Skip to main content
SpringerLink
Log in

A Formal Concept View of Abstract Argumentation

  • Conference paper
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7958))

Abstract

The paper presents a parallel between two important theories for the treatment of information which address questions that are apparently unrelated and that are studied by different research communities: an enriched view of formal concept analysis and abstract argumentation. Both theories exploit a binary relation (expressing object-property links, attacks between arguments). We show that when an argumentation framework rather considers the complementary relation does not attack, then its stable extensions can be seen as the exact counterparts of formal concepts. This leads to a cube of oppositions, a generalization of the well-known square of oppositions, between eight remarkable sets of arguments. This provides a richer view for argumentation in cases of bi-valued attack relations and fuzzy ones.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amgoud, L., Cayrol, C.: On the acceptability of arguments in preference-based argumentation. In: Cooper, G.F., Moral, S. (eds.) Proc. 14th Conf. on Uncertainty in Artif. Intellig. (UAI 1998), Madison, W.I., July 24-26, pp. 1–7. Morgan Kaufmann (1998)

    Google Scholar 

  2. Amgoud, L., Devred, C.: Argumentation frameworks as constraint satisfaction problems. In: Benferhat, S., Grant, J. (eds.) SUM 2011. LNCS, vol. 6929, pp. 110–122. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  3. Amgoud, L., Prade, H.: Towards a logic of argumentation. In: Hüllermeier, E., Link, S., Fober, T., Seeger, B. (eds.) SUM 2012. LNCS (LNAI), vol. 7520, pp. 558–565. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  4. Barbut, M., Monjardet, B.: Ordre et Classification. Algèbre et Combinatoire. Tome 2, Hachette, Paris (1970)

    Google Scholar 

  5. Belohlavek, R.: Fuzzy Galois connections. Math. Logic Quart. 45, 497–504 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Belohlavek, R.: Fuzzy Relational Systems. Foundations and principles. Kluwer (2002)

    Google Scholar 

  7. Belohlavek, R., Vychodil, V.: What is a fuzzy concept lattice. In: Proc. CLA 2005, Olomounc, Czech Republic, pp. 34–45 (2005)

    Google Scholar 

  8. Besnard, P., Doutre, S.: Characterization of semantics for argument systems. In: Dubois, D., Welty, C.A., Williams, M.A. (eds.) Proc. of the 9th Inter. Conf. on Principles of Knowledge Representation and Reasoning (KR 2004), Whistler, Canada, June 2-5, pp. 183–193. AAAI Press (2004)

    Google Scholar 

  9. Besnard, P., Doutre, S.: Checking the acceptability of a set of arguments. In: Proc. 10th Inter. Workshop on Non-Monotonic Reasoning (NMR 2004), Whistler, Canada, June 6-8, pp. 59–64 (2004)

    Google Scholar 

  10. Béziau, J.-Y.: The power of the hexagon. Logica Universalis 6 (2012)

    Google Scholar 

  11. Blanché, R.: Structures Intellectuelles. Essai sur l’Organisation Systématique des Concepts, Vrin, Paris (1966)

    Google Scholar 

  12. Burusco, A., Fuentes-Gonzalez, R.: The study of the L-fuzzy concept lattice. Mathware & Soft Comput. 3, 209–218 (1994)

    MathSciNet  Google Scholar 

  13. Couturat, L.: L’Algèbre de la Logique. Gauthier-Villars, Paris (1905)

    Google Scholar 

  14. Djouadi, Y., Dubois, D., Prade, H.: Différentes extensions floues de l’analyse formelle de concepts. In: Actes Rencontres Francophones sur la Logique Floue et ses Applications (LFA 2009), Annecy, Cépaduès, November 5-6, pp. 141–148 (2009)

    Google Scholar 

  15. Djouadi, Y., Dubois, D., Prade, H.: Possibility theory and formal concept analysis: Context decomposition and uncertainty handling. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS(LNAI), vol. 6178, pp. 260–269. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  16. Djouadi, Y., Dubois, D., Prade, H.: Graduality, uncertainty and typicality in formal concept analysis. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds.) 35 Years of Fuzzy Set Theory. STUDFUZZ, vol. 261, pp. 127–147. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  17. Djouadi, Y., Prade, H.: Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices. Fuzzy Optimization and Decision Making 10(4), 287–309 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dubois, D., Dupin de Saint Cyr, F., Prade, H.: A possibilty-theoretic view of formal concept analysis. Fundamenta Informaticae 75(1-4), 195–213 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Dubois, D., Prade, H.: Possibility Theory. Plenum Press (1988)

    Google Scholar 

  20. Dubois, D., Prade, H.: Possibilistic logic: a retrospective and prospective view. Fuzzy Sets and Systems 144, 3–23 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dubois, D., Prade, H.: Bridging gaps between several frameworks for the idea of granulation. In: Proc. Symp. on Foundations of Computational Intelligence (in IEEE Symposium Series on Computational Intelligence - SSCI 2011) (FOCI 2011), Paris, April 11-15, pp. 59–65 (2011)

    Google Scholar 

  22. Dubois, D., Prade, H.: Possibility theory and formal concept analysis: Characterizing independent sub-contexts. Fuzzy Sets and Systems 196, 4–16 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dubois, D., Prade, H.: From Blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Universalis 6 (2012)

    Google Scholar 

  24. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77, 321–358 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dunne, P.E., Hunter, A., McBurney, P., Parsons, S., Wooldridge, M.: Weighted argument systems: Basic definitions, algorithms, and complexity results. Artificial Intelligence 175, 457–486 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fan, S.-Q., Zhang, W.-X., Xu, W.: Fuzzy inference based on fuzzy concept lattice. Fuzzy Sets & Syst. 157, 3177–3187 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ferré, S., Ridoux, O.: Introduction to logical information systems. Information Processing and Mgmt. 40, 383–419 (2004)

    Article  MATH  Google Scholar 

  28. Gabbay, D.M.: Introducing equational semantics for argumentation networks. In: Liu, W. (ed.) ECSQARU 2011. LNCS, vol. 6717, pp. 19–35. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  29. Ganter, B., Wille, R.: Formal Concept Analysis. Springer (1999)

    Google Scholar 

  30. Georgescu, G., Popescu, A.: Non-dual fuzzy connections. Archive for Mathematical Logic 43, 1009–1039 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gorogiannis, N., Hunter, A.: Instantiating abstract argumentation with classical logic arguments: Postulates and properties. Artificial Intellig. 175, 1479–1497 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lai, H., Zhang, D.: Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory. Inter. J. Approx. Reason. (2009)

    Google Scholar 

  33. Medina, J., Ojeda-Aciego, M., Ruiz-Calvino, J.: Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets and Systems 160(2), 130–144 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht (1982)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRIT University of Toulouse, 118 rte de Narbonne, Toulouse, France

    Leila Amgoud & Henri Prade

Authors
  1. Leila Amgoud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henri Prade
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    Linda C. van der Gaag

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Amgoud, L., Prade, H. (2013). A Formal Concept View of Abstract Argumentation. In: van der Gaag, L.C. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2013. Lecture Notes in Computer Science(), vol 7958. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39091-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39091-3_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39090-6

  • Online ISBN: 978-3-642-39091-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation