar
  8. [Ho90]
    W. van der Hoek, Some Considerations on the Logic PFD, (a Logic Combining modalities and Probabilities), Report Free University IR-227, Amsterdam (1990). To appear in Proc. 2nd Russian Conference on Logic Programming, LNCS (1992).Google Scholar
  9. [Ho91a]
    W. van der Hoek, Qualitative Modalities, proceedings of the Scandinavian Conference on Artificial Intelligence-91, B. Mayoh (ed.) IOS Press, Amsterdam (1991), 322–327.Google Scholar
  10. [Ho91b]
    W. van der Hoek. On the semantics of graded modalities. Technical Report IR-246, Free University of Amsterdam, 1991. To appear in The Journal of Applied Non Classical Logic, vol I, 2 (1992).Google Scholar
  11. [Ho91c]
    W. van der Hoek, Systems for Knowledge and Beliefs, in: In J. van Eijck, editor, Logics in AI-JELIA'90, Lecture Notes in Artificial Intelligence 478, Springer, Berlin, 1991, pp. 267–281. Extended version to appear in Journal of Logic and Computation.Google Scholar
  12. [Ho92]
    W. van der Hoek, Modalities for Reasoning about Knowledge and Quantities, Ph.D. thesis, Amsterdam, 1992.Google Scholar
  13. [HoM88]
    W. van der Hoek and J.-J.Ch. Meyer. Possible logics for belief. Technical Report IR-170, Free University of Amsterdam, 1988. To appear in Logique et Analyse.Google Scholar
  14. [HoM90]
    W. van der Hoek & J.-J.Ch. Meyer, Making Some Issues of Implicit Knowledge Explicit, Report Free University IR-222 (1990). To appear in Foundations of Computer Science.Google Scholar
  15. [HR91]
    W. van der Hoek & M. de Rijke, Generalized Quantifiers amd Modal Logic, in: Generalized Quantifiers Theory and Applications, J. van der Does and J. van Eijck (eds), Dutch Network for Logic, Language and Information (1991), pp. 115–142.Google Scholar
  16. [Le80]
    W. Lenzen. Glauben, Wissen und Warscheinlichkeit. Springer Verlag, Wien, 1980.Google Scholar
  17. [MH91]
    J.-J.Ch. Meyer and W. van der Hoek. Non-monotonic reasoning by monotonic means. In J. van Eijck, editor, Logics in AI-JELIA'90, Lecture Notes in Artificial Intelligence 478, Springer, Berlin, 1991, pages 399–411.Google Scholar
  18. [MHV91]
    J.-J.Ch. Meyer, W. van der Hoek, and G.A.W. Vreeswijk. Epistemic logic for computer science: A tutorial. EATCS bulletin, 44:242–270, 1991. (Part I), and EATCS bulletin, 45:256–287, 1991. (Part II).Google Scholar
  19. [Pe86]
    D. Perlis. On the consistency of commonsense reasoning. Computational Intelligence, 2:180–190, 1986.Google Scholar
  20. [Re76]
    N. Rescher. Plausible Reasoning, an Introduction to the Theory and Practice of Plausibilistic Inference. Van Gorcum, Assen, 1976.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • W. van der Hoek
    • 1
  • J. -J. Ch. Meyer
    • 1
    • 2
  1. 1.Department of Mathematics and Computers ScienceFree University, AmsterdamHV AmsterdamThe Netherlands
  2. 2.University of Nijmegen, ToernooiveldED NijmegenThe Netherlands

Personalised recommendations