Constructing argument graphs with deductive arguments: a tutorial

argumentation, as proposed by Dung (1995), provides a good starting point for formalizing argumentation. Dung proposed that a set of arguments and counterarguments could be represented by a directed graph. Each node in the graph denotes an argument and each arc denotes one argument attacking another. So if there is an arc from node A to node B, then A attacks B, or equivalently A is a counterargument to B. See Figure 1 for an example of an abstract argument graph. Even though abstract argumentation provides a clear and precise approach to formalizing aspects of argumentation, the arguments are treated as atomic. There is no formalized content to an argument, and so all arguments are treated as equal. Therefore if we want to understand individual arguments, we need to provide content for them. This leads to the idea of “instantiating” abstract argumentation with deductive arguments. Each deductive argument has some premises from which a claim is derived by deductive reasoning. In deductive reasoning, we start with some premises, and we derive a conclusion using one or more inference steps. Each inference step is infallible in the sense that it does not introduce uncertainty. In other words, if we accept the premises are valid, ∗Corresponding author. Email: [email protected] ISSN: 1946-2166 print/ISSN 1946-2174 online c © 2009 Taylor & Francis DOI: 10.1080/1946216YYxxxxxxxx http://www.informaworld.com October 30, 2013 21:0 Argument & Computation deductivefinal 2 Taylor & Francis and I.T. Consultant then we should accept that the intermediate conclusion of each inference step is valid, and therefore we should accept that the conclusion is valid. For example, if we accept that Philippe and Tony are having tea together in London is valid, then we should accept that Philippe is not in Toulouse (assuming the background knowledge that London and Toulouse are different places, and that nobody can be in different places at the same time). As another example, if we accept that Philippe and Tony are having an ice cream together in Toulouse is valid, then we should accept that Tony is not in London. Note, however, we do not need to believe or know that the premises are valid to apply deductive reasoning. Rather, deductive reasoning allows us to obtain conclusions that we can accept contingent on the validity of their premises. So for the first example above, the reader might not know whether or not Philippe and Tony are having tea together in London. However, the reader can accept that Philippe is not in Toulouse, contingent on the validity of these premises. Important alternatives to deductive reasoning in argumentation, include inductive reasoning, abductive reasoning, and analogical reasoning. In this tutorial, we assume that deductive reasoning is formalized by a monotonic logic. Each deductive argument is a pair where the first item is a set of premises that logically entails the second item according to the choice of monotonic logic. So we have a logical language to express the set of premises, and the claim, and we have a logical consequence relation to relate the premises to the claim. Key benefits of deductive arguments include: (1) Explicit representation of the information used to support the claim of the argument; (2) Explicit representation of the claim of the argument; and (3) A simple and precise connection between the support and claim of the argument via the consequence relation. What a deductive argument does not provide is a specific proof of the claim from the premises. There may be more than one way of proving the claim from the premises, but the argument does not specify which is used. It is therefore indifferent to the proof used. Deductive argumentation is formalized in terms of deductive arguments and counterarguments, and there are various choices for defining this (Besnard and Hunter (2008)). Deductive argumentation offers a simple route to instantiating abstract argumentation which we will consider in this tutorial paper. Perhaps the first paper to consider this is by Cayrol who instantiated Dung’s proposal with deductive arguments based on classical logic (Cayrol (1995)). In the rest of this tutorial, we will investigate some of the choices we have for defining arguments and counterarguments, and for how they can be used in modelling argumentation. We will focus on two choices for base logic. These are simple logic (which has a language of literals and rules of the form α1 ∧ . . . ∧ αn → β where α1, . . . , αn, β are literals, and modus ponens is the only proof rule) and classical logic (propositional and first-order classical logic). Then for instantiating A1 = Patient has hypertension so prescribe diuretics A2 = Patient has hypertension so prescribe betablockers A3 = Patient has emphysema which is a contraindication for betablockers Figure 1. Example of an abstract argument graph which captures a decision making scenario where there are two alternatives for treating a patient, diuretics or betablockers. Since only one treatment should be given for the disorder, each argument attacks the other. There is also a reason to not give betablockers, as the patient has emphysema which is a contraindication for this treatment. October 30, 2013 21:0 Argument & Computation deductivefinal Argument & Computation 3 Generative graphs Descriptive graphs