Abstract
In this paper we show that some versions of Dung’s abstract argumentation frames are equivalent to classical propositional logic. In fact, Dung’s attack relation is none other than the generalised Peirce–Quine dagger connective of classical logic which can generate the other connectives \({\neg, \wedge, \vee, \to}\) of classical logic. After establishing the above correspondence we offer variations of the Dung argumentation frames in parallel to variations of classical logic, such as resource logics, predicate logic, etc., etc., and create resource argumentation frames, predicate argumentation frames, etc., etc. We also offer the notion of logic proof as a geometrical walk along the nodes of a Dung network and thus we are able to offer a geometrical abstraction of the notion of inference based argumentation. Thus our paper is also a contribution to the question:
“What is a logical system”
in as much as it integrates logic with abstract argumentation networks.
Similar content being viewed by others
References
Barringer H., Gabbay D., Woods J.: Temporal dynamics of argumentation networks. In: Hutter, D., Stephan, W. (eds) Volume Dedicated to Joerg Siekmann. Mechanising Mathematical Reasoning. LNCS, vol. 2605, pp. 59–98. Springer, Berlin (2005)
Barringer, H., Gabbay, D., Woods, J.: Network modalities. In: Gross, G., Schulz, K.U. (eds.) Linguistics, Computer Science and Language Processing. Festschrift for Franz Guenthner on the Occasion of his 60th Birthday, pp. 79–102. College Publications (2008)
Bench-Capon T.J.M.: Persuasion in practical argument using value-based argumentation frameworks. J. Logic Comput. 13(3), 429–448 (2003)
Bench-Capon T.J.M., Atkinson K.: Abstract argumentation and values. In: Rahwan, I., Simari, G. (eds) Argumentation in AI, Chap. 3, pp. 45–64. Springer, Berlin (2009)
Besnard, P., Doutre, S.: Characterization of semantics for argument systems. In: Dubois, D., Welty, C., Williams, OM.-A (eds.) KR 2004, pp. 183–193. AAAI Press (2004)
Besnard, P., Hunter, A.: Elements of Argumentation. MIT Press (2008)
Boella, G., Gabbay, D., van der Torre, L., Villata, S.: Support in Abstract Argumentation (n383) Expanded Version. This paper is an expansion of an earlier original paper under the same title published in 2010: In: Baroni, P., Cerutti, F., Giacomi, M., Simari, G. (eds.) Computational Models of Argument, COMMA 2010, pp. 111–122. IOS press (2010)
Baroni P, Cerutti F., Giacomin M., Guida G.: Encompassing attacks to attacks in abstract argumentation frameworks. In: ECSQARU 2009, LNAI, vol. 5590, pp. 83–94. Springer, Berlin (2009)
Brewka, G., Woltran, S.: Abstract dialectical frameworks. In: Proceedings of KR 2010. AAAI Press (2010)
Brewka, G., Dunne, P., Woltran, S.: Relating the semantics of abstract dialectical frameworks and standard AFs. In: Proceedings of IJCAI-11, pp. 780–786 (2011)
Caminada, M.: On the issue of reinstatement in argumentation. In: Fischer, M., van der Hoek, W., Konev, B., Lisitsa, A. (eds.) Logics in Artificial Intelligence: 10th European Conference, JELIA 2006. LNAI, vol. 4160, pp. 111–123. Springer, Berlin (2006)
Caminada, M.: Semi-stable semantics. In: Dunne, P.E., Bench-Capon, T.J.M. (eds.) Computational Models of Argument. Proceedings of COMMA 2006, pp. 121–130. IOS Press (2006)
Caminada, M., Wu, Y.: On the limitations of abstract argumentation. In: Proceedings of BNAIC 2011. http://users.numericable.lu/martincaminada/publications/BNAIC_limitations_abstract.pdf
Caminada M., Amgoud L.: On the evaluation of argumentation formalisms. Artif. Intell. 171(5–6), 286–310 (2007)
Caminada M., Gabbay D.: A logical account of formal argumentation Studia Logica 93, 109–145 (2009)
Chalamish, M., Gabbay, D., Schild, U.: Intelligent evaluation of evidence using Wigmore diagrams. In: ICAIL-11, pp. 61–65 (2011)
Chesnevar C., Maguitman A., Loui R.: Logical models of argument. ACM Comput. Surv. 32, 337–383 (2000)
Dung P.M.: On the acceptability of arguments and its fundamental role in non- monotonic reasoning, logic programming and n-person games. Artif. Intell. 77, 321–357 (1995)
Gabbay, D.: Labelled Deductive Systems. Oxford University Press (1996)
Gabbay, D.: Fibring Logics. Oxford University Press (1998)
Gabbay D.: Modal provability foundations for argumentation networks. Studia Logica 93(2–3), 181–189 (2009)
Gabbay D.: Fibering argumentation networks. Studia Logica 93(2–3), 231–296 (2009)
Gabbay D.: Semantics for higher level attacks in extended argumentation fames. Part 1, overview–Studia Logica 39, 355379 (2009)
Gabbay, D.: Equational Approach to Argumentation Networks (2011)
Gabbay, D.: Meta-Logical Investigations in Argumentation Networks. Monograph. Springer
Gabbay D., d’Avila Garcez A.S.: Logical modes of attack in argumentation networks. Studia Logica 93(2–3), 199–230 (2009)
Gamut, L.T.F.: Logic, Language, and Meaning. University of Chicago Press (1991)
Grossi, D.: On the logic of abstract argumentation. In: AAMAS’10 (2010)
Hunter, A.: Base logics in argumentation. In: Proceedings of COMMA 2010, pp. 275–286 (2010)
Leisenring, A.C.: Mathematical logic and Hilbert’s epsilon-symbol. McDonald, London (1969)
Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer (2008)
Modgil S.: Reasoning about preferences in argumentation frameworks. Artif. Intell. 173(9–10), 901–993 (2009)
Peirce, C.S.: A Boolean algebra with one constant. In: Hartshorne, C., Weiss, P. (eds.) Collected Papers of Charles Sanders Peirce, vol. 4, pp. 12–20. Harvard University Press (1931–1935)
Prakken H.: An abstract framework for argumentation with structured arguments. Argum. Comput. 1(2), 93–124 (2010)
Price R.: The Stroke function in natural deduction. Zeitsehr. f.math. Logik und Grundlagen d. Math. 7, 117–128 (1961)
Seto, Y.: Proofs of some axioms by stroke function. In: Proc. Japan Acad. vol. 44, pp. 1024–1026 (1968)
Sheffer H.M.: A set of five independent postulates for Boolean algebras, with application to logical constants. Trans. Am. Math. Soc. 14, 481–488 (1913)
Strasser S.: Towards the proof-theoretic unification of Dung’s argumentation framework: an adaptive logic approach. J. Logic Comput. 21, 133–156 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research done under ISF project “Integrating Logic and Network Reasoning”.
Rights and permissions
About this article
Cite this article
Gabbay, D.M. Dung’s Argumentation is Essentially Equivalent to Classical Propositional Logic with the Peirce–Quine Dagger. Log. Univers. 5, 255–318 (2011). https://doi.org/10.1007/s11787-011-0036-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-011-0036-3