Bipolar Argumentation Frameworks, Modal Logic and Semantic Paradoxes

Abstract

Bipolar Argumentation Frameworks (BAF) are a natural extension of Dung's Argumentation Frameworks (AF) where a relation of support between arguments is added to the standard attack relation. Despite their interest, BAF present several difficulties and their semantics are quite complex. This paper provides a definition of semantic concepts for BAF in terms of fixpoints of the functions of neutrality and defense, thus preserving most of the fundamental properties of Dung's AF. From this angle it becomes easy to show that propositional dynamic logic provides an adequate language to talk about BAF. Finally, we illustrate how this framework allows to encode the structure of the referential discourse involved in semantic paradoxes such as the Liar. It turns out that such paradoxes can be seen as BAF without a stable extension. A further notion is that of evidential support [?] that we will not deal with here. 6 Carlo Proietti gratefully acknowledges funding received from the European Commission (Marie Skłodowska-Curie Individual Fellowship 2016, 748421) for his research. The authors would like to thank the anonymous reviewers of LORI'19 for many helpful suggestions. 4 Indeed, the only support available in standard AF is the "defense" relation: argument a supports argument b by attacking one of its attackers. This is too restrictive in most real-life debates, where arguments providing direct support are commonly used. 5 We limit ourselves to binary necessary support. Indeed this notion of support is often introduced as a more general relation between a set of arguments and an argument [?,?]. 6 Evidential support can be seen as a special kind of necessary support where an argument cannot be accepted unless it is ultimately supported by "evidence", the latter being a special