Abstract
This paper provides a “liberal paradox” that applies to the framework of abstract argumentation and complements the liberal paradox in preference aggregation. In abstract argumentation, arguments are viewed as abstract entities whose validities are determined according to a binary attack relation. When forming a collective attack relation, parts of it may be reserved to members of the society who hold expert knowledge. I prove that when only a binary evaluation of each argument is permitted, even under a minimal condition of rationality, the assignment of expert rights to two or more agents may be inconsistent with the condition of strong unanimity. Since argumentation aggregation is a particular case of judgement aggregation, this result might be a corollary of Dietrich and List (Soc Choice Welf 31(1):59–78, 2008), if the agenda I consider turns out to be connected in their sense, an issue that this paper has not been able to settle.
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Notes
For an overall summary of the state-of-the-art achievements in this field, see Rahwan and Simari (2009).
Sen’s paper, especially his formulation of the notion of rights, has faced various challenges since its publication. For some representative work, see Nozick (1974), Bernholz (1974), Gärdenfors (1981), Sugden (1981) and Gaertner et al. (1992), among others. For recent developments, see Deb et al. (1997), van Hees (1999, 2004), and Dowding and van Hees (2003). It is not my intent to clarify the notion of rights in the current paper. Even so, as we will see, because the right involved is that of determining the entirety of the social point of view regarding the attack relation instead of only part of it, Sen’s approach may be considered to be more appropriate in this field than it is in preference aggregation.
That is, the conflict between individual rights and Pareto optimality, a concept that is arguably highly relevant to the property “strong unanimity” defined in Sect. 4 (emphasis and footnote added).
The Liar Paradox is a famous example that concerns the problem of self-attack that derives from the classical statement “This sentence is false.” For convenience, we call this argument \(\kappa \). If \(\kappa \) is true, then by that argument, \(\kappa \) is false. If \(\kappa \) is false, however, because the argument states precisely that (namely, that it is false), then \(\kappa \) is true. We have thus shown that \(\kappa \) is true if and only if it is false. Although the Liar Paradox invokes the most fundamental challenge of self-reference in language and logic, it is irrelevant to my concerns here. Thus, the constraint that there may be no self-attacking arguments is desirable in my context. However, this postulate is not the general case in abstract argumentation theory, Dung (1995) included.
Thus, not being symmetric means that \(\alpha \) attacks \(\beta \) while \(\beta \) does not attack \(\alpha \), which may be the case that \(\alpha \) is much powerful than \(\beta \) in argumentation. The two arguments \(\gamma \) and \(\alpha \) in Example 1 below vividly illustrate this point.
Thus, not being antisymmetric means that \(\exists \,\alpha , \beta :(\alpha \rightharpoonup \beta )\wedge (\beta \rightharpoonup \alpha )\wedge (\alpha \ne \beta )\). This requirement is reasonable because it is common for two arguments to attack each other in real argumentations, which is especially true for debates that concern moral values. For example, let \(\alpha =\) “Newspapers should not publish the presidential candidates’ health information if they do not accede to the request because it concerns their private lives.” and \(\beta =\) “Newspapers should publish the presidential candidates’ health information even if they do not accede to the request because it has public significance.” Thus, we can see that \(\alpha \rightharpoonup \beta \) and \(\beta \rightharpoonup \alpha \) simultaneously. In fact, the foundation of this argumentation is the conflict between private life and public interest.
For a comprehensive treatise on digraph theory, see Bang-Jensen and Gutin (2010).
In the artificial intelligence literature, in addition to the notions of in and out, like Caminada, many authors also adopt the label undec to denote the labelling of an argument whose status, i.e., justified or unjustified, cannot be decided. In some scenarios of the real life, e.g., judicial practices, however, such an undecided argument is not acceptable. In the present work, just as we classify arguments only as justified or unjustified, we do not adopt the label undec. In Sect. 4, it will be clear that this constraint is crucial to my paradox. For further discussion of argument labelling, see also Caminada and Pigozzi (2011).
Distinct from my focus in the present work, Bodanza and Auday (2009) analyse the problem of aggregating individual attack relations over a common set of arguments to “obtain a unique socially justified set of arguments.” They articulate the difference between the aggregation methods involved. That is, their work “can be done in two different ways: a social attack relation is built up from the individual ones, and then is used to produce a set of justified arguments, or this set is directly obtained from the sets of individually justified arguments.” Their primary concern is “whether these two procedures can coincide or under what conditions this could happen.” My task here starts from the first step of the first approach, although with a completely different destination.
This formulation was suggested by the Editor.
This was reminded by Juan Carlos García-Bermejo Ochoa.
Bodanza and Auday (2009) define this concept in a similar manner. The difference are as follows: (1) they do not define the social argumentation function explicitly. Instead, they call it an aggregation of individual argumentation frameworks “according to some specified mechanism M”; (2) they impose no requirement of irreflexivity on either individual or social level such as that implied by the concept of attack relation here.
On the other hand, Dunne et al. (2012) call this procedure “argument aggregation” and this mechanism “argument aggregation function”. In our daily usage, argument means both a reason given in proof or rebuttal, and the act or process of arguing. Since we use “argument” in the first sense from the outset of the current work, we adopt “argumentation” to avoid ambiguity. In fact, just as footnote 10 above shows, there are two ways to obtain socially justified set of arguments. We feel that “argument aggregation” risks the implication of the second way of Bodanza and Auday (2009). However, that is not what we want to explore.
We indicate the force that determines the social attack relation between two arguments by a label next to the corresponding harpoon, where DU denotes Condition DU and R with a subscript the expert right of the corresponding agent.
To be specific, they define these three properties as follows: Universal Domain: the domain of the aggregation function F is the set of all possible profiles of consistent and complete individual judgement sets; Minimal Rights: there exist (at least) two agents who are each decisive regarding (at least) one proposition-negation pair \(\{p,\lnot p\}\subseteq X\); Unanimity Principle: for any profile \((A_1,\ldots ,A_n)\) in the domain of F and any proposition \(p\in X\), if \(p\in A_i\) for all agents i, then \(p\in F(A_1,\ldots ,A_n)\), where \(A_i\) is the judgement set of agent i.
This perspective and the related literature were suggested by a referee.
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Acknowledgements
My greatest debt is to Salvador Barberà. Although in a form different from that presented in this paper, it was he who introduced me to the problem of argumentation. I have benefited immensely from the many helpful discussions I have had with him. I am also grateful for discussions with Juan Carlos García-Bermejo Ochoa, Franz Dietrich and Xavier Vilà; comments from Carmen Beviá, Richard Booth, Martin Caminada, Gregory Gutin, Paola Manzini, Marco Mariotti, Jordi Massó, Prasanta Pattanaik, Ariel Rubinstein, Leon van der Torre, Jörgen Weibull, Yongsheng Xu, and Jesús Zamora Bonilla; and advice concerning references from Paul E. Dunne and Iyad Rahwan. I would like to thank the three anonymous referees of the IJCAI 2011 Workshop on Social Choice and Artificial Intelligence for their insightful comments on a previous version of this paper with the title “A Liberal Impossibility of Abstract Argumentation” as well as the participants of presentations given in Madrid, London, Stockholm, Luxembourg, Barcelona and New Delhi for their helpful suggestions. In the final stage of preparation, I am very grateful to two anonymous referees for their thoughtful suggestions. Especially I owe a great deal to an editor for his extraordinarily detailed critique and unusual professional dedication. The study is based on one part of my dissertation. I gratefully acknowledge support from the Ministerio de Ciencia e Innovación de España and FEDER through the project ECO2008-04756. All errors are my own.
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Li, N. A paradox of expert rights in abstract argumentation. Soc Choice Welf 51, 737–752 (2018). https://doi.org/10.1007/s00355-018-1136-2
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DOI: https://doi.org/10.1007/s00355-018-1136-2