On generalized notions of consistency and reinstatement and their preservation in formal argumentation

We present a conceptualization providing an original domain-independent perspective on two crucial properties in reasoning: consistency and reinstatement. They emerge as a pair of dual characteristics, representing complementary requirements on the outcomes of reasoning processes. Central to our formalization are two underlying parametric relations: incompatibility and reinstatement violation. Different instances of these relations give rise to a spectrum of consistency and reinstatement scenarios. As a demonstration of versatility and expressive power of our approach we provide a characterization of various abstract argumentation semantics which are expressed as combinations of distinct consistency and reinstatement constraints. Moreover, we conduct an investigation into preserving these essential properties across different reasoning stages. Specifically, we delve into scenarios where a labelling is derived from other labellings through a synthesis function, using the synthesis of argument justification as an illustrative instance. We achieve a general characterization of consistency preservation synthesis functions, while we unveil an impossibility result concerning reinstatement preservation, leading us to explore an alternative notion to ensure feasibility. Our exploration reveals a weakness in the traditional definition of argument justification, for which we propose a refined version overcoming this limitation.