Studia Logica

, Volume 53, Issue 2, pp 161–201 | Cite as

Nonmonotonic reasoning: from finitary relations to infinitary inference operations

  • Michael Freund
  • Daniel Lehmann


A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there.


General Operation Mathematical Logic Computational Linguistic Central Topic Interesting Family 
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]
    Jürgen Dix andDavid Makinson,The relationship between klm andmak models for nonmonotonic inference operations,Journal of Logic, Language and Information 1(2) 1992, pp. 131–140.Google Scholar
  2. [2]
    Michael Freund,Supracompact inference operations, In J. Dix, K. P. Jantke, and P.H. Schmitt, editors,Nonmonotonic andInductive Logic, First International Workshop, Karlsruhe, Germany, pp. 59 – 73. Springer Verlag, December 1990. Lecture Notes in Artificial Intelligence, Vol. 543.Google Scholar
  3. [3]
    Michael Freund,Supracompact inference operations Studia Logica 52 (1993), pp. 457–481.Google Scholar
  4. [4]
    Michael Freund andDaniel Lehmann,Deductive inference operations In J. van Eijck, editor,Proceedings of JELIA 1990 European Workshop on Logical Methods in Artificial Intelligence Lecture Notes in AI Vol. 478 pp. 227–233, Amsterdam, the Netherlands, September 1990, Springer Verlag.Google Scholar
  5. [5]
    Michael Freund andDaniel Lehmann,Nonmonotonic inference operations Bulletin of the IGPL 1(1), pp. 23–68, July 1993.Google Scholar
  6. [6]
    Michael Freund, Daniel Lehmann andDavid Makinson,Canonical extensions to the infinite case of finitary nonmonotonic inference operations, InWorkshop on Nomonotonic Reasoning, pp. 133 – 138, Sankt Augustin, FRG, December 1989. Arbeitspapiere der GMD no. 443.Google Scholar
  7. [7]
    Michael Freund, Daniel Lehmann andPaul Morris,Rationality, transitivity, and contraposition, Artificial Intelligence, 52(2), pp. 191 – 203, December 1991. Research Note.Google Scholar
  8. [8]
    Dov M. Gabbay,Theoretical foundations for non-monotonic reasoning in expert systems In Krzysztof R. Apt, editor,Proc. of the NATO Advanced Study Institute on Logics and Models of Concurrent Systems pp. 439–457, La Colle-sur-Loup, France, October 1985. Springer-Verlag.Google Scholar
  9. [9]
    Gerhard Gentzen,The Collected Papers of Gerhard Gentzen edited by M. E. Szabo, North Holland, Amsterdam, 1969.Google Scholar
  10. [10]
    Matthew L. Ginsberg,Counterfactuals Artificial Intelligence 30, pp. 35–79, 1986.Google Scholar
  11. [11]
    George Grätzer,Lattice Theory W. H. Freeman, San Francisco, 1971.Google Scholar
  12. [12]
    Sarit Kraus, Daniel Lehmann andMenachem Magidor,Nonmonotonic reasoning, preferential models and cumulative logics Artifical Intelligence 44(1 – 2), pp. 167–207, July 1990.Google Scholar
  13. [13]
    Daniel Lehmann,What does a conditional knowledge base entail? In Ron Brachman and Hector Levesque, editors,Proceedings of the First International Conference on Principles of Knowledge Representation and Reasoning Toronto, Canada, May 1989. Morgan Kaufmann.Google Scholar
  14. [14]
    Daniel Lehmann,Plausibility logic In Egon Boerger, Gerhard Jäger, Hans Kleine Buening, and Michael M. Richter, editors,Proceedings of Computer Science Logic '91 volume 626, pp. 227–241, Berne, Swutzerland, October 1991. Lecture Notes in Computer Science, Springer Verlag.Google Scholar
  15. [15]
    Daniel Lehmann andMenachem Magidor,Preferential logics: the predicate calculus case In Rohit Parikh, editor,Proceedings of the Third Conference on Theoretical Aspects of Reasoning About Knowledge pp. 57–72, Monterey, California, March 1990. Morgan Kaufmann.Google Scholar
  16. [16]
    Daniel Lehmann andMenachem Magidor,What does a conditional knowledge base entail? Artificial Intelligence 55(1), pp. 1–60, May 1992.Google Scholar
  17. [17]
    David Makinson,General theory of cumulative inference In M. Reinfrank, J. de Kleer, M. L. Ginsberg, and E. Sandewall, editors,Proceedings of the Second International Workshop on Non-Monotonic Reasoning pp. 1–18, Grassau, Germany, June 1988. Springer Verlag. Volume 346, Lecture Notes in Artificial Intelligence.Google Scholar
  18. [18]
    David Makinson,General patterns in nonmonotonic reasoning, In D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors,Handbook of Logic in Artificial Intelligence and Logic Programming Vol. 2, Nonmonotonic and Uncertain Reasoning, Oxford University Press, due 1993. in preparation.Google Scholar
  19. [19]
    David Poole,A logical framework for default reasoning Artificial Intelligence 36, pp. 27–47, 1988.Google Scholar
  20. [20]
    Karl Schlechta,Results on infinite extensions Journal of Applied Non-Classical Logics 1(1) 1991, pp. 65–72.Google Scholar
  21. [21]
    Karl Schlechta,Some results on classical preferential models, Journal of Logic and Computation, 2(6), 1992, in print.Google Scholar
  22. [22]
    Alfred Tarski,Logic, Semantics, Metamathematics. Papers from 1923 – 1938 Clarendon Press, Oxford 1956.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Michael Freund
    • 1
  • Daniel Lehmann
    • 2
  1. 1.Département de MathématiquesUniversité d'OrléansOrléans, Cédex 2France
  2. 2.Institute of Computer ScienceHebrew UniversityJerusalemIsrael

Personalised recommendations