A dialectical formalisation of preferred subtheories reasoning under resource bounds

Dialectical Classical Argumentation (Dialectical Cl-Arg) has been shown to satisfy rationality postulates under resource bounds. In particular, the consistency and non-contamination postulates are satisfied despite dropping the assumption of logical omniscience and the consistency and subset minimality checks on arguments' premises that are deployed by standard approaches to Cl-Arg. This paper studies Dialectical Cl-Arg's formalisation of Preferred Subtheories (PS) non-monotonic reasoning under resource bounds. The contribution of this paper is twofold. First, we establish soundness and completeness for Dialectical Cl-Arg's credulous consequence relation under the preferred semantics and credulous PS consequences. This result paves the way for the use of argument game proof theories and dialogues that establish membership of arguments in admissible (and so preferred) extensions, and hence the credulous PS consequences of a belief base. Second, we refine the non-standard characteristic function for Dialectical Cl-Arg, and use this refined function to show soundness for Dialectical Cl-Arg consequences under the grounded semantics and resource-bounded sceptical PS consequence. We provide a counterexample that shows that completeness does not hold. However, we also show that the grounded consequences defined by Dialectical Cl-Arg strictly subsume the grounded consequences defined by standard Cl-Arg formalisations of PS, so that we recover sceptical PS consequences that one would intuitively expect to hold.