Structured Argumentation, Indirect Arguments and “Attack” Relation

In contemporary studies on argumentation theory, the so-called “structured argumentation” or “structured argumentative approach” plays a prominent role. This methodology of studying argumentation can be seen as a refinement of another and perhaps more fundamental “abstract argumentation framework” developed by P.M. Dung. Structured argumentation is significant in that it is one of those areas where argumentation can be effectively investigated using the techniques of modern symbolic logic. The key point in this respect is that from the point of view of the structured argumentative approach the notion of “argument” is usually reduced to the notion of “deductive argument” in its usual logical sense. Most often, “deductive argument” here means a rather limited class of “direct arguments” (direct transitions from sentences to sentences). However, the practice of logically oriented argumentation is not limited to direct arguments and is very often based on the use of “indirect arguments” (e.g. “reductio ad absurdum”, “proof by contradiction”, “reasoning by cases”, etc.). In this paper, we will show that this class of arguments cannot be analyzed using the traditional structured argumentative approach. To address this problem, we will propose a new notion of “argument” whose extension will include not only direct but also indirect deductive arguments. In addition, we will show that using the new approach allows us to significantly enhance the expressive power of the “argumentative attack” relation. We will formulate eight new varieties of ‘refuting’ and “undercutting” arguments, with the help of which it becomes possible to formally analyze argumentative situations where indirect arguments can attack each other, can be used to attack direct arguments, or even can be attacked by direct arguments.