Annals of Mathematics and Artificial Intelligence

, Volume 87, Issue 3, pp 187–226 | Cite as

A review of the relations between logical argumentation and reasoning with maximal consistency

  • Ofer ArieliEmail author
  • AnneMarie Borg
  • Jesse Heyninck


This is a survey of some recent results relating Dung-style semantics for different types of logical argumentation frameworks and several forms of reasoning with maximally consistent sets (MCS) of premises. The related formalsims are also examined with respect to some rationality postulates and are carried on to corresponding proof systems for non-monotonic reasoning.


Logical argumentation Structured argumentation Reasoning with maximal consistency Defeasible reasoning Extension-based semantics Dynamic proof systems 

Mathematics Subject Classification (2010)

68-02 68T27 68T37 


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We thank Christian Straßer and the anonymous reviewers for many comments and helpful suggestions. The work on this paper is supported by the Israel Science Foundation (Grant No.817/15). AnneMarie Borg and Jesse Heyninck are also supported by the Alexander von Humboldt Foundation and the German Ministry for Education and Research.


  1. 1.
    Amgoud, L.: Postulates for logic-based argumentation systems. Int. J. Approx. Reason. 55(9), 2028–2048 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Amgoud, L., Besnard, P.: Bridging the gap between abstract argumentation systems and logic. In: Proceedings of the SUM’09, LNCS 5785, pp 12–27. Springer (2009)Google Scholar
  3. 3.
    Amgoud, L., Besnard, P.: A formal analysis of logic-based argumentation systems. In: Proceedings of the SUM’10, LNCS 6379, pp 42–55. Springer (2010)Google Scholar
  4. 4.
    Amgoud, L., Besnard, P.: Logical limits of abstract argumentation frameworks. Journal of Applied Non-Classical Logics 23(3), 229–267 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Amgoud, L., Cayrol, C.: Inferring from inconsistency in preference-based argumentation frameworks. J. Autom. Reason. 29(2), 125–169 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Amgoud, L., Vesic, S.: Handling inconsistency with preference-based argumentation. In: Proceedings SUM’10, pp 56–69. Springer (2010)Google Scholar
  7. 7.
    Arieli, O., Borg, A., Straßer, C.: Prioritized sequent-based argumentation. In: Proceedings of the AAMAS’18, pp 1105–1113. ACM (2018)Google Scholar
  8. 8.
    Arieli, O., Borg, A., Straßer, C.: Reasoning with maximal consistency by argumentative approaches. J. Log. Comput. 28(7), 1523–1563 (2018)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Arieli, O., Borg, A., Straßer, C.: A proof theoretic perspective of logical argumentation. submitted (2019)Google Scholar
  10. 10.
    Arieli, O., Straßer, C.: Dynamic derivations for sequent-based logical argumentation. In: Proceedings of the COMMA’14, vol. 266 of Frontiers in Artificial Intelligence and Applications, pp 89–100. IOS Press (2014)Google Scholar
  11. 11.
    Arieli, O., Straßer, C.: Sequent-based logical argumentation. Journal of Argument and Computation 6(1), 73–99 (2015)Google Scholar
  12. 12.
    Arieli, O., Straßer, C.: Deductive argumentation by enhanced sequent calculi and dynamic derivations. Electron. Notes Theor. Comput. Sci. 323, 21–37 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Arieli, O., Straßer, C.: Logical argumentation by dynamic proof systems. Theoretical Computer Science, in press ( (2019)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Asenjo, F.G.: A calculus of antinomies. Notre Dame Journal of Formal Logic 7, 103–106 (1966)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Avron, A.: The method of hypersequents in the proof theory of propositional non-classical logics. In: Logic: Foundations to Applications, pp 1–32. Oxford Science Publications (1996)Google Scholar
  16. 16.
    Baral, C., Kraus, S., Minker, J.: Combining multiple knowledge bases. IEEE Trans. Knowl. Data Eng. 3(2), 208–220 (1991)Google Scholar
  17. 17.
    Baroni, P., Caminada, M., Giacomin, M.: An introduction to argumentation semantics. Knowl. Eng. Rev. 26(4), 365–410 (2011)Google Scholar
  18. 18.
    Baroni, P., Caminada, M., Giacomin, M.: Abstract argumentation frameworks and their semantics. In: Baroni, P., Gabay, D., Giacomin, M., van der Torre, L. (eds.) Handbook of Formal Argumentation, pp 159–236. College Publications (2018)Google Scholar
  19. 19.
    Baroni, P., Giacomin, M.: Semantics for abstract argumentation systems. In: Rahwan, I., Simary, G.R. (eds.) Argumentation in Artificial Intelligence, pp 25–44 (2009)Google Scholar
  20. 20.
    Beirlaen, M., Heyninck, J., Pardo, P., Straßer, C.: Argument strength in formal argumentation. Journal of Applied Logics-IfCoLog Journal of Logics and their Applications 5(3), 629–675 (2018)MathSciNetGoogle Scholar
  21. 21.
    Benferhat, S., Dubois, D., Prade, H.: Representing default rules in possibilistic logic. In: Proceedings of the KR’92, pp 673–684 (1992)Google Scholar
  22. 22.
    Benferhat, S., Dubois, D., Prade, H.: A local approach to reasoning under incosistency in stratified knowledge bases. In: Proceedings of the ECSQARU’95, LNCS 946, pp 36–43. Springer (1995)Google Scholar
  23. 23.
    Benferhat, S., Dubois, D., Prade, H.: Some syntactic approaches to the handling of inconsistent knowledge bases: a comparative study part 1: the flat case. Stud. Logica. 58(1), 17–45 (1997)zbMATHGoogle Scholar
  24. 24.
    Besnard, P., García, A., Hunter, A., Modgil, S., Prakken, H., Simari, G., Toni, F.: Introduction to structured argumentation. Argument & Computation 5(1), 1–4 (2014)Google Scholar
  25. 25.
    Besnard, P., Hunter, A.: A logic-based theory of deductive arguments. J. Artif. Intell. 128(1–2), 203–235 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Besnard, P., Hunter, A.: Argumentation based on classical logic. In: Rahwan, I., Simary, G.R. (eds.) Argumentation in artificial intelligence, pp 133–152. Springer (2009)Google Scholar
  27. 27.
    Besnard, P., Hunter, A.: A review of argumentation based on deductive arguments. In: Baroni, P., Gabay, D., Giacomin, M., van der Torre, L. (eds.) Handbook of Formal Argumentation, pp 437–484. College Publications (2018)Google Scholar
  28. 28.
    Bondarenko, A., Dung, P.M., Kowalski, R., Toni, F.: An abstract, argumentation-theoretic approach to default reasoning. J. Artif. Intell. 93(1), 63–101 (1997)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Borg, A.: Equipping sequent-based argumentation with defeasible assumptions. In: Proceedings of the COMMA’18, vol. 305 of Frontiers in Artificial Intelligence and Applications. IOS Press (2018)Google Scholar
  30. 30.
    Borg, A., Arieli, O.: Hypersequential argumentation frameworks: an instantiation in the modal logic S5. In: Proceedings of the AAMAS’18, pp 1097–1104. ACM (2018)Google Scholar
  31. 31.
    Borg, A., Arieli, O., Straßer, C.: Hypersequent-based argumentation: an instantiation in the relevance logic RM. In: Black, E., Modgil, S., Oren, N. (eds.) Proceedings of the TAFA’17, LNCS 10757, pp 17–34. Springer (2017)Google Scholar
  32. 32.
    Borg, A., Straßer, C., Arieli, O.: A generalized proof-theoretic approach to structured argumentation by hypersequent calculi. Submitted (2019)Google Scholar
  33. 33.
    Brewka, G.: Preferred subtheories: an extended logical framework for default reasoning. In: Sridharan, N.S. (ed.) Proceedings of the IJCAI’89, pp 1043–1048. Morgan Kaufmann (1989)Google Scholar
  34. 34.
    Caminada, M., Amgoud, L.: On the evaluation of argumentation formalisms. J. Artif. Intell. 171, 286–310 (2007)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Caminada, M., Modgil, S., Oren, N.: Preferences and unrestricted rebut. In: Proceedings of the COMMA’14, pp 209–220. IOS Press (2014)Google Scholar
  36. 36.
    Cayrol, C.: On the relation between argumentation and non-monotonic coherence-based entailment. In: Proceedings IJCAI’95, pp 1443–1448. Morgan Kaufmann (1995)Google Scholar
  37. 37.
    Čyras, K., Toni, F.: Non-monotonic inference properties for assumption-based argumentation. In: Proceedings of the TAFA’15, pp 92–111. Springer (2015)Google Scholar
  38. 38.
    D’Agostino, M., Modgil, S.: Classical logic, argumentation and dialectic. Artif. Intell. 262, 15–51 (2018)zbMATHGoogle Scholar
  39. 39.
    D’Agostino, M., Modgil, S.: A study of argumentative characterisations of preferred subtheories. In: Proceedings of the IJCAI’18, pp 1788–1794 (2018)Google Scholar
  40. 40.
    Dauphin, J., Cramer, M.: Aspic-end: structured argumentation with explanations and natural deduction. In: Proceedings of the TAFA’17, pp 51–66. Springer (2017)Google Scholar
  41. 41.
    Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. J. Artif. Intell. 77, 321–357 (1995)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Dung, P.M., Kowalski, R., Toni, F.: Dialectic proof procedures for assumption-based, admissible argumentation. J. Artif. Intell. 170(2), 114–159 (2006)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Elvang-Gøransson, M., Krause, P., Fox, J.: Acceptability of arguments as ‘logical uncertainty’. In: Proceedings of the ECSQARU’93, pp 85–90. Springer (1993)Google Scholar
  44. 44.
    García, A., Simari, G.: Defeasible logic programming: an argumentative approach. Theory Pract. Logic Program. 4(1–2), 95–138 (2004)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Gärdenfors, P., Rott, H.: Belief revision. In: Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp 35–132. Oxford University Press (1995)Google Scholar
  46. 46.
    Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) German. An English Translation Appears in ‘The Collected Works of Gerhard Gentzen’. North-Holland (1969)Google Scholar
  47. 47.
    Gorogiannis, N., arguments, A. Hunter.: Instantiating abstract argumentation with classical logic Postulates and properties. J. Artif. Intell. 175(9–10), 1479–1497 (2011)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Heyninck, J., Arieli, O.: On the semantics of simple contrapositive assumption-based argumentation frameworks. In: Proceedings of the COMMA’18, vol. 305 of Frontiers in Artificial Intelligence and Applications. IOS Press (2018)Google Scholar
  49. 49.
    Heyninck, J., Arieli, O.: Simple contrapositive assumption-based frameworks. Accepted to LPNMR’19 (extended abstract in AAMAS’19) (2019)Google Scholar
  50. 50.
    Heyninck, J., Straßer, C.: Relations between assumption-based approaches in nonmonotonic logic and formal argumentation. In: Proceedings of the NMR’16, pp 65–76 (2016)Google Scholar
  51. 51.
    Heyninck, J., Straßer, C.: Revisiting unrestricted rebut and preferences in structured argumentation. In: Proceedings of the IJCAI’17, pp 1088–1092. AAAI Press (2017)Google Scholar
  52. 52.
    Kaci, S., van der Torre, L., Villata, S.: Preference in abstract argumentation. In: Proceedings of the COMMA’18, vol. 305 of Frontiers in Artificial Intelligence and Applications. IOS Press (2018)Google Scholar
  53. 53.
    Konieczny, S., Marquis, P., Vesic, S.: New inference relations from maximal consistent subsets. In: Proceedings of the KR’18, pp 649–650. AAAI Press (2018)Google Scholar
  54. 54.
    Konieczny, S., Pino Pérez, R.: Merging information under constraints: a logical framework. Log. Comput. 12(5), 773–808 (2002)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Lin, J.: Integration of weighted knowledge bases. J. Artif. Intell. 83(2), 363–378 (1996)MathSciNetGoogle Scholar
  56. 56.
    Malouf, R.: Maximal consistent subsets. Comput. Linguist. 33(2), 153–160 (2007)Google Scholar
  57. 57.
    Modgil, S., Prakken, H.: A general account of argumentation with preferences. J. Artif. Intell. 195, 361–397 (2013)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Modgil, S., Prakken, H.: The ASPIC+ framework for structured argumentation: a tutorial. Argument and Computation 5(1), 31–62 (2014)Google Scholar
  59. 59.
    Pigozzi, G., Tsoukias, A., Viappiani, P.: Preferences in artificial intelligence. Ann. Math. Artif. Intell. 77(3-4), 361–401 (2016)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Pollock, J.: How to reason defeasibly. J. Artif. Intell. 57(1), 1–42 (1992)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Prakken, H.: An abstract framework for argumentation with structured arguments. Argument and Computation 1(2), 93–124 (2010)Google Scholar
  62. 62.
    Prakken, H.: Historical overview of formal argumentation. In: Baroni, P., Gabay, D., Giacomin, M., van der Torre, L. (eds.) Handbook of Formal Argumentation, pp 75–143. College Publications (2018)Google Scholar
  63. 63.
    Prakken, H., Vreeswijk, G.: Logical systems for defeasible argumentation. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosochical Logic 14, pp 219–318. Kluwer (2002)Google Scholar
  64. 64.
    Priest, G.: Logic of paradox. J. Philos. Log. 8, 219–241 (1979)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Priest, G.: Reasoning about truth. J. Artif. Intell. 39, 231–244 (1989)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Rescher, N., Manor, R.: On inference from inconsistent premises. Theor. Decis. 1, 179–217 (1970)zbMATHGoogle Scholar
  67. 67.
    Simari, G., Loui, R.P.: A mathematical treatment of defeasible reasoning and its implementation. J. Artif. Intell. 53(2–3), 125–157 (1992)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Straßer, C., Arieli, O.: Sequent-based argumentation for normative reasoning. In: Proceedings of the DEON’14, LNCS 8554. An extended version will appear in the Journal of Logic and Computation (, pp 224–240. Springer (2014)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Thimm, M., Wallner, J.P.: Some complexity results on inconsistency measurement. In: Proceedings of the KR’16, pp 114–124 (2016)Google Scholar
  70. 70.
    Tonim, F.: Assumption-based argumentation for epistemic and practical reasoning. Computable Models of the Law, Languages, Dialogues, Games, Ontologies 4884, 185–202 (2008)Google Scholar
  71. 71.
    Toni, F.: A tutorial on assumption-based argumentation. Argument and Computation 5(1), 89–117 (2014)Google Scholar
  72. 72.
    Vesic, S.: Identifying the class of maxi-consistent operators in argumentation. J. Artif. Intell. Res. 47, 71–93 (2013)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Vesic, S., van der Torre, L.: Beyond maxi-consistent argumentation operators. In: Proceedings of the JELIA’12, LNCS 7519, pp 424–436. Springer (2012)Google Scholar

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Authors and Affiliations

  1. 1.School of Computer ScienceThe Academic College of Tel-AvivTel AvivIsrael
  2. 2.Institute of Philosophy IIRuhr University BochumBochumGermany

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