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Towards the Proof-Theoretic Unification of Dung’s Argumentation Framework: An Adaptive Logic Approach

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Adaptive Logics for Defeasible Reasoning

Part of the book series: Trends in Logic ((TREN,volume 38))

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Abstract

This chapter presents a unifying adaptive logic framework for abstract argumentation. It consists of a core system for abstract argumentation and various adaptive logics based on it. These logics represent in an accurate sense all standard extensions defined within Dung?s abstract argumentation framework with respect to skeptical and credulous acceptance. The models of our logics correspond exactly to specific extensions of given argumentation frameworks. Additionally, the dynamics of adaptive proofs mirror the argumentative reasoning of a rational agent. In particular, the presented logics allow for external dynamics, i.e., they are able to deal with the arrival of new arguments and are therefore apt to model open-ended argumentations by providing provisional conclusions.

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Notes

  1. 1.

    A former version of the content of this chapter has been published under the name “Towards the Proof-Theoretic Unification of Dung’s Argumentation Framework: An Adaptive Logic Approach” in the “Journal of Logic and Computation”, 2010, [1]. The paper is co-authored by Dunja Šešelja.

  2. 2.

    We restrict the discussion in this chapter to the finite case, i.e., to argumentation frameworks with a finite number of propositional letters.

  3. 3.

    Semi-stable semantics were defined by Caminada in [22] and are equivalent to Verhijs’ admissible stage extensions in [29].

  4. 4.

    Our notion of defeat differs from the way defeat is defined in various preference or value based enhancements of Dung’s abstract argumentation framework. Defeat is there usually defined as a binary relation between arguments which is a subset of the attack relation: \(a_1\) defeats \(a_2\) iff \(a_1\) attacks \(a_2\) and \(a_2\) is not ‘preferable’ to \(a_1\). The preferability of one argument over another is modeled in different ways: in terms of a preference relation between arguments in [30], by allowing for arguments to attack an attack in [31], or in terms of mapping arguments into partially ordered values in [19].

  5. 5.

    The following fact offers an alternative definition of semi-stable extensions in terms of admissible sets \(S\) for which \(S \cup S^+\) is maximal: Let \({\mathsf {A}} = \langle {\mathcal {A}}, \rightarrow \rangle \) be an AF and \(S \subseteq {\mathcal {A}}\). \(S\) is a semi-stable extension iff \(S\) is an admissible set of arguments for which there is no admissible set of arguments \(T \subseteq {\mathcal {A}}\) such that \(T \cup T^+ \supset S \cup S^+\). The statement is proven in Appendix F, Fact F.3.1.

  6. 6.

    We also allow for \(\bot \) on the left hand side of \(\twoheadrightarrow \). We will comment on this in a moment.

  7. 7.

    More precisely we would have to express this by “as many propositional letters as possible that represent arguments of the given AF”.

  8. 8.

    Actually, as we will see in Sect. 8.3.5, the \({\mathbf {L_A}}\)-models of \(\varGamma ^n_{\mathsf {A}}\) are a superset of the models corresponding to the admissible extensions of \({\mathsf {A}}\). Due to this we will perform a pre-selection on the \({\mathbf {L_A}}\)-models before selecting the \(\varOmega _P\)-minimally abnormal models (see Sect. 8.3.6).

  9. 9.

    See Chap. 3 for a detailed discussion of their meta-theory.

  10. 10.

    The reader should not be confused by the fact that for both strategies, simple strategy resp. minimal abnormality, we apply the same semantic selection, namely the selection of minimally abnormal \({\mathbf {L_A}}\)-models with respect to the abnormalities in \(\varOmega _\twoheadrightarrow \) resp. \(\varOmega _P\). The reason for this is that the simple strategy is equivalent to the minimal abnormality strategy for a lower limit logic \({\mathbf {LLL}}\), abnormalities \(\varOmega \) and a class of premise sets \(\varvec{\varGamma }\) if the following fact holds:

    (\(F\star \)): For all \(\varGamma \in \varvec{\varGamma }\) and all finite and non-empty \(\varDelta \subseteq \varOmega \), \(\varGamma \vdash _\mathbf{LLL}\;\mathsf{Dab}(\varDelta )\), then there is a \(\varphi \in \varDelta \) such that \(\varGamma \vdash _\mathbf{LLL} \varphi \).

    This is the case for our \({\mathbf {L_A}}\), \(\varOmega _\twoheadrightarrow \) and premise sets defined by \(\varGamma ^n_{\mathsf {A}}\) (as shown in Appendix F). Hence, in this case the simple strategy, as we will see, allows for a simplified marking strategy (see Definition 8.3.3) compared to the one for minimal abnormality (which is defined in Sect. 8.3.8, Definition 8.3.4). Of course, due to (\(F\star \)) the semantic selection for the simple strategy can also be characterized as follows: selected are all \({\mathbf {L_A}}\)-models of \(\varGamma ^n_{{\mathsf {A}}}\) that validate only those abnormalities in \(\varOmega _\twoheadrightarrow \) that are \({\mathbf {L_A}}\)-derivable from \(\varGamma ^n_{{\mathsf {A}}}\) (or equivalently, that are validated by all other \({\mathbf {L_A}}\)-models of \(\varGamma ^n_{{\mathsf {A}}}\)). Note, that in the case that fact (\(F\star \)) does not hold, such models are not guaranteed to exist. See also the discussion in Sect. 2.4.3.

  11. 11.

    The representational results are stated in Sect. 8.4 (see Theorem 8.4.1 and Corollary 8.4.1) and proven in Appendix F.

  12. 12.

    Let for instance \(\varSigma = \bigl \{\{1,2\},\{1,3\}\bigr \}\). Choice sets are \(\{1\}, \{1,2\}, \{1,3\}\), \(\{2,3\}\) and \(\{1,2,3\}\). Minimal are \(\{1\}\) and \(\{2,3\}\).

  13. 13.

    \({\mathsf {F}}\) is considered to be \({\mathbf {LLL}}\)-contingent, i.e., neither \(\vdash \) LLL F nor \(\vdash \) LLL \(\lnot \) F.

  14. 14.

    We have not characterized the marking conditions for minimal abnormality for logics that employ the minimal abnormality strategy for the flat case such as \({\mathbf {AL_2'}}\). They are a straightforward specification of our Definition 8.3.4. See also the characterization of sequential ALs in Chap. 3 where the proof theory is presented in generic terms.

  15. 15.

    As is well-known in the adaptive logic research, in case all minimally abnormal models validate the same set of abnormalities, the minimal abnormality strategy and the simple strategy are equivalent (cf. Footnote 10). See Sect. 2.4.3.

  16. 16.

    In view of our discussion it is straightforward to define the marking conditions for the simple strategy for \(\varOmega _G\) in \({\mathbf {AL_G}}\): A line with condition \(\varDelta \) is marked at stage s if a \(p_i \in \varDelta \cap \varOmega _G\) has been derived at an unmarked line on a condition \(\varDelta ' \subseteq \varOmega _\twoheadrightarrow \).

  17. 17.

    In accordance with Footnote 5, \({\mathbf {AL_S}}\) can easily be shown to be equivalent to \(\langle {\mathbf {L_A}}, [\varOmega _\twoheadrightarrow , \varOmega _S]\), \([\)simple strategy, minimal abnormality strategy\(]\rangle \).

  18. 18.

    The normal selections strategy was first introduced in [37]. See also [38] and Sect. 2.8 for a more elaborated representation. In Sect. 3.4 normal selections are introduced in sequential combinations of ALs and in Sect. 5.7 normal selections are presented in a more general way.

  19. 19.

    Usually the semantic consequence relation has to be defined in terms of equivalence classes of \(\varOmega \)-minimally abnormal \({\mathbf {AL_X}}\)-models. For two \({\mathbf {AL_X}}\)-models \(M \sim N\) iff \(\mathrm Ab ^{{\mathbf {L_X}}}_\varOmega (M) = \mathrm Ab ^{{\mathbf {L_X}}}_\varOmega (N)\). The semantic consequence relation is then defined by \(\varGamma \Vdash _{{\mathbf {AL^n}}} \varphi \) iff there is an \(\varOmega \)-minimally abnormal \({\mathbf {AL_X}}\)-model \(M\) of \(\varGamma \) such that for all \(\varOmega \)-minimally abnormal \({\mathbf {AL_X}}\)-models \(N\) of \(\varGamma \) for which \(N \sim M\), \(N \models _{{\mathbf {L_X}}} \varphi \) (see Definition 2.8.1). However, the nature of our abnormalities and of our premise sets allows for the simplification in Definition 8.5.1 since it can easily be shown that for all AFs \({\mathsf {A}}\) and for all \(\varOmega \)-minimally abnormal \({\mathbf {AL_X}}\)-models of \(\varGamma ^n_{\mathsf {A}}\), \(M\) and \(N\),

    $$\begin{aligned} (M \sim N)\,\text {iff}\,(\text {for all}\,\varphi \in {\mathcal {W}}_n, M\models _{{\mathbf {L_X}}}\varphi \,\text {iff}\,N\models _{{\mathbf {L_X}}}\varphi ) \end{aligned}$$

    The simplification is explicated in a more detailed way in Appendix F.4.

  20. 20.

    In particular they lack e.g. the following properties which hold for ALs in the standard format:

    1. (i)

      fixed-point property – \( Cn _{\mathbf {LLL}}( Cn _{\mathbf {AL}}(\varGamma )) = Cn _{\mathbf {AL}}(\varGamma )\),

    2. (ii)

      closure of the consequence set with respect to the \({\mathbf {LLL}}\)  – \( Cn _{\mathbf {LLL}}( Cn _{\mathbf {AL}}(\varGamma )) = Cn _{\mathbf {AL}}(\varGamma )\),

  21. 21.

    A different meta-level approach is [41]. Here, arguments are presented by propositions and extensions are presented by primitives. The authors in [39] offer, besides the modal systems, also a classical logic meta-level approach (using circumscription).

  22. 22.

    Note that neither of the following is derivable: \(p_1, p_2, \mathop {\mathsf {def}}p_1, \mathop {\mathsf {def}}p_2\).

  23. 23.

    See [42] for different variants of argumentation games proposed for ALs, and for instance [24] for the interpretation of abstract argumentation in form of games.

  24. 24.

    We will present the technical details in a future paper.

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Acknowledgments

I am thankful to Diderik Batens, Joke Meheus and the anonymous referees of the Journal of Logic and Computation for valuable comments which helped to improve the paper.

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    Christian Straßer

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Straßer, C. (2014). Towards the Proof-Theoretic Unification of Dung’s Argumentation Framework: An Adaptive Logic Approach. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_8

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