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A formal framework for deliberated judgment

Abstract

While the philosophical literature has extensively studied how decisions relate to arguments, reasons and justifications, decision theory almost entirely ignores the latter notions. In this article, we elaborate a formal framework to introduce in decision theory the stance that decision-makers take towards arguments and counter-arguments. We start from a decision situation, where an individual requests decision support. We formally define, as a commendable basis for decision-aid, this individual’s deliberated judgment, a notion inspired by Rawls’ contributions to the philosophical literature, and embodying the requirement that the decision-maker should carefully examine arguments and counter-arguments. We explain how models of deliberated judgment can be validated empirically. We then identify conditions upon which the existence of a valid model can be taken for granted, and analyze how these conditions can be relaxed. We then explore the significance of our framework for the practice of decision analysis. Our framework opens avenues for future research involving both philosophy and decision theory, as well as empirical implementations.

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Notes

  1. 1.

    We remain at a fairly abstract level in our conceptualization of the topic. We accordingly set aside all the issues concerning the construction of problems and the evolution of their meaning as the decision process unfolds in concrete decision situations (Rosenhead and Mingers 2001).

  2. 2.

    Since no one has a concrete access to such an idealistic set of all the arguments, we expect that this concept will be mainly useful for philosophical explorations, and that the pragmatic interpretation will prevail in practical applications.

  3. 3.

    Even under the pragmatic interpretation, claiming that \(S= S^*\) would mean that there is no relevant knowledge beyond what the analyst can find by studying the literature and consulting experts and stakeholders, but also that the list of arguments she has found captures all the semantic and linguistic subtleties that could distinguish alternative formulations of arguments.

  4. 4.

    Our approach to formalize this concept is inspired by formal argumentation theory in artificial intelligence (Dung 1995; Rahwan and Simari 2009). However, the latter approach is not sufficient to empirically investigate i’s attitude towards arguments, because it neglects two crucial tasks. First, this literature does not investigate the role that the decision analyst plays when she interacts with a decision-maker: should she remain a neutral observer, or should she interact more tightly with the decision-maker by providing him with arguments and counter-arguments liable to lead him to change his mind? Second, this literature does not put emphasis on the specific challenges involved in interacting with a decision-maker to identify empirically the arguments he endorses. Most of the time, this literature considers situations where the relation between arguments can be computed from a given logical representation of the arguments (Besnard and Hunter 2008) or is given a priori (Baroni and Giacomin 2009), possibly integrating uncertainties (Hunter 2014) and dynamics (Rotstein et al. 2010; Marcos et al. 2011; Dimopoulos et al. 2018).

    Its most common use assumes that it is possible to establish the objective relations between arguments. In our example, \(s_3\) would be considered to objectively attack \(s_2\) and \(s_2\) to objectively attack \(s_1\). However, in some cases, it might be difficult, or perhaps even impossible, to determine such objective relations. In any case, this distinction is superfluous if the goal is to inquire about i’s opinion about these relations between arguments. Other proposals in formal argumentation theory (Amgoud and Cayrol 2002; Bench-Capon 2003; Amgoud et al. 2008; Amgoud and Prade 2009; Bench-Capon and Atkinson 2009; Ferretti et al. 2017) supplement an objective attack relation with information representing i’s subjectivity, such as his values or his preference over arguments. Such approaches seem closer to our aim, but they also use an objective attack relation, in addition to the subjective information. Furthermore, this approach assumes that it is possible to distinguish between, on the one hand, cases where \(s_3\) attacks \(s_2\) but i does not deem this attack important, and on the other hand situations where \(s_3\) does not attack \(s_2\). This assumption is also unnecessary for our purpose. Because our aim is mainly empirical, we propose to use another formalism, more adapted to our specific purpose, and leave aside here the task of more fully exploring the relations with proposals in formal argumentation theory such as dynamic argumentation.

  5. 5.

    Note that, contrary to the usual assumption in formal argumentation theory, we do not consider it possible that both \(s_2\) trumps \(s_1\) and \(s_1\) trumps \(s_2\)in a given perspective. This is a choice of modelization, and not an hypothesis about the way i thinks: for to hold, by definition of our “trump” relation, \(s_2\) must be a sufficiently strong argument to turn \(s_1\) into an ineffective argument. If, on the contrary, i considers that \(s_2\) is a plausible argument defending some claim incompatible with \(s_1\), but not sufficiently strong to defeat \(s_1\), then we model it by \(s_2 \mathbin {\ntriangleright _\exists }s_1\) and \(s_1 \mathbin {\ntriangleright _\exists }s_2\). Our choice permits to reduce our informational requirements, as there are fewer cases to be distinguished (our framework treats in the same way situations where two arguments trump each other and situations where none trumps the other). Note, however, that we do allow for the possibility that and : this can happen by i adopting each of those two attitudes in two different perspectives. Hence, our choice of modelization does not translate in any formal restriction. This note only serves to make the semantics of the notion encapsulated by our “trump” relation clear.

  6. 6.

    Another way of viewing the relations and \(\mathbin {\ntriangleright _\exists }\) goes as follows. Given a perspective p, define as a binary relation over \(S^*\): iff, when i is in the perspective p, \(s_2\) turns \(s_1\) into an invalid argument. Define P as the set of all possible perspectives. Then, define , and \(s_2 \mathbin {\ntriangleright _\exists }s_1\) iff . We favor another presentation because it emphasizes that we consider that we have direct access to and \(\mathbin {\ntriangleright _\exists }\), rather than to .

  7. 7.

    Relatedly, notice that there is an important asymmetry between the notions of justifiable and untenable. Since t and some incompatible \(t'\) can both be justifiable, the fact that t is justifiable does not necessarily imply that the fate of t in i’s view is entirely settled by its justifiability. By contrast, there is no way an untenable proposition could come back into the scene.

  8. 8.

    Somewhat similar distinctions are discussed in formal argumentation theory about skeptical versus credulous justification (Prakken 2006). Delving into the details of a comparative analysis falls beyond the scope of the present article.

  9. 9.

    That said, our notion of DJ does not claim to reflect faithfully all the aspects of the notion of “reflective equilibrium” as used by the authors mentioned above. A thorough exploration of the links between our formal framework and these philosophical theories falls beyond the scope of the present article.

  10. 10.

    This would amount to assume that i already knows all the arguments and can aggregate them successfully. If this were possible, i would probably not need help from an analyst.

  11. 11.

    Note that the replacer may be more powerful than the argument it replaces, in the sense that it may trump arguments or support propositions that the replaced argument did not trump or support.

  12. 12.

    Such a configuration of arguments, where \(s_3\) trumps \(s_2\) which in turns trumps \(s_1\), recalls the notion of “strong defense” in argumentation theory (Baroni and Giacomin 2007). A further discussion of this issue falls beyond the scope of this paper.

  13. 13.

    This procedure could be considered as a persuasion dialogue (Prakken 2009).

  14. 14.

    Cycles in our sense have to be distinguished from cycles involving an attack relation as defined in formal argumentation theory. We do not deny that cycles of attacks in the formal argumentation sense often happen, and Condition 9 does not exclude cycles understood in that sense: these cycles are generally not cycles in “trump” relations. We consider that an argument \(s_2\) trumps another one only when i considers that the first one is strong enough to render the second one ineffective. This definition relies on an asymmetry, \(s_2\) being, in a sense, “favored over” \(s_1\). Our trump relation is therefore somewhat analogical to a strict preference relation, for which an assumption of acyclicity is commonplace in the literature.

  15. 15.

    Readers used to decision theoretic axiomatizations might find this condition odd, since axioms usually mandate conditions considered more “basic”, such as transitivity and irreflexivity, and derive from them the conclusion that cycles are forbidden. This strategy does not work for our setting (or is not applicable in a simple way), because “basic” conditions such as transitivity would be unreasonable to impose here. For example, given and , it is easy to think about situations where i would consider that \(s_3 \mathbin {\ntriangleright _\forall }s_1\), and to think about situations where i would consider that . Neither anti-transitivity nor transitivity can thus be reasonably imposed (and our current condition avoids such requirements). Studying which conditions exactly are necessary to ban cycles (or make them innocuous) in our setting would be interesting, but it does not seem crucial at this stage. Indeed, in concrete settings we consider that cycles involving arguments from \(S_\gamma \) are unlikely to occur. (This claim should be backed up by empirical studies.)

  16. 16.

    Theorem 2 has an interesting corollary which permits to view our proposal as providing useful means to take account of the fact that knowledge evolves. In some cases it might be important, for example for efficiency reasons in contexts of limited resources, to investigate if a decision-aid provided before some discovery of new knowledge is still valid after the discovery. Take a decision-aid which has been provided using a set of argument \(S_\gamma \) which is CAC with respect to the set of known arguments before the discovery \(S^*_\text {before}\) and using a \(S_\gamma \)-operationally valid model \(\eta \). Theorem 2 shows that, if we can prove that \(S_\gamma \) is CAC with respect to the set of all the arguments \(S^*_\text {after}\) supplemented thanks to the new discovery, then there is no need to check the validity of \(\eta \) again. We thank an anonymous reviewer for this observation.

References

  1. Allais M. (1979) The so-called allais paradox and rational decisions under uncertainty. In M. Allais, O. Hagen (Eds.), Expected Utility Hypotheses and the Allais Paradox, number 21 in Theory and Decision Library (pp. 437–681). Springer. ISBN 978-94-015-7629-1. https://doi.org/10.1007/978-94-015-7629-1_17.

  2. Amgoud, L., & Cayrol, C. (2002). A reasoning model based on the production of acceptable arguments. Annals of Mathematics and Artificial Intelligence 34(1–3):197–215. ISSN 1012-2443. https://doi.org/10.1023/A:1014490210693.

  3. Amgoud, L., & Prade, H. (2009) Using arguments for making and explaining decisions. Artificial Intelligence 173(3–4):413–436. ISSN 0004-3702. https://doi.org/10.1016/j.artint.2008.11.006.

  4. Amgoud, L., Dimopoulos, Y., & Moraitis, P. (2008). Making decisions through preference-based argumentation. In Principles of Knowledge Representation and Reasoning: Proceedings of the Eleventh International Conference, KR 2008 (pp. 113–123), Sydney, Australia. http://www.aaai.org/Library/KR/2008/kr08-012.php.

  5. Baroni, P., & Giacomin, M. (2007). On principle-based evaluation of extension-based argumentation semantics. Artificial Intelligence 171(10–15):675–700. ISSN 0004-3702. https://doi.org/10.1016/j.artint.2007.04.004.

  6. Baroni, P., & Giacomin, M. (2009). Semantics of abstract argument systems. In G. Simari , & I. Rahwan (Eds.), Argumentation in Artificial Intelligence (pp. 25–44). Springer. ISBN 978-0-387-98197-0. https://doi.org/10.1007/978-0-387-98197-0_2.

  7. Bench-Capon, T., & Atkinson, K. (2009) Abstract argumentation and values. In G. Simari, & I. Rahwan (Eds), Argumentation in Artificial Intelligence (pp. 45–64). Springer. ISBN 978-0-387-98196-3 978-0-387-98197-0. https://doi.org/10.1007/978-0-387-98197-0_3.

  8. Bench-Capon, T. J. M. (2003). Persuasion in practical argument using value-based argumentation frameworks. Journal of Logic and Computation 13(3):429–448. ISSN 0955-792X. https://doi.org/10.1093/logcom/13.3.429.

  9. Besnard, P., & Hunter, A. (2008) Elements of argumentation. MIT Press . ISBN 978-0-262-02643-7. https://doi.org/10.7551/mitpress/9780262026437.001.0001.

  10. Carenini, G., & Moore, J. D., (2006). Generating and evaluating evaluative arguments. Artificial Intelligence 170(11):925–952. ISSN 0004-3702. https://doi.org/10.1016/j.artint.2006.05.003.

  11. Dietrich, F., & List, C. (2013). A reason-based theory of rational choice. Noûs 47(1):104–134. ISSN 1468-0068. https://doi.org/10.1111/j.1468-0068.2011.00840.x.

  12. Dietrich, F., & List, C. (2016). Reason-based choice and context-dependence: an explanatory framework. Economics & Philosophy 32(2):175–229. ISSN 1474-0028. https://doi.org/10.1017/S0266267115000474.

  13. Dimopoulos, Y., Mailly, J.-G., & Moraitis, P. (2018). Control argumentation frameworks. In S. A. McIlraith, K. Q. Weinberger (Eds.), Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, Louisiana, USA, February 2–7, 2018. AAAI Press. https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16639.

  14. Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77(2):321 – 357. ISSN 0004-3702. https://doi.org/10.1016/0004-3702(94)00041-X.

  15. Ellsberg, D. (1961). Risk, ambiguity, and the savage axiomes. The Quarterly Journal of Economics, 75, 643–669. https://doi.org/10.2307/1884324.

  16. Estlund, D. (2009). Democratic Authority: A Philosophical Framework. Princeton: Princeton University Press. http://press.princeton.edu/titles/8571.html.

  17. Ferretti, E., Tamargo, L. H., García, A. J., Errecalde, M. L., & Simari, G. R. (2017). An approach to decision making based on dynamic argumentation systems. Artificial Intelligence 242:107–131. ISSN 0004-3702. https://doi.org/10.1016/j.artint.2016.10.004.

  18. Goodman, N. (1983). Fact, Fiction, and Forecast. Cambridge: Harvard University Press, 4th Edn. http://www.hup.harvard.edu/catalog.php?isbn=9780674290716.

  19. Habermas, J. (1981). Theorie des kommunikativen Handelns. Suhrkamp. http://www.suhrkamp.de/buecher/theorie_des_kommunikativen_handelns-juergen_habermas_28775.html.

  20. Hausman, D. M. (2011) Preference, Value, Choice, and Welfare. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9781139058537.

  21. Hunter, A. (2014). Probabilistic qualification of attack in abstract argumentation. International Journal of Approximate Reasoning 55(2):607–638. ISSN 0888-613X. https://doi.org/10.1016/j.ijar.2013.09.002.

  22. Kontoleon, A., Pascual, U., & Swanson, T. M. (Eds.). (2007). Biodiversity Economics. Cambridge: Cambridge University Press. ISBN 978-0-521-86683-5. https://doi.org/10.1017/CBO9780511551079.

  23. Labreuche, C. (2011). A general framework for explaining the results of a multi-attribute preference model. Artificial Intelligence, 175(7–8):1410–1448. ISSN 0004-3702. https://doi.org/10.1016/j.artint.2010.11.008.

  24. Landry, M., Malouin, J., & Oral, M. (1983). Model validation in operations research. European Journal of Operational Research, 14, 207–220. https://doi.org/10.1016/0377-2217(83)90257-6.

  25. Landry, M., Banville, C., & Oral, M. (1996). Model legitimisation in operational research. European Journal of Operational Research, 92(3), 443–457. https://doi.org/10.1016/0377-2217(96)00003-3.

  26. MacCrimmon, K. R., & Larsson, S. (1979). Utility theory: Axioms versus ‘paradoxes’. In M. Allais, O. Hagen (Eds.), Expected Utility Hypotheses and the Allais Paradox, number 21 in Theory and Decision Library (pp. 333–409). Springer. ISBN 978-94-015-7629-1. https://doi.org/10.1007/978-94-015-7629-1_15.

  27. Marcos, M. J., Falappa, M. A., & Simari, G. R. (2011). Dynamic argumentation in abstract dialogue frameworks. In P. McBurney, I. Rahwan, & S. Parsons (Eds.) Argumentation in Multi-Agent Systems, Lecture Notes in Computer Science (pp. 228–247). Berlin: Springer. ISBN 978-3-642-21940-5. https://doi.org/10.1007/978-3-642-21940-5_14.

  28. Meinard, Y., Remy, A., & Schmid, B. (2017). Measuring impartial preference for biodiversity. Ecological Economics 132:45–54 ISSN 0921-8009. https://doi.org/10.1016/j.ecolecon.2016.10.007.

  29. Morgenstern, O. (1979). Some reflections on utility. In M. Allais & O. Hagen (Eds.) Expected Utility Hypotheses and the Allais Paradox, number 21 in Theory and Decision Library (pp. 175–183). Springer. ISBN 978-94-015-7629-1. https://doi.org/10.1007/978-94-015-7629-1_6.

  30. Prakken, H. (2006). Combining sceptical epistemic reasoning with credulous practical reasoning. In Proceedings of the 2006 Conference on Computational Models of Argument: COMMA 2006 (pp. 311–322), Amsterdam: IOS Press. ISBN 978-1-58603-652-2. http://www.cs.uu.nl/groups/IS/archive/henry/b+a.pdf. Corrected version, April 2008.

  31. Prakken, H. (2009). Models of persuasion dialogue. In G. Simari & I. Rahwan (Eds.) Argumentation in Artificial Intelligence (pp. 281–300). Springer. ISBN 978-0-387-98197-0. https://doi.org/10.1007/978-0-387-98197-0_14.

  32. Rahwan, I., & Simari, G. R. (eds.) (2009). Argumentation in Artificial Intelligence. Berlin: Springer. https://doi.org/10.1007/978-0-387-98197-0.

  33. Raiffa, H. (1985). Back from prospect theory to utility theory. In M. Grauer, M. Thompson, A. P. Wierzbicki (Eds.) Plural Rationality and Interactive Decision Processes, number 248 in Lecture Notes in Economics and Mathematical Systems (pp. 100–113). Springer. ISBN 978-3-540-15675-8. https://doi.org/10.1007/978-3-662-02432-4_8.

  34. Rawls, J. (1999). A Theory of Justice. Cambridge: Harvard University Press. http://www.hup.harvard.edu/catalog.php?isbn=9780674000780.

  35. Rawls, J. (2005). Political Liberalism (Expanded ed.). New York: Columbia University Press. ISBN 978-0-231-52753-8. https://cup.columbia.edu/book/political-liberalism/9780231527538.

  36. Rosenhead, J., & Mingers, J. (eds.) (2001). Rational Analysis for a Problematic World Revisited: Problem Structuring Methods for Complexity, Uncertainty and Conflict, 2nd Edn. Hoboken: Wiley. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471495239.html.

  37. Rotstein, N. D., Moguillansky, M. O., García, A. J., & Simari, G. R. (2010). A dynamic argumentation framework. In P. Baroni, F. Cerutti, M. Giacomin, & G. R. Simari (Eds.) Computational Models of Argument: Proceedings of COMMA 2010, Desenzano del Garda, Italy, September 8–10, 2010, volume 216 of Frontiers in Artificial Intelligence and Applications, (pp. 427–438). IOS Press. ISBN 978-1-60750-618-8. https://doi.org/10.3233/978-1-60750-619-5-427.

  38. Roy, B. (1996). Methodology, multicriteria, & for Decision Aiding. Dordrecht: Kluwer Academic. ISBN 978-1-4757-2500-1. https://doi.org/10.1007/978-1-4757-2500-1.

  39. Savage, L. J. (1972). The Foundations of Statistics. Courier Corporation, 2nd Edn. ISBN 978-0-486-62349-8. http://store.doverpublications.com/0486623491.html.

  40. Scanlon, T. M. (2000). What We Owe to Each Other. Cambridge: Harvard University Press. http://www.hup.harvard.edu/catalog.php?isbn=9780674004238.

  41. Sen, A. (2009). The Idea of Justice. Cambridge: Harvard University Press.

  42. Slovic, P., & Tversky, A. (1974). Who accepts Savage’s axiom? Behavioral Science, 19, 368–373. https://doi.org/10.1002/bs.3830190603.

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Acknowledgements

We thank Denis Bouyssou, Cyril Hédoin, Jean-Sébastien Gharbi, André Lapied, Bernard Roy, Stéphane Deparis and two anonymous reviewers for very helpful comments.

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Correspondence to Olivier Cailloux.

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A Proofs, and additional explanatory results

A Proofs, and additional explanatory results

Our main goal in this section is to prove Theorem 2. We do this by first proving that if a set \(S_\gamma \) is CAC, then it includes enough decisive arguments to settle the issue (we will call such a set \(S\subseteq S^*\)efficient). This requires a few intermediate lemmas. Efficiency will bring a number of consequences of interest to us, among which Theorem 2. As a second goal, we want to give some further results that help understand the relationship between the notions of clear-cut, validity and operational validity, existence of a CAC set of arguments, and efficiency.

Let us start with the formal definition of efficiency.

Definition 13

(Efficiency) Given a decision situation , given \(S\subseteq S^*\), \(S\) is efficient iff , and .

Recall that designates the arguments not trumped by any argument, thus, the decisive arguments, and hence, designates the arguments always trumped by some decisive argument in \(S\).

In all this section, we assume we are given a decision situation and a subset of arguments \(S_\gamma \subseteq S^*\) (except in Theorem 5).

Our strategy for proving that CAC implies efficiency, roughly speaking, involves excluding “undecided” situations from \(S_\gamma \). For example, we want to show that it is impossible that an argument has no decisive argument trumping it in \(S_\gamma \), but also fails to be defended in \(S_\gamma \). We will do this by progressively promoting or degrading arguments, e.g., show that, in \(S_\gamma \), if an argument is resistant (has no argument that decisively trumps it), then it must also be defended, and if it is defended, it must be replaceable by decisive arguments.

Define as the decisive arguments from \(S_\gamma \).

Define an argument \(s\) as finitely defended iff some finite set of arguments from \(S_{\gamma \text {dec}}\) defends it, thus, iff \(\exists S\subseteq S_{\gamma \text {dec}}\) such that , \(S\) finite. Define \(S_{\gamma \text {def}}\) as the arguments from \(S_\gamma \) that are finitely defended.

Define \(R_{\gamma \text {dec}}\subseteq S^*\) as the arguments that are replaceable by \(S_{\gamma \text {dec}}\). Recall that \(S\) replaces \(s\) iff and .

Define as the resistant arguments from \(S_\gamma \), namely, those not trumped by any argument from \(S_{\gamma \text {dec}}\).

Define \(E_{\gamma \text {res}}\subseteq S^*\) as the arguments that are essentially replaceable by \(S_{\gamma \text {res}}\). Recall that \(S\) essentially replaces \(s\) iff and .

Similarly, \(E_{\gamma \text {dec}}\) are the arguments essentially replaceable by \(S_{\gamma \text {dec}}\).

Lemma 1

(\(S_{\gamma \text {def}}\subseteq R_{\gamma \text {dec}}\)) If \(S_\gamma \) is Closed under reinstatement and Answerable, then the arguments from \(S_\gamma \) that are finitely defended are replaceable by decisive arguments from \(S_\gamma \); formally: \(S_{\gamma \text {def}}\subseteq R_{\gamma \text {dec}}\).

Proof

The strategy for this proof is the following. If \(s\in S_{\gamma \text {def}}\), some finite set of arguments defends \(s\). We wish to pick defenders one by one, replacing \(s\) by applying Closed under reinstatement to \(s\) and the chosen defender, obtaining an argument that fewer arguments trump, and then show that iterating the process yields a decisive argument replacing \(s\).

We need the following intermediate result. Assume a set of arguments \(S\subseteq S_{\gamma \text {dec}}\) is given, together with an argument \(s_1 \in S\) and an argument \(s^\text {r}_1 \in S_\gamma \) defended by \(S\). Then, there exists an argument \(s^\text {r}_2 \in S_\gamma \) replacing \(s^\text {r}_1\) and defended by \(S{\setminus } \{s_1\}\).

Indeed, from Answerability, because \(s_1 \in S_{\gamma \text {dec}}\), . Also, as \(s_1 \in S_{\gamma \text {dec}}\), we can assume that \(s_1 \ne s^\text {r}_1\), otherwise \(s^\text {r}_1 \in S_{\gamma \text {dec}}\) and the result is obtained by taking \(s^\text {r}_2 = s^\text {r}_1\). And \(s_1\) does not trump \(s^\text {r}_1\), otherwise \(s^\text {r}_1\) is trumped by a decisive argument and thus not defended. We can thus apply Closed under reinstatement to \((s_1, s^\text {r}_1)\). We obtain that for some \(s^\text {r}_2 \in S_\gamma \), and \(s^\text {r}_2\) replaces \(s^\text {r}_1\). Thus, \(S{\setminus } \{s_1\}\) defends \(s^\text {r}_2\): any argument trumping \(s^\text {r}_2\) already trumped \(s^\text {r}_1\), hence, is trumped by \(S\) (because that set defends \(s^\text {r}_1\)), and is not trumped by \(s_1\). This proves our intermediate result.

Coming back to the main point, we know that a finite coalition \(S\subseteq S_{\gamma \text {dec}}\) defends \(s\in S_\gamma \). Define \(s^\text {r}_1 = s\) and apply the intermediate result repetitively to obtain an argument \(s^\text {r}_2 \in S_\gamma \) replacing \(s\) and defended by \(S\) minus one element, then \(s^\text {r}_3 \in S_\gamma \) replacing \(s^\text {r}_2\), thus, replacing \(s\) (because replacement is transitive) and defended by \(S\) minus two elements, and so on, until obtaining a replacer defended by \(\emptyset \), thus, decisive. \(\square \)

Lemma 2

() If \(S_\gamma \) is covering, then any argument is either essentially replaceable by \(S_{\gamma \text {res}}\), or attacked by an argument from \(S_{\gamma \text {res}}\); formally: .

Proof

We consider in turn three sets whose union yields \(S^*\): \({\overline{S_\gamma }}\), and .

First, : from covering, if \(s\notin S_\gamma \), \(s\) is unnecessary, and by definition, \(s\) is unnecessary iff \(s\in E_{\gamma \text {res}}\) or .

Second, , because \(S_{\gamma \text {dec}}\subseteq S_{\gamma \text {res}}\).

Third, , because \(S_\gamma \cap \overline{\mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})} = S_{\gamma \text {res}}\) by definition.

We have considered all three possible cases, and the conclusion obtains in all cases. \(\square \)

Lemma 3

(\(S_{\gamma \text {res}}\subseteq S_{\gamma \text {def}}\)) If \(S_\gamma \) is CAC, any argument in \(S_\gamma \) that has no argument that decisively trumps it is finitely defended; formally: \(S_{\gamma \text {res}}\subseteq S_{\gamma \text {def}}\).

Proof

Recall that the relation Q is defined in Bounded length (Condition 9) as \(Q = (\mathbin {\vartriangleright _\exists }\cup (\mathbin {\vartriangleright _\exists }\circ \mathbin {\vartriangleright _\exists })) \cap (S_\gamma \times S_\gamma )\). Observe that, given any set \(S\ne \emptyset \), Bounded Length forbids that \(\forall s\in S: S\cap Q^{-1}(s) \ne \emptyset \). Otherwise, applying \(Q^{-1}\) to an element of \(S\) would always yield some element in \(S\), and \(Q^{-1}\) could then be applied any desired number of times starting from any \(s\in S\), thereby building a chain as long as desired. Accordingly, for any set S, Bounded Length imposes that if \(\forall s\in S: S\cap Q^{-1}(s) \ne \emptyset \), then \(S= \emptyset \).

Define \(S= S_{\gamma \text {res}}\cap {\overline{S_{\gamma \text {def}}}}\). We show that, given any \(s\in S\), \(S\cap Q^{-1}(s) \ne \emptyset \). This suffices to obtain \(S= \emptyset \) and, therefore, our desired conclusion.

Pick any \(s\in S\). Towards exhibiting an argument in \(S\cap Q^{-1}(s)\), we want first to exhibit some argument \(s'\) that is a) trumped by some argument \(s^* \in S_{\gamma \text {res}}\), thus \(s' \in \mathbin {\vartriangleright _\exists }(S_{\gamma \text {res}})\); b) not trumped by any argument in \(S_{\gamma \text {dec}}\), thus \(s' \notin \mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})\); c) equal to \(s\) or trumping \(s\). As a second step, from the existence of such an \(s'\) we will then prove that \(s^*\), the particular trumping argument in part a), belongs to \(S\) (thanks to parts a) and b)), and belongs to \(Q^{-1}(s)\) (thanks to part c)).

Our first step thus amounts to show that some \(s'\) satisfies our three conditions above.

From \(s\notin S_{\gamma \text {def}}\) and \(s\in S_\gamma \), we know that \(s\) is not finitely defended, and using the contrapositive of Bounded width, we obtain that \(s\) is not infinitely defended either. Hence, by definition of defense, there exists some \(s_1 \in \overline{\mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})} \cap \mathbin {\vartriangleright _\exists ^{-1}}(s)\). And, applying [\(S^*= E_{\gamma \text {res}}\cup \mathbin {\vartriangleright _\exists }(S_{\gamma \text {res}})\)], either \(s_1 \in E_{\gamma \text {res}}\), or \(s_1 \in \mathbin {\vartriangleright _\exists }(S_{\gamma \text {res}})\).

If \(s_1 \in E_{\gamma \text {res}}\), \(s\in \mathbin {\vartriangleright _\exists }(S_{\gamma \text {res}})\). Besides, because \(s\in S\), \(s\in S_{\gamma \text {res}}\). Thus taking \(s' = s\) satisfies our three conditions.

And if \(s_1 \in \mathbin {\vartriangleright _\exists }(S_{\gamma \text {res}})\), because \(s_1 \in \overline{\mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})}\)), taking \(s' = s_1\) satisfies our three conditions.

For our second step, consider an argument \(s^* \in S_{\gamma \text {res}}\) that trumps \(s'\) (we know this is possible thanks to part a)). Thanks to part b), we know that \(s'\) is not trumped by any argument in \(S_{\gamma \text {dec}}\), and from [\(S_{\gamma \text {def}}\subseteq R_{\gamma \text {dec}}\)], we know that if \(s'\) was trumped by an argument in \(S_{\gamma \text {def}}\), it would be trumped by an argument in \(S_{\gamma \text {dec}}\), thus, \(s'\) is not trumped by any argument in \(S_{\gamma \text {def}}\). Because \(s^* \mathbin {\vartriangleright _\exists }s'\), we know that \(s^* \notin S_{\gamma \text {def}}\). Thus, \(s^* \in S\). Finally, \(s^* \mathbin {\vartriangleright _\exists }s\) or \(s^* \mathbin {\vartriangleright _\exists }s' \mathbin {\vartriangleright _\exists }s\) (thanks to part c)), thus, \(s^* \in Q^{-1}(s)\). \(\square \)

Lemma 4

(\(S^*= E_{\gamma \text {dec}}\cup \mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})\)) If \(S_\gamma \) is CAC, any argument is either essentially replaceable by decisive arguments from \(S_\gamma \), or attacked by a decisive argument from \(S_\gamma \); formally: \(S^*= E_{\gamma \text {dec}}\cup \mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})\).

Proof

This follows from [\(S^*= E_{\gamma \text {res}}\cup \mathbin {\vartriangleright _\exists }(S_{\gamma \text {res}})\)], [\(S_{\gamma \text {res}}\subseteq S_{\gamma \text {def}}\)] and [\(S_{\gamma \text {def}}\subseteq R_{\gamma \text {dec}}\)]. \(\square \)

Theorem 3

(CAC implies efficiency) If \(S_\gamma \) is CAC, \(S_\gamma \) is efficient.

Proof

We prove that .

This proves the point, as it shows that

  1. (i)

    , and

  2. (ii)

    , because .

That follows from the definitions of \(E_{\gamma \text {dec}}\) and \(T_i\).

The next subset relation holds because if some decisive argument supports t, that argument is not in \(\mathbin {\vartriangleright _\forall }(S_{\gamma \text {dec}})\).

Finally, Answerability mandates that \(\mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}}) \subseteq \mathbin {\vartriangleright _\forall }(S_{\gamma \text {dec}})\), hence, \(\overline{\mathbin {\vartriangleright _\forall }(S_{\gamma \text {dec}})} \subseteq \overline{\mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})}\), and using [\(S^*= E_{\gamma \text {dec}}\cup \mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})\)], \(\overline{\mathbin {\vartriangleright _\exists }(S_{\gamma \text {dec}})} \subseteq E_{\gamma \text {dec}}\). \(\square \)

Theorem 4

(Validity of \(\eta )\) Assume \(S_\gamma \) is efficient and \(\eta \), a model of the decision situation, is \(S_\gamma \)-operationally valid. Then \(T_i = T_\eta \).

Proof

Recall that a model is \(S_\gamma \)-operationally valid iff for all , \(s\in S^*\), we have and \([\forall s_c \in S_\gamma : (s_c \mathbin {\ntriangleright _\exists }s) \vee (\exists s_{cc} \in S^*\;|\;s_{cc} \mathbin {\vartriangleright _{\eta }}s_c \wedge s_{cc} \mathbin {\vartriangleright _\exists }s_c)]\), and when t is not supported by \(\eta \), .

Consider \(t \in T_\eta \). By definition, some . From operational validity of \(\eta \), we obtain that and \(\forall s_c \mathbin {\vartriangleright _\forall }s: s_c \notin S_\gamma \cap \overline{\mathbin {\vartriangleright _\exists ^{-1}}(S^*)}\) (because \([s_c \mathbin {\vartriangleright _\forall }s\wedge s_c \in S_\gamma ] \Rightarrow s_c \in \mathbin {\vartriangleright _\exists ^{-1}}(S^*)\)). Hence, \(s\notin \mathbin {\vartriangleright _\forall }(S_\gamma \cap \overline{\mathbin {\vartriangleright _\exists ^{-1}}(S^*)})\), thus . Efficiency of \(S_\gamma \) brings \(t \in T_i\).

If \(t \notin T_\eta \), from operational validity of \(\eta \), no decisive argument in \(S_\gamma \) may support t, equivalently, , and from efficiency, \(t \notin T_i\). \(\square \)

We can now prove Theorem 2.

Proof of theorem 2

From [CAC implies efficiency], we obtain that \(S_\gamma \) is efficient. It then follows from the efficiency of \(S_\gamma \) that the decision situation is clear-cut and that a \(S_\gamma \)-operationally valid model exists. The last consequence is given by Theorem 4. \(\square \)

The following theorem may help clarify the relationship between efficiency, existence of CAC arguments, and the situation admitting a model as we conceive it.

Theorem 5

(CAC subset equivalent to efficiency) Given a decision situation and a subset of arguments \(S\subseteq S^*\), there exists a set \(S_\gamma \subseteq S\) that is CAC iff \(S\) is efficient.

Proof

From [CAC implies efficiency], if some set \(S_\gamma \subseteq S\) is CAC, then \(S_\gamma \) is efficient, and because efficiency propagates to supersets, \(S\) is efficient.

If \(S\) is efficient (thus, the decision situation is clear-cut), then a CAC subset \(S_\gamma \) exists: suffices to choose as members of \(S_\gamma \) only the decisive arguments required to support the justifiable propositions and trump the supporters of untenable propositions. Observing that no arguments trump any argument in the resulting set (thus \(s\mathbin {\vartriangleright _\exists }s_\gamma \) for no \(s\in S^*, s_\gamma \in S_\gamma \)), most of the conditions for \(S_\gamma \) to be CAC are immediately seen to be satisfied. About arguments \(s\in S^*{\setminus } S_\gamma \) being unnecessary, we only have to show that when , either \(s\) is trumped by an argument from \(S_\gamma \) that is not decisively trumped, or \(s\) is essentially replaceable by arguments from \(S_\gamma \). Indeed, by our construction of \(S_\gamma \), if \(s\) supports an accepted t, it is essentially replaceable, and otherwise, it is trumped by a decisive argument. \(\square \)

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Cailloux, O., Meinard, Y. A formal framework for deliberated judgment. Theory Decis 88, 269–295 (2020). https://doi.org/10.1007/s11238-019-09722-7

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Keywords

  • Decision analysis
  • Justification
  • Empirical validation
  • Methodology