基于证据的高阶集合论辩框架
Evidential-Based Higher-Order Set Argumentation Framework
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中文总结 AI 辅助
本文提出EHSAF框架，统一处理证据支持、高阶关系与集体交互，开发两种完备语义并证明其等价性条件，通过编码实现推理，弥合证据定性与定量推理的差距。
中文摘要 AI 辅助
证据论辩通过要求论证及交互由植根于表面证据的证据链提供支持，扩展了邓格（Dung）的抽象论辩。然而，现有形式体系缺乏对证据支持、高阶关系（针对任意元素的攻击与支持）及集体交互（源为集合）的统一处理。本文提出基于证据的高阶集合论辩框架（EHSAF），其在单一表达性设置中保守概括了若干现有框架。我们为EHSAF开发了两种完备语义：相邻完备标签语义，其对支持循环中的论证允许多个真值（真、假、未决），反映对未来证据的开放认知态度；以及基于外延的完备语义，其遵循严格证据论立场，仅接受具有充分依据的支持链的论证。我们表明这两种语义在存在支持循环时存在差异，并证明在支持无循环时二者等价。为实现计算推理，我们提供了EHSAF的标准命题编码，并证明在三值卢卡西维茨（Łukasiewicz）逻辑中，其模型恰好对应相邻完备标签。我们进一步将该编码扩展至连续模糊逻辑（哥德尔、乘积及卢卡西维茨），定义了连续模糊标准编码语义。我们证明该模糊语义满足关键性质——连续性、单调性、边界条件及解的存在性，且其三值化在自然t-范数条件下可恢复相邻完备标签。因此，我们的框架将表达性论辩与有原则的三值及模糊语义相统一，弥合了关于证据的定性与定量推理之间的差距。
英文摘要
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 \emph{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 \emph{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.
