On the acceptance of loops in argumentation frameworks

Current approaches for giving semantics to abstract argumentation frameworks dismiss altogether any possibility of having conflicts among accepted arguments by requiring that the latter should be ‘conflict free’. In reality, however, contradictory phenomena coexist, or it may happen that one cannot make a choice between conflicting indications but still would like to keep track to all of them. For this purpose we introduce in this article a new kind of argumentation semantics, called ‘conflict-tolerant’, in which all the accepted arguments must be justified (in the sense that each one of them can be defended), but some of them may still attack each other. In terms of graphical representation of argumentation systems, where attacks are represented by directed edges, this means that the possibility of accepting ‘loops’ of arguments is not automatically ruled out without any further considerations. To provide conflict-tolerant semantics, we enhance the two standard approaches for defining coherent (conflict-free) semantics for argumentation frameworks. The extension-based approach is generalized by relaxing the ‘conflict-freeness’ requirement of the chosen sets of arguments, and the three-valued labelling approach is replaced by a four-valued labelling system that allows to capture mutual attacks among accepted arguments. We show that our setting is not a substitute of standard (conflict-free) semantics, but rather a generalized framework that accommodates both conflict-free and conflict-tolerant semantics. Moreover, the one-to-one relationship between extensions and labellings of conflict-free semantics is carried on to a similar correspondence between the extended approaches for providing conflict-tolerant semantics. Thus, in our setting as well, these are essentially two points of views for the same thing.