Argumentation theory
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Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion; studying rules of inference, logic, and procedural rules in both artificial and real world settings. Argumentation is concerned primarily with reaching conclusions through logical reasoning, that is, claims based on premises. Although including debate and negotiation which are concerned with reaching mutually acceptable conclusions, argumentation theory also encompasses eristic dialog, the branch of social debate in which victory over an opponent is the primary goal. This art and science is often the means by which people protect their beliefs or self-interests in rational dialogue, in common parlance, and during the process of arguing. Argumentation is used in law, for example in trials, in preparing an argument to be presented to a court, and in testing the validity of certain kinds of evidence. Also, argumentation scholars study the post hoc rationalizations by which organizational actors try to justify decisions they have made irrationally.

Key components of argumentation
Understanding and identifying arguments, either explicit or implied, and the goals of the participants in the different types of dialogue.
Identifying the premises from which conclusions are derived
Establishing the "burden of proof" — determining who made the initial claim and is thus responsible for providing evidence why his/her position merits acceptance
For the one carrying the "burden of proof", the advocate, to marshal evidence for his/her position in order to convince or force the opponent's acceptance. The method by which this is accomplished is producing valid, sound, and cogent arguments, devoid of weaknesses, and not easily attacked.
In a debate, fulfillment of the burden of proof creates a burden of rejoinder. One must try to identify faulty reasoning in the opponent’s argument, to attack the reasons/premises of the argument, to provide counterexamples if possible, to identify any logical fallacies, and to show why a valid conclusion cannot be derived from the reasons provided for his/her argument.


Components of Argument
In The Uses of Argument (1958), Toulmin proposed a layout containing six interrelated components for analyzing arguments:
1. Claim
Conclusions whose merit must be established. For example, if a person tries to convince a listener that he is a British citizen, the claim would be “I am a British citizen.” (1)
2. Data
The facts we appeal to as a foundation for the claim. For example, the person introduced in 1 can support his claim with the supporting data “I was born in Bermuda.” (2)
3. Warrant
The statement authorizing our movement from the data to the claim. In order to move from the data established in 2, “I was born in Bermuda,” to the claim in 1, “I am a British citizen,” the person must supply a warrant to bridge the gap between 1 & 2 with the statement “A man born in Bermuda will legally be a British Citizen.” (3)
4. Backing
Credentials designed to certify the statement expressed in the warrant; backing must be introduced when the warrant itself is not convincing enough to the readers or the listeners. For example, if the listener does not deem the warrant in 3 as credible, the speaker will supply the legal provisions as backing statement to show that it is true that “A man born in Bermuda will legally be a British Citizen.”
5. Rebuttal
Statements recognizing the restrictions to which the claim may legitimately be applied. The rebuttal is exemplified as follows, “A man born in Bermuda will legally be a British citizen, unless he has betrayed Britain and has become a spy of another country.”
6. Qualifier
Words or phrases expressing the speaker’s degree of force or certainty concerning the claim. Such words or phrases include “possible,” “probably,” “impossible,” “certainly,” “presumably,” “as far as the evidence goes,” or “necessarily.” The claim “I am definitely a British citizen” has a greater degree of force than the claim “I am a British citizen, presumably.”
The first three elements “claim,” “data,” and “warrant” are considered as the essential components of practical arguments, while the second triad “qualifier,” “backing,” and “rebuttal” may not be needed in some arguments

Kinds of argumentation :
1.)Conversational argumentation
Main articles: Conversation Analysis and Discourse Analysis
The study of naturally-occurring conversation arose from the field of sociolinguistics. It is usually called conversational analysis. Inspired by ethnomethodology, it was developed in the late 1960s and early 1970s principally by the sociologist Harvey Sacks and, among others, his close associates Emanuel Schegloff and Gail Jefferson. Sacks died early in his career, but his work was championed by others in his field, and CA has now become an established force in sociology, anthropology, linguistics, speech-communication and psychology.It is particularly influential in interactional sociolinguistics, discourse analysis and discursive psychology, as well as being a coherent discipline in its own right. Recently CA techniques of sequential analysis have been employed by phoneticians to explore the fine phonetic details of speech.
Empirical studies and theoretical formulations by Sally Jackson and Scott Jacobs, and several generations of their students, have described argumentation as a form of managing conversational disagreement within communication contexts and systems that naturally prefer agreement.
2.) Mathematical argumentation
Main article: Philosophy of mathematics
The basis of mathematical truth has been the subject of long debate. Frege in particular sought to demonstrate (see Gottlob Frege, The Foundations of Arithemetic, 1884, and Logicism in Philosophy of mathematics) that arithmetical truths can be derived from purely logical axioms and therefore are, in the end, logical truths. The project was developed by Russell and Whitehead in their Principia Mathematica. If an argument can be cast in the form of sentences in Symbolic Logic, then it can be tested by the application of accepted proof procedures. This has been carried out for Arithmetic using Peano axioms. Be that as it may, an argument in Mathematics, as in any other discipline, can be considered valid just in case it can be shown to be of a form such that it cannot have true premises and a false conclusion.
3.) Scientific argumentation
Main article: Philosophy of Science
Perhaps the most radical statement of the social grounds of scientific knowledge appears in Alan G.Gross "The Rhetoric of Science." Cambridge: Harvard University Press, 1990. Gross holds that science is rhetorical "without remainder," meaning that scientific knowledge itself cannot be seen as an idealized ground of knowledge. Scientific knowledge is produced rhetorically, meaning that it has special epistemic authority only insofar as its communal methods of verification are trustworthy. This thinking represents an almost complete rejection of the foundationalism on which argumentation was first based.
4.) Legal argumentation
Main articles: Oral argument and Closing argument
Legal arguments (or oral arguments) are spoken presentations to a judge or appellate court by a lawyer (or parties when representing themselves) of the legal reasons why they should prevail. Oral argument at the appellate level accompanies written briefs, which also advance the argument of each party in the legal dispute. A closing argument (or summation) is the concluding statement of each party's counsel (often called an attorney in the United States) reiterating the important arguments for the trier of fact, often the jury, in a court case. A closing argument occurs after the presentation of evidence.
5.) Political argumentation
Main article: Political argument
Political arguments are used by academics, media pundits, candidates for political office and government officials. Political arguments are also used by citizens in ordinary interactions to comment about and understand political events. The rationality of the public is a major question in this line of research. A robust political science research tradition seems to prove that the American public is largely irrational and ignorant of even the most basic knowledge of national or world affairs. Political scientist S. Popkin coined the expression "low information voters" to describe most voters who know very little about politics or the world in general.
Some theorists have inferred from this that only comprehensively trained elites can debate public issues. They point as additional proof to the practice of academic debate in the United States, an activity almost exclusively involving children of the upper middle classes, future lawyers and graduate students, and not ordinary citizens.

Sources :
J. Robert Cox and Charles Arthur Willard, eds. Advances in Argumentation Theory and Research 1982.
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).
Bondarenko, A., Dung, P. M., Kowalski, R., and Toni, F. , "An abstract, argumentation-theoretic approach to default reasoning", Artificial Intelligence 93(1-2) 63-101 (1997).
Dung, P. M., Kowalski, R., and Toni, F. "Dialectic proof procedures for assumption-based, admissible argumentation." Artificial Intelligence. 170(2), 114-159 (2006).
Frans van Eemeren, Rob Grootendorst, Sally Jackson, and Scott Jacobs, Reconstructing Argumentative Discourse 1993.