Relevance in Structured Argumentation

Abstract

We study properties related to relevance in non-monotonic consequence relations obtained by systems of structured argumentation. Relevance desiderata concern the robustness of a consequence relation under the addition of irrelevant information. For an account of what (ir)relevance amounts to we use syntactic and semantic considerations. Syntactic criteria have been proposed in the domain of relevance logic and were recently used in argumentation theory under the names of non-interference and crashresistance. The basic idea is that the conclusions of a given argumentative theory should be robust under adding information that shares no propositional variables with the original database. Some semantic relevance criteria are known from non-monotonic logic. For instance, cautious monotony states that if we obtain certain conclusions from an argumentation theory, we may expect to still obtain the same conclusions if we add some of them to the given database. In this paper we investigate properties of structured argumentation systems that warrant relevance desiderata.Our attack form is sometimes called premise-attack [20] or directed undercut [6]. In Section 4 we will show that by adjusting · and · adequately we are able to accommodate many other attack forms defined in the literature.Definition 4 (Attack Diagram). Given a setting AF ⊢ (S), its attack diagram is the directed graph with the set of nodes Arg ⊢ (S) and edges between a and b iff a attacks b.Definition 5 (Dung Semantics, [12]). Where AF ⊢ (S) is a setting and A ⊆ Arg ⊢ (S) we define: A is conflict-free iff there are no a, b ∈ A such that a attacks b. A defends a ∈ Arg ⊢ (S) iff for each attacker b ∈ Arg ⊢ (S) of a there is a c ∈ A that attacks b. A is admissible iff it is conflict-free and it defends every a ∈ A. A is complete iff it is admissible and it contains every a ∈ Arg ⊢ (S) it defends. A is preferred iff it is ⊆-maximal complete. A is grounded iff it is ⊆-minimal complete. A is stable iff it is admissible and for all a ∈ Arg ⊢ (S) \ A there is a b ∈ A that attacks a.We denote the set of all admissible [complete, preferred, stable] sets A (also called "extensions") by Adm(AF ⊢ (S)) [Cmp(AF ⊢ (S)), Prf(AF ⊢ (S)), Stb(AF ⊢ (S))] and the grounded set by Grd(AF ⊢ (S)).Definition 6 (Consequence Relations). Where Sem ∈ {Grd, Prf, Stb}, and given a setting AF ⊢ we define:Where the setting AF ⊢ is clear from the context we will simply write |∼ Sem to avoid clutter.Remark 2. For reasons of space we restrict our focus in this paper on skeptical consequence as defined in Definition 6. Note that |∼ AF ⊢ Cmp coincides with |∼ AF ⊢ Grd .3 The Relevance Properties Syntactic RelevanceA syntactical relevance property that has been proposed in the context of structured argumentation is noninterference [9]. Let us call two sets of formulas syntactically disjoint if no atom that occurs in a formula

Rationality and maximal consistent sets for a fragment of ASPIC + without undercut

“…Their motivation, however, has been questioned by [10]. 14 In Definition 5 we follow what is the most frequent approach in the ASPIC-family, namely to assume that undercutting attacks are preference-independent. However, since our main results concern ASPIC-frameworks without undercut this choice is insignificant for this study.…”

“…We use this term in order to indicate that the addition of "irrelevant" information leads to contaminating effects (see also[14]). "Irrelevance" is disambiguated differently in the two versions considered.…”

A Generalized Proof-Theoretic Approach to Logical Argumentation Based on Hypersequents

“…For instance, in [54] such issues are considered and paraconsistent logics are taken to overcome trivialization, a weaker version of crash-resistance from [41]. Recently, in [37], properties of some well-known structured argumentation systems (including sequent-based argumentation) that warrant several relevance desiderata are investigated. 34 In [9] similar problems are discussed and resolved by introducing relevant attack rules.…”

“…By doing so, any completeness-based semantics can be used for the framework. Recently, in [37] a general framework is defined, in which several well-known structured argumentation frameworks can be represented. It is shown that for this general framework and under a few further assumptions, both crash-resistance and non-interference are obtained for many completeness-based semantics.…”

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