On Elitist Lifting and Consistency in Structured Argumentation

We address the question of how to lift an ordering over rules to an ordering over arguments (sets of rules) that is well-behaved. It has been shown that so-called elitist lifting may lead to inconsistencies. We give restrictions on the underlying rule-ordering that avoid inconsistency. Then we show that a recently proposed solution – so-called disjoint strict lifting – that was introduced to address conceptual objections, also leads to inconsistency. We show that another recent proposal, telling us to reorder rule-orderings to take argument structure into account before lifting, is able to avoid conceptual problems without leading to any new inconsistencies. We generalise this approach by defining what we call structural rule-orderings and show a correspondence between weakest link and last link lifting of such orderings, which has interesting consequences for the question of consistency. We arrive at our results using a signature-based approach to structured argumentation. Instead of settling on a given framework, such as ASPIC+, we define an argumentation language that allows us to express only those properties of argumentation systems we need to establish our results. This abstract approach simplifies and clarifies the technical work while making our contribution more general.