Higher-Order Quadripolar Argumentation Framework and Encoded Semantics

This paper introduces the Higher-Order Quadripolar Argumentation Framework (HQAF), which extends bipolar frameworks by allowing attacks and three types of supports—necessary, deductive, and evidential—to interact at higher order. In HQAF, each interaction may have as its source and target an argument, an attack, or a support. We provide a 3-valued semantics via adjacent complete labellings and equational systems, and define encoded semantics directly in propositional logics, mainly Łukasiewicz three-valued logic for the 3-valued case and fuzzy logics based on continuous t-norms (Gödel, Product, Łukasiewicz) for the fuzzy case. Model equivalence between HQAF and its normal encoding is established. The fuzzy equational semantics is shown to be a specific formalism of the fuzzy encoded semantics, and ternarization connects fuzzy models to 3-valued complete labellings. The study is confined to non-set HQAF; collective interactions are reserved for future work.