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Weakest Link in Formal Argumentation: Lookahead and Principle-Based Analysis

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Logic and Argumentation (CLAR 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14156))

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Abstract

In this paper, we introduce a new definition of weakest link attack relation assignment based on lookahead, and compare this new lookahead definition with two existing ones in the literature using a principle-based analysis. We adopt a formal framework for such attack relation assignments that was introduced by Dung in 2016. We show that our lookahead definition does not satisfy context independence, we introduce a new principle called weak context independence, and we show that lookahead weakest link satisfies weak context independence. We also show that lookahead weakest link is the closest approximation to Brewka’s prioritised default logic PDL, also known as the greedy approach. For PDL, we prove an impossibility result under Dung’s axioms. Our results generalise earlier findings restricted to total orders to the more general case of modular orders.

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Notes

  1. 1.

    Structured argumentation builds arguments from the rules and facts of a knowledge base. Abstract argumentation just assumes an attack relation to define sets of arguments that are collectively acceptable, while ignoring the underlying logic that defines attacks as logical conflicts.

  2. 2.

    These priorities give the same outputs for the fitness-loving Scot and snoring professor [23, 24], which are just variants of Example 1 with facts. Other variants of Example 1 with facts and strict rules [6, 7] give the (non-)teaching dean professor scenario [13], see Example 5 below. For further variants of Example 1 defined by partial orders we refer to Dung’s paper in 2018 [16]. A brief discussion for the case of partial orders can be found in Sect. 7.

  3. 3.

    As a consequence, the atoms in \(\mathscr {L}\) here only consist of domain atoms representing propositions about the concerned domains. Dung also considers non-domain atoms \(ab_d\) for the non-applicability of a defeasible rule d, and undercuts as strict rules \(b_1, \ldots , b_n\rightarrow ab_{d}\) that act against the applicability of a defeasible rule d in RD [13]. We leave for future work the extension of our current results to knowledge bases with strict rules and undercutting arguments.

  4. 4.

    A reader might wonder why Definition 15 does not simply state: \((A,B) \in att_{lwl}(K)\) iff \((A,B) \in att_{dwl}(K)\) and A is \(\sqsubseteq \)-maximal with \((\cdot , B) \in att_{dwl}(K)\). The reason is that, under these attacks, one can define some K whose stable belief sets include logically contradictory sets.

  5. 5.

    The notion of undercut from Principle 7 is the same as in Pollock [27] and ASPIC+ [24]: an argument A undercuts B at \(B' \in sub(B)\) iff the last rule \(d = last(B') \in RD\) and A states that this defeasible rule d is not applicable \(cnl(A) = ab_{d}\).

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Acknowledgements

The authors are thankful to the three anonymous reviewers for their helpful comments and suggestions. L. van der Torre is financially supported by FNR through the project OPEN O20/14776480, the G.A. INTER/CHIST/19/14589586 Horizon 2020 grant, and EU’s Justice programme under grant 101007420 (ADELE).

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Authors and Affiliations

  1. Zhejiang University, Hangzhou, China

    Chen Chen & Leendert van der Torre

  2. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Pere Pardo, Leendert van der Torre & Liuwen Yu

  3. University of Bologna, Bologna, Italy

    Liuwen Yu

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  2. Pere Pardo
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Correspondence to Pere Pardo .

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  1. IRIT CNRS, Toulouse, France

    Andreas Herzig

  2. Zhejiang University, Hangzhou, China

    Jieting Luo

  3. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Pere Pardo

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Chen, C., Pardo, P., van der Torre, L., Yu, L. (2023). Weakest Link in Formal Argumentation: Lookahead and Principle-Based Analysis. In: Herzig, A., Luo, J., Pardo, P. (eds) Logic and Argumentation. CLAR 2023. Lecture Notes in Computer Science(), vol 14156. Springer, Cham. https://doi.org/10.1007/978-3-031-40875-5_5

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