L. Amgoud, Postulates for logic-based argumentation systems, ECAI Workshop on Weighted Logics for ai (WL4AI'12), 2012.
DOI : 10.1016/j.ijar.2013.10.004

URL : https://hal.archives-ouvertes.fr/hal-01123709

L. Amgoud and P. Besnard, Bridging the Gap between Abstract Argumentation Systems and Logic, Scalable Uncertainty Management, 3rd International Conference, SUM 2009, pp.12-27, 2009.
DOI : 10.1016/0004-3702(92)90069-A

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

L. Amgoud and P. Besnard, A Formal Analysis of Logic-Based Argumentation Systems, Scalable Uncertainty Management, 4th International Conference, SUM 2010, pp.42-55, 2010.
DOI : 10.1007/978-3-642-15951-0_10

L. Amgoud, P. Besnard, and S. Vesic, Identifying the Core of Logic-Based Argumentation Systems, 2011 IEEE 23rd International Conference on Tools with Artificial Intelligence, pp.633-636, 2011.
DOI : 10.1109/ICTAI.2011.100

L. Amgoud, M. Caminada, C. Cayrol, M. C. Lagasquie, and H. Prakken, Towards a consensual formal model: inference part (Tech, Deliverable D2.2: Draft Formal Semantics for Inference and Decision-Making. ASPIC project. Retrieved from www.argumentation.org, 2004.

L. Amgoud and C. Cayrol, Inferring from inconsistency in preference-based argumentation frameworks, Journal of Automated Reasoning, vol.29, issue.2, pp.125-169, 2002.
DOI : 10.1023/A:1021603608656

L. Amgoud, N. Maudet, and S. Parsons, Modelling dialogues using argumentation, Proceedings Fourth International Conference on MultiAgent Systems, pp.10-12, 2000.
DOI : 10.1109/ICMAS.2000.858428

L. Amgoud and H. Prade, Using arguments for making and explaining decisions, Artificial Intelligence, vol.173, issue.3-4, pp.413-436, 2009.
DOI : 10.1016/j.artint.2008.11.006

L. Amgoud and M. Serrurier, Agents that argue and explain classifications. Autonomous Agents and Multi-Agents Systems, pp.187-209, 2008.

S. Benferhat, D. Dubois, and H. Prade, Some syntactic approaches to the handling of inconsistent knowledge bases: A comparative study part 1: The at case. Studia Logica, pp.17-45, 1997.

P. Besnard and A. Hunter, Elements of argumentation, 2008.
DOI : 10.7551/mitpress/9780262026437.001.0001

E. Black and A. Hunter, An inquiry dialogue system, Autonomous Agents and Multi-Agent Systems, vol.15, issue.6, pp.173-209, 2009.
DOI : 10.1007/s10458-008-9074-5

D. G. Bobrow, Editor's preface, Artificial Intelligence, vol.13, issue.1-2, pp.1-2, 1980.
DOI : 10.1016/0004-3702(80)90009-0

B. Bonet and H. Geffner, Arguing for decisions: A qualitative model of decision making, UAI'96: Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence, Reed College, pp.98-105, 1996.

M. Caminada, On the Issue of Reinstatement in Argumentation, Logics in Artificial Intelligence, 10th European Conference, pp.111-123, 2006.
DOI : 10.1007/11853886_11

M. Caminada, Semi-stable semantics, Computational Models of Argument: Proceedings of COMMA 2006, pp.121-130, 2006.
DOI : 10.1093/logcom/exr033

URL : http://logcom.oxfordjournals.org/cgi/content/short/22/5/1207

M. Caminada and L. Amgoud, On the evaluation of argumentation formalisms, Artificial Intelligence, vol.171, issue.5-6, pp.286-310, 2007.
DOI : 10.1016/j.artint.2007.02.003

C. Cayrol, On the relation between argumentation and non-monotonic coherence-based entailment, Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence, IJCAI 95, pp.2-1443, 1995.

S. Coste, C. Devred, and P. Marquis, Symmetric argumentation frameworks, Symbolic and Quantitative Approaches to Reasoning with Uncertainty, 8th European Conference, pp.317-328, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00014624

P. M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games, Artificial Intelligence, vol.77, issue.2, pp.321-357, 1995.
DOI : 10.1016/0004-3702(94)00041-X

P. M. Dung, P. Mancarella, and F. Toni, Computing ideal sceptical argumentation, Artificial Intelligence, vol.171, issue.10-15, pp.642-674, 2007.
DOI : 10.1016/j.artint.2007.05.003

URL : http://doi.org/10.1016/j.artint.2007.05.003

M. Elvang-gøransson, J. Fox, and P. Krause, Acceptability of arguments as ???logical uncertainty???, Symbolic and Quantitative Approaches to Reasoning and Uncertainty, European Conference, ECSQARU'93, pp.85-90, 1993.
DOI : 10.1007/BFb0028186

N. Gorogiannis and A. Hunter, Instantiating abstract argumentation with classical logic arguments: Postulates and properties, Artificial Intelligence, vol.175, issue.9-10, pp.1479-1497, 2011.
DOI : 10.1016/j.artint.2010.12.003

URL : http://doi.org/10.1016/j.artint.2010.12.003

G. Governatori, M. J. Maher, G. Antoniou, and D. Billington, Argumentation Semantics for Defeasible Logic, Journal of Logic and Computation, vol.14, issue.5, pp.675-702, 2004.
DOI : 10.1093/logcom/14.5.675

M. Mozina, J. Zabkar, and I. Bratko, Argument based machine learning, Artificial Intelligence, vol.171, issue.10-15, pp.922-937, 2007.
DOI : 10.1016/j.artint.2007.04.007

S. Parsons, M. Wooldridge, and L. Amgoud, Properties and Complexity of Some Formal Inter-agent Dialogues, Journal of Logic and Computation, vol.13, issue.3, pp.347-376, 2003.
DOI : 10.1093/logcom/13.3.347

H. Prakken, An abstract framework for argumentation with structured arguments, Argument & Computation, vol.4, issue.2, pp.93-124, 2010.
DOI : 10.1016/S0004-3702(96)00041-0

H. Prakken and G. Sartor, Argument-based extended logic programming with defeasible priorities, Journal of Applied Non-Classical Logics, vol.53, issue.1993, pp.25-75, 1997.
DOI : 10.1080/11663081.1997.10510900

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

N. Rescher and R. Manor, On inference from inconsistent premisses, Theory and Decision, vol.1, issue.2, pp.179-219, 1970.
DOI : 10.1007/BF00154005

G. Simari and R. Loui, A mathematical treatment of defeasible reasoning and its implementation, Artificial Intelligence, vol.53, issue.2-3, pp.125-157, 1992.
DOI : 10.1016/0004-3702(92)90069-A

K. Sycara, Persuasive argumentation in negotiation, Theory and Decision, vol.12, issue.3, pp.203-242, 1990.
DOI : 10.1007/BF00162699

A. Tarski, On some fundamental concepts of metamathematics Logic, semantics, metamathematics: Papers from, Trans., pp.30-37, 1923.

S. Zabala, I. Lara, and H. Geffner, Beliefs, reasons and moves in a model for argumentative dialogues, Proc. 25th latino-american conf. on computer science, 1999.

P. Let, X. , X. ?. , and X. ??-?-l, Assume that X ? X ? . According to the compactness axiom, it holds that CN(X ) = Y ? f X CN(Y )

C. Assume and . Cn, According to the expansion axiom Moreover, X ? X ? X ?? thus CN(X ) ? CN(X ? X ?? ) (a') Similarly From (a), it follows that, )). Finally, by the idempotence axiom, the inclusion CN(X ? ? X ?? ) ? CN(X ? X ?? ) holds. To show that CN(X ? X ?? ) ? CN(X ? ? X ?? ), the same reasoning is applied by starting with X instead of X ?

P. Let and X. , Assume that X is consistent, then CN(X ) = L (1)

?. ?. Assume, X. Such, and . Inconsistent, This means that CN(X ? ) = L. However, since X ? ? X then CN(X ? ) ? CN(X ) (according to the monotonicity axiom, Thus, L ? CN(X ). Since CN(X ) ? L, then CN(X ) = L. Thus, X is inconsistent. Contradiction

T. Since and . Concs, Thus, there exists x ? ? C such that x ?, GE ConcsGE

?. Then and . Ext, Base(E) is consistent. Consequently, T satisfies consistency. From Amgoud (2012), T is also closed under sub-arguments