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Hypersequent-Based Argumentation: An Instantiation in the Relevance Logic RM

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Theory and Applications of Formal Argumentation (TAFA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10757))

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Abstract

In this paper we introduce hypersequent-based frameworks for the modeling of defeasible reasoning by means of logic-based argumentation. These frameworks are an extension of sequent-based argumentation frameworks, in which arguments are represented not only by sequents, but by more general expressions, called hypersequents. This generalization allows us to overcome some of the weaknesses of logical argumentation reported in the literature and to prove several desirable properties, stated in terms of rationality postulates. For this, we take the relevance logic RM as the deductive base of our formalism. This logic is regarded as “by far the best understood of the Anderson-Belnap style systems” (Dunn and Restall, Handbook of Philosophical Logic, vol. 6). It has a clear semantics in terms of Sugihara matrices, as well as sound and complete Hilbert- and Gentzen-type proof systems. The latter are defined by hypersequents and admit cut elimination. We show that hypersequent-based argumentation yields a robust defeasible variant of RM with many desirable properties (e.g., rationality postulates and crash-resistance).

The first two authors are supported by the Israel Science Foundation (grant 817/15).

The first and the third author are supported by the Alexander von Humboldt Foundation and the German Ministry for Education and Research.

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Notes

  1. 1.

    Set signs in arguments are omitted.

  2. 2.

    See [4] for further advantages of this approach.

  3. 3.

    By requiring that both the attacking and the attacked argument should be in \(\text {Arg}_\mathsf{L}(\mathcal {S})\) we prevent “irrelevant attacks”, that is: situations in which, e.g., \(\lnot p\Rightarrow \lnot p\) attacks \(p\Rightarrow p\) (by Undercut), although \(\mathcal {S}= \{p\}\).

  4. 4.

    It is well-known (see [16]) that the grounded extension of a framework is unique.

  5. 5.

    This follows since any attacker of \(q\Rightarrow q\) has an inconsistent support.

  6. 6.

    The common, intuitive interpretation of the sign “\(\mid \)” is disjunction.

  7. 7.

    Strictly speaking, RM is a semi-relevance logic: it does satisfy the basic relevance criterion (introduced in [3]) and the minimal semantic relevance criterion [9], but it does not have the variable sharing property (introduced in [3]), see, e.g., [9].

  8. 8.

    Unlike R, RM does satisfy the mingle axiom \(\phi \supset (\phi \supset \phi )\).

  9. 9.

    Since a sequent is a particular case of a hypersequent and hypersequent calculi generalize sequent calculi, arguments in the sense of the previous sections are particular cases of the arguments according to the new definition.

  10. 10.

    Intuitively, this is so due to the possibility of splitting hypersequents into different components. A formal justification will be given in the next subsection.

  11. 11.

    Where free arguments are those arguments that are based only on premises that are not involved in minimally inconsistent subsets of \(\mathcal {S}\) (see Definition 10).

  12. 12.

    Note that if \(\mathcal {T}\) is consistent, then so are \(\mathsf{CN}_\mathsf{L}(\mathcal {T})\) and \(\mathcal {T}'\) for every \(\mathcal {T}'\subseteq \mathcal {T}\). If \(\mathcal {T}\) is inconsistent, then so is every superset of \(\mathcal {T}\).

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Borg, A., Arieli, O., Straßer, C. (2018). Hypersequent-Based Argumentation: An Instantiation in the Relevance Logic RM . In: Black, E., Modgil, S., Oren, N. (eds) Theory and Applications of Formal Argumentation. TAFA 2017. Lecture Notes in Computer Science(), vol 10757. Springer, Cham. https://doi.org/10.1007/978-3-319-75553-3_2

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