Skip to main content

On a Formal Connection between Truth, Argumentation and Belief

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8607)

Abstract

Building on recent connections established between formal models used to study truth and argumentation, we define logics for reasoning about them that we then go on to axiomatize, relying on a link with three-valued Łukasiewicz logic. The first set of logics we introduce are based on formalizing so called skeptical reasoning, and our result shows that a range of semantics that are distinct for particular models coincide at the level of validities. Then, responding to the challenge that our logics do not capture credulous reasoning, we explore modal extensions, leading us to introduce models of three-valued belief induced by argument. We go on to take a preliminary look at some formal properties of this framework, offer a conjecture, then conclude by presenting some challenges for future work.

Keywords

  • Formal Connection
  • Kripke Model
  • Argumentation Theory
  • Argumentation Framework
  • Modal Extension

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-662-44116-9_6
  • Chapter length: 22 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   44.99
Price excludes VAT (USA)
  • ISBN: 978-3-662-44116-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   59.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aczel, P.: Non-wellfounded sets. Technical Report 14, CSLI (1988)

    Google Scholar 

  2. Arieli, O., Caminada, M.W.A.: A QBF-based formalization of abstract argumentation semantics. Journal of Applied Logic 11(2), 229–252 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Barwise, J., Moss, L.: Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena. CSLI, Stanford (1996)

    MATH  Google Scholar 

  4. Beall, J.C.: Revenge of the Liar: New Essays on the Paradox. Oxford University Press (2007)

    Google Scholar 

  5. Bezem, M., Grabmayer, C., Walicki, M.: Expressive power of digraph solvability. Ann. Pure Appl. Logic 163(3), 200–213 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Béziau, J.-Y.: A sequent calculus for Łukasiewicz’s three-valued logic based on Suszko’s bivalent semantics. Bulletin of the Section of Logic 28(2), 89–97 (1998)

    Google Scholar 

  7. Boros, E., Gurvich, V.: Perfect graphs, kernels and cooperative games. Discrete Mathematics 306, 2336–2354 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Caminada, M.: On the issue of reinstatement in argumentation. In: Fisher, M., van der Hoek, W., Konev, B., Lisitsa, A. (eds.) JELIA 2006. LNCS (LNAI), vol. 4160, pp. 111–123. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  9. Caminada, M.W.A., Gabbay, D.M.: A logical account of formal argumentation. Studia Logica 93(2-3), 109–145 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Caminada, M.W.A.: Semi-stable semantics. In: Proceedings of the 2006 Conference on Computational Models of Argument: Proceedings of COMMA 2006, pp. 121–130. IOS Press, Amsterdam (2006)

    Google Scholar 

  11. Cook, R.: Patterns of paradox. The Journal of Symbolic Logic 69(3), 767–774 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Dimopoulos, Y., Torres, A.: Graph theoretical structures in logic programs and default theories. Theoretical Computer Science 170(1-2), 209–244 (1996)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Doutre, S.: Autour de la sématique préférée des systèmes d’argumentation. PhD thesis, Université Paul Sabatier, Toulouse (2002)

    Google Scholar 

  14. Duchet, P.: Graphes noyau-parfaits, II. Annals of Discrete Mathematics 9, 93–101 (1980)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Duchet, P., Meyniel, H.: Une généralisation du théorème de Richardson sur l’existence de noyaux dans les graphes orientés. Discrete Mathematics 43(1), 21–27 (1983)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77, 321–357 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Dyrkolbotn, S., Walicki, M.: Kernels in digraphs that are not kernel perfect. Discrete Mathematics 312(16), 2498–2505 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Dyrkolbotn, S., Walicki, M.: Propositional discourse logic. Synthese 191(5), 863–899 (2014)

    CrossRef  MathSciNet  Google Scholar 

  19. Gabbay, D.: Modal provability foundations for argumentation networks. Studia Logica 93(2-3), 181–198 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Galeana-Sánchez, H., Guevara, M.-K.: Some sufficient conditions for the existence of kernels in infinite digraphs. Discrete Mathematics 309(11), 3680–3693 (2009); 7th International Colloquium on Graph Theory (ICGT) (2005)

    Google Scholar 

  21. Galeana-Sánchez, H., Neumann-Lara, V.: On kernels and semikernels of digraphs. Discrete Mathematics 48(1), 67–76 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Gottlob, G.: Complexity results for nonmonotonic logics. Journal of Logic and Computation 2(3), 397–425 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Grossi, D.: Argumentation in the view of modal logic. In: McBurney, P., Rahwan, I., Parsons, S. (eds.) ArgMAS 2010. LNCS (LNAI), vol. 6614, pp. 190–208. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  24. Grossi, D.: On the logic of argumentation theory. In: van der Hoek, W., Kaminka, G.A., Lespérance, Y., Luck, M., Sen, S. (eds.) AAMAS, pp. 409–416. IFAAMAS (2010)

    Google Scholar 

  25. Kripke, S.: Outline of a theory of truth. The Journal of Philosophy 72(19), 690–716 (1975)

    CrossRef  Google Scholar 

  26. Łukasiewicz, J.: Selected works, edited by L. Borkowski. Studies in Logic and the Foundations of Mathematics. North Holland, Amsterdam (1970)

    Google Scholar 

  27. Mercier, H., Sperber, D.: Why do humans reason? Arguments for an argumentative theory. Behavioral and Brain Sciences 34, 57–74 (2011)

    CrossRef  Google Scholar 

  28. Milner, E.C., Woodrow, R.E.: On directed graphs with an independent covering set. Graphs and Combinatorics 5, 363–369 (1989)

    CrossRef  MATH  MathSciNet  Google Scholar 

  29. Minari, P.: A note on Łukasiewicz’s three-valued logic. Annali del Dipartimento di Filosofia dell’Universitá di Firenze 8(1) (2002)

    Google Scholar 

  30. Neumann-Lara, V.: Seminúcleos de una digráfica. Technical report, Anales del Instituto de Matemáticas II, Universidad Nacional Autónoma México (1971)

    Google Scholar 

  31. Pollock, J.L.: How to reason defeasibly. Artif. Intell. 57(1), 1–42 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. Priest, G.: The logic of paradox. Journal of Philosophical Logic 8, 219–241 (1979)

    CrossRef  MATH  MathSciNet  Google Scholar 

  33. Quine, W.V.: The ways of paradox and other essays. Random House, New York (1966)

    Google Scholar 

  34. Rabern, L., Rabern, B., Macauley, M.: Dangerous reference graphs and semantic paradoxes. Journal of Philosophical Logic 42(5), 727–765 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  35. Rahwan, I., Simari, G.R. (eds.): Argumentation in artificial intelligence. Springer (2009)

    Google Scholar 

  36. Reiter, R.: A logic for default reasoning. Artif. Intell. 13(1-2), 81–132 (1980)

    CrossRef  MATH  MathSciNet  Google Scholar 

  37. Richardson, M.: On weakly ordered systems. Bulletin of the American Mathematical Society 52, 113–116 (1946)

    CrossRef  MATH  MathSciNet  Google Scholar 

  38. Richardson, M.: Solutions of irreflexive relations. The Annals of Mathematics, Second Series 58(3), 573–590 (1953)

    CrossRef  MATH  Google Scholar 

  39. Tarski, A.: The concept of truth in formalised languages. In: Corcoran, J. (ed.) Logic, Semantics, Metamathematics, papers from 1923 to 1938, Hackett Publishing Company (1983) (translation of the Polish original from 1933)

    Google Scholar 

  40. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press (1944, 1947)

    Google Scholar 

  41. Walicki, M., Dyrkolbotn, S.: Finding kernels or solving SAT. Journal of Discrete Algorithms 10, 146–164 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  42. Wu, Y., Caminada, M.W.A., Gabbay, D.M.: Complete extensions in argumentation coincide with 3-valued stable models in logic programming. Studia Logica 93(2-3), 383–403 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  43. Yablo, S.: Paradox without self-reference. Analysis 53(4), 251–252 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dyrkolbotn, S. (2014). On a Formal Connection between Truth, Argumentation and Belief. In: Colinet, M., Katrenko, S., Rendsvig, R.K. (eds) Pristine Perspectives on Logic, Language, and Computation. ESSLLI ESSLLI 2013 2012. Lecture Notes in Computer Science, vol 8607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44116-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44116-9_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44115-2

  • Online ISBN: 978-3-662-44116-9

  • eBook Packages: Computer ScienceComputer Science (R0)