Skip to main content
SpringerLink
Log in

Well-founded semantics for defeasible logic

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

Fixpoint semantics are provided for ambiguity blocking and propagating variants of Nute’s defeasible logic. The semantics are based upon the well-founded semantics for logic programs. It is shown that the logics are sound with respect to their counterpart semantics and complete for locally finite theories. Unlike some other nonmonotonic reasoning formalisms such as Reiter’s default logic, the two defeasible logics are directly skeptical and so reject floating conclusions. For defeasible theories with transitive priorities on defeasible rules, the logics are shown to satisfy versions of Cut and Cautious Monotony. For theories with either conflict sets closed under strict rules or strict rules closed under transposition, a form of Consistency Preservation is shown to hold. The differences between the two logics and other variants of defeasible logic—specifically those presented by Billington, Antoniou, Governatori, and Maher—are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Antonelli G.A. (2005) Grounded consequence for defeasible logic. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Antoniou G. (2006) Defeasible reasoning: a discussion of some intuitions. International Journal of Intelligent Systems 21(6): 545–558

    Article  Google Scholar 

  • Antoniou, G., Billington, D., Governatori, G., & Maher, M. J. (2000, April). A flexible framework for defeasible logics. In Proceedings of the 8th workshop on non-monotonic reasoning, Breckenridge, Colorado.

  • Antoniou G., Billington D., Governatori G., Maher M.J. (2001) Representation results for defeasible logic. ACM Transactions on Computational Logic 2(2): 255–287

    Article  Google Scholar 

  • Antoniou G., Billington D., Governatori G., Maher M.J. (2006) Embedding defeasible logic into logic programming. Theory and Practice of Logic Programming 6(6): 703–735

    Article  Google Scholar 

  • Antoniou, G., Billington, D., Governatori, G., Maher, M. J., & Rock, A. (2000). A family of defeasible reasoning logics and its implementation. In W. Horn (Ed.), Proceedings of the 14th European conference on artificial intelligence, Berlin, Germany (pp. 459–463). Amsterdam: IOS Press.

  • Billington D. (1993) Defeasible logic is stable. Journal of Logic and Computation 3(4): 379–400

    Article  Google Scholar 

  • Billington D., De Coster K., Nute D. (1990) A modular translation from defeasible nets to defeasible logics. Journal of Experimental and Theoretical Artificial Intelligence 2: 151–177

    Article  Google Scholar 

  • Brewka G. (1996) Well-founded semantics for extended logic programs with dynamic preferences. Journal of Artificial Intelligence Research 4: 19–36

    Google Scholar 

  • Brewka, G. (2001). On the relationship between defeasible logic and well-founded semantics. In T. Eiter, W. Faber & T. Truszczynski (Eds.), Proceedings of the 6th international conference on logic programming and nonmonotonic reasoning. Lecture Notes in Computer Science (Vol. 2173, pp. 121–132). Berlin: Springer.

  • Caminada, M. (2006, May). Well-founded semantics for semi-normal extended logic programs. In Proceedings of the 11th international workshop on non-monotonic reasoning, Lake District, UK.

  • Covington M.A., Nute D., Vellino A. (1997) Prolog programming in depth. Prentice-Hall, Upper Saddle River, NJ

    Google Scholar 

  • Donnelly, S. (1999). Semantics, soundness, and incompleteness for a defeasible logic. Master Thesis, The University of Georgia.

  • Gabbay D. (1985) Theoretical foundations for non-monotonic reasoning in expert systems. In: Apt K. (eds) Logics and models of concurrent systems. Springer-Verlag, New York, pp 439–457

    Google Scholar 

  • Gelfond, M., & Lifschitz, V. (1988). The stable model semantics for logic programming. In K. Bowen & A. Kowalski (Eds.), Proceedings of the fifth international conference on logic programming, Seattle, Washington (pp. 1070–1080). Cambridge, MA: MIT Press.

  • Gelfond M., Lifschitz V. (1991) Classical negation in logic programs and disjunctive databases. New Generation Computing 9(3/4): 365–386

    Article  Google Scholar 

  • Governatori G., Maher M., Antoniou G., Billington D. (2004) Argumentation semantics for defeasible logic. Journal of Logic and Computation 14(5): 675–702

    Article  Google Scholar 

  • Horty J. (2002) Skepticism and floating conclusions. Artificial Intelligence 135(1–2): 55–72

    Article  Google Scholar 

  • Horty J., Thomason R., Touretzky D. (1990) A skeptical theory of inheritance in nonmonotonic semantic networks. Artificial Intelligence 42: 311–348

    Article  Google Scholar 

  • Maher, M., & Governatori, G. (1999). A semantic decomposition of defeasible logics. In Proceedings of the 16th national conference on artificial intelligence, Orlando, FL (pp. 299–305). Menlo Park, CA: AAAI Press.

  • Maher M., Rock A., Antoniou G., Billington D., Miller T. (2001) Efficient defeasible reasoning systems. International Journal on Artificial Intelligence Tools 10(4): 483–501

    Article  Google Scholar 

  • Maier, F., & Nute, D. (2006). Ambiguity propagating defeasible logic and the well-founded semantics. In M. Fisher, W. van der Hoek, B. Konev, & A. Lisitsa (Eds.), JELIA 2006. Lecture Notes in Computer Science (Vol. 4160, pp. 306–318). Berlin: Springer.

  • Makinson, D. (1994). General patterns in nonmonotonic reasoning. In D. Gabbay & C. Hogger (Eds.), Handbook of logic for artificial intelligence and logic programming (Vol. III, pp. 35–110). Oxford: Oxford University Press.

  • Makinson D., Schlechta K. (1991) Floating conclusions and zombie paths: two deep difficulties in the ‘directly skeptical’ approach to inheritance nets. Artificial Intelligence 48: 199–209

    Article  Google Scholar 

  • Moore R.C. (1985) Semantical considerations on nonmonotonic logic. Artificial Intelligence 25(1): 75–94

    Article  Google Scholar 

  • Nute, D. (1986). LDR: A logic for defeasible reasoning. ACMC research report 01-0013, University of Georgia.

  • Nute, D. (1987). Defeasible reasoning. In Proceedings of the 20th Hawaii international conference on system science, University of Hawaii (pp. 470–477).

  • Nute D. (1992) Basic defeasible logic. In: del Cerro L.F., Penttonen M. (eds) Intensional logics for programming.. Oxford University Press, Oxford, pp 125–154

    Google Scholar 

  • Nute, D. (1994). Defeasible logic. In D. Gabbay & C. Hogger (Eds.), Handbook of logic for artificial intelligence and logic programming (Vol. III, pp. 353–395). Oxford: Oxford University Press.

  • Nute D. (1999) Norms, priorities, and defeasibility. In: McNamara P., Prakken H. (eds) Norms, logics and information systems. IOS Press, Amsterdam, pp 201–218

    Google Scholar 

  • Nute, D. (2001). Defeasible logic: Theory, implementation, and applications. In Proceedings of the 14th international conference on applications of prolog (pp. 87–114). Tokyo: IF Computer Japan.

  • Nute, D., Billington, D., & De Coster, K. (1989). Defeasible logic and inheritance hierarchies with exceptions. In Proceedings of the Tübingen workshop on semantic nets and nonmonotonic reasoning, University of Tübingen (Vol. I, pp. 69–82).

  • Pollock J.L. (1974) Knowledge and justification. Princeton University Press, Princeton, NJ

    Google Scholar 

  • Pollock J.L. (1987) Defeasible reasoning. Cognitive Science 11(4): 481–518

    Article  Google Scholar 

  • Prakken, H. (2002). Intuitions and the modelling of defeasible reasoning: Some case studies. In S. Benferhat & E. Giunchiglia (Eds.), Proceedings of the ninth international workshop on non-monotonic reasoning (pp. 91–99).

  • Reiter R. (1980) A logic for default reasoning. Artificial Intelligence 13: 81–132

    Article  Google Scholar 

  • Schaub, T., & Wang, K. (2001). Preferred well-founded semantics for logic programming by alternating fixpoints: Preliminary report. In S. Benferhat & E. Giunchiglia (Eds.), Proceedings of the ninth international workshop on non-monotonic reasoning, Toulouse, France (pp. 238–246).

  • Stein L.A. (1991). Resolving ambiguity in nonmonotonic reasoning. Dissertation, Brown University.

  • Tarski A. (1955) A lattice theoretic fixpoint theorem and its application. Pacific Journal of Mathematics 5: 285–309

    Google Scholar 

  • Van Gelder A., Ross K.A., Schlipf J. (1991) The well-founded semantics for general logic programs. Journal of the ACM 38(3): 620–650

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Florida Institute for Human and Machine Cognition, 40 S Alcaniz St., Pensacola, FL, 32502, USA

    Frederick Maier

  2. Institute for Artificial Intelligence, The University of Georgia, Athens, GA, 30605, USA

    Donald Nute

Authors
  1. Frederick Maier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donald Nute
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frederick Maier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maier, F., Nute, D. Well-founded semantics for defeasible logic. Synthese 176, 243–274 (2010). https://doi.org/10.1007/s11229-009-9492-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-009-9492-1

Keywords

Search

Navigation