Interval Methods for Judgment Aggregation in Argumentation

Given a set of conflicting arguments, there can exist multiple plausible opinions about which arguments should be accepted, rejected, or deemed undecided. Recent work explored some operators for deciding how multiple such judgments should be aggregated. Here, we generalize this line of study by introducing a family of operators called interval aggregation methods, which contain existing operators as instances. While these methods fail to output a complete labelling in general, we show that it is possible to transform a given aggregation method into one that does always yield collectively rational labellings. This employs the downadmissible and up-complete constructions of Caminada and Pigozzi. For interval methods, collective rationality is attained at the expense of a strong Independence postulate, but we show that an interesting weakening of the Independence postulate is retained.