cs.AIAug 28, 2026

Evidential-Based Higher-Order Set Argumentation Framework

Authors: Shuai Tang

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Abstract

Evidential argumentation extends Dung's abstract argumentation by requiring arguments and interactions to be backed by chains of evidence rooted in prima-facie elements. However, existing formalisms lack a unified treatment of evidential support, higher-order relations (attacks and supports targeting arbitrary elements), and collective interactions (sources as sets). In this paper, we introduce the Evidential-Based Higher-Order Set Argumentation Framework (EHSAF), which conservatively generalises several existing frameworks within a single expressive setting. We develop two complete semantics for EHSAFs: an adjacent complete labelling semantics that admits multiple truth values (true, false, undecided) for arguments in support cycles, reflecting an open epistemic attitude toward future evidence; and an extension-based complete semantics that follows a strict evidentialist stance, accepting only arguments with well-founded support chains. We show that these two semantics diverge in the presence of support cycles, and prove their equivalence under support-acyclicity. To enable computational reasoning, we provide a normal propositional encoding of EHSAFs and prove that, in three-valued Łukasiewicz logic, its models correspond precisely to the adjacent complete labellings. We further extend this encoding to continuous fuzzy logics (G{ö}del, Product, and Łukasiewicz), defining a continuous fuzzy normal encoded semantics. We establish that this fuzzy semantics satisfies key properties---continuity, monotonicity, boundary conditions, and solution existence---and that its ternarisation recovers the adjacent complete labellings under natural t-norm conditions. Our framework thus unifies expressive argumentation with principled three-valued and fuzzy semantics, bridging the gap between qualitative and quantitative reasoning about evidence.

Figures & tables

⇒\Rightarrow012\frac{1}{2}1
0111
12\frac{1}{2}12\frac{1}{2}11
1012\frac{1}{2}1
Table 1: Truth table for ⇒\Rightarrow in PL3L\mathcal{PL}_{3}^{L} .
⇔\Leftrightarrow012\frac{1}{2}1
0112\frac{1}{2}0
12\frac{1}{2}12\frac{1}{2}112\frac{1}{2}
1012\frac{1}{2}1
Table 2: Truth table for ⇔\Leftrightarrow in PL3L\mathcal{PL}_{3}^{L} .
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