Rational Aggregation

Politics, Philosophy & Economics

Volumn 1, Issue 3, 2002, Pages 337-354

  Chapman, Bruce ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

collective rationality; Condorcet jury theorem; discursive dilemma; doctrinal paradox; judgement; public reason; social choice

Indexed keywords

EID: 85004281871     PISSN: 1470594X     EISSN: None     Source Type: Journal    
DOI: 10.1177/1470594X02001003004     Document Type: Article

References (13)

1. C. List and P. Pettit, ‘Aggregating Sets of Judgements: An Impossibility Result’, Economics and Philosophy 18 (2002): p. 89, and ‘Aggregating Sets of Judgements: Two Impossibility Results Compared’, Synthese (forthcoming).
2. K.J. Arrow, Social Choice and Individual Values, second edition (New Haven, CT: Yale University Press, 1963).
3. Brennan, ‘Collective Coherence?’, International Review of Law and Economics 21 (2001): p. 197 offers an argument for being sceptical that deductive closure should necessarily be a property of even individual rationality. For suppose that belief in, or assent to, a proposition is represented by a degree of confidence in the truth of which is greater than some threshold P, where P is thought of as the probability that is true. Then it might be perfectly rational to believe and to believe t (each in isolation) on such a basis, but not to believe in their conjunction (s&t) even though the conjunction represents the deductive closure on what the individual otherwise believes. Such would be the case if the compound probability of the conjunction fell below P. Brennan thinks that this argument supports our giving more credence to our judgements over outcomes than our judgements over the underlying reasons supporting those outcomes. However, I consider a similar sort of situation below, at n. 20, and there my argument is that we should insist that the individual organize her probability assessments under the relevant reasons, and not allow the individual to make a credible claim simply because she can point in some unreasoned way (even probabilistically) to an outcome supported by the possibility of some reason. The difference in our views goes to the sort of claim that is being made, and the precise nature of the burden of rationality that it must carry. Some claims, even if they are right, or right more probably than not, may not be right for the right reasons, and it is an open question, at least, whether they are then both right and rational.
4. The first part of this requirement closely resembles Arrow's condition of ‘the independence of irrelevant alternatives’, and the second part adds something very close to Kenneth May's condition of ‘neutrality’. See Arrow, Social Choice and Individual Values and Kenneth May, ‘A Set of Independent, Necessary and Sufficient Conditions for Simple Majority Decision’, Econometrica 20 (1952): p. 680. May also refers to an anonymity condition that, in the social choice context, is fully analogous to the one required by List and Pettit for judgement aggregation functions.
5. Kornhauser, ‘Modelling Collegial Courts II: Legal Doctrine’, Journal of Law, Economics and Organization 8 (1992): p. 441 seems to have been the first to analyse fully this sort of problem in a legal context and coined the term ‘doctrinal paradox’ for it. However, the phenomenon, without a name and for only a very brief discussion, also appeared in Kornhauser and Sager, ‘Unpacking the Court’, Yale Law Journal 96 (1986): p. 82. The doctrinal paradox was first introduced into a social choice framework in Bruce Chapman, ‘More Easily Done Than Said: Rules, Reasons, and Rational Social Choice’, Oxford Journal of Legal Studies 18 (1998): p. 293.
6. List and Pettit, ‘Aggregating Sets of Judgements: An Impossibility Result,’ p. 95.
7. Marquis de Condorcet, Condorcet: Selected Writings (Indianapolis, IN: Bobbs- Merrill, 1976). I borrow the illustration that follows this footnote in the text from P. Pettit, ‘Deliberative Democracy and the Discursive Dilemma’, Nous 11, Philosophical Issues supplement (2001).
8. If two members of a majority are judging, or voting on, two inconsistent propositions x and not-x, then, contrary to what the jury theorem requires, it cannot be that each member of the majority is more likely to be right than wrong in the judgements he or she makes, since that would imply that the sum of the probabilities of x and not-x being true is greater than one. (Note, however, that two individuals can judge the same proposition differently, or inconsistently, even if each is more likely to be right than wrong on that proposition; that possibility is allowed because each is only probably, and not certainly, right.) The same difficulty would plague those versions of the jury theorem that require only that the average of the probabilities that each member of the group of voters is right be greater than. 5. See Theorem 5 in Grofman, Owen and Feld, ‘Thirteen Theorems in Search of the Truth’, Theory and Decision 15 (1983): p. 261. The average for the group cannot be greater than. 5 for both the x and not-x propositions. At most, the average would have to be. 5, in which case majority voting adds nothing to the credibility of individual voting.
9. Of course, it is possible that these two ‘atomic’ propositions might themselves be compounds of further ‘sub-atomic’ propositions, thus generating another iteration of the judgement aggregation paradox at this lower level. Some might argue that this attests to a general indeterminacy in the structure that makes the paradox possible, something that renders it less important. Others might argue that this only goes to show that the paradox occurs with greater frequency than one first realizes, something that might magnify its significance. For debate around this question, see J. Rogers, ‘Issue Voting by Multimember Appellate Courts: A Response to Some Radical Proposals’, Vanderbilt Law Review 49 (1996): p. 993 and D. Post and S. Salop, ‘Issues and Outcomes, Guidance, and Indeterminacy: A Reply to Professor John Rogers and Others’, Vanderbilt Law Review 49 (1996): p. 1069.
10. R. Nozick, The Nature of Rationality (Princeton: Princeton University Press, 1993), p. 159 makes this same point.
11. List and Pettit, ‘Aggregating Sets of Judgements: An Impossibility Result’, p. 105.
12. Of course, the distinction is not really between reasoning based on modus ponens and reasoning based on modus tollens. These are logically equivalent. Rather, the distinction is between beginning with the atomic propositions and beginning with the compound proposition.
13. Here is another example of the disjunctive version of the paradox that should really set a lawyer's teeth on edge. (The example is adapted from one used by Levmore, ‘Conjunction and Aggregation’, Michigan Law Review 99 (2001): p. 723 to make a different point about the conjunction of independent probabilities.) Suppose that a plaintiff was injured while using some product and that she advanced two separate and independent claims for the recovery of damages from the defendant manufacturer. The plaintiff might argue that the product was either defectively manufactured or sold with an inadequate warning, where either one of these two arguments, call them J1 and J2 to match the columns in Table 2, would be sufficient, if successful, to win a judgement J for damages. Now suppose that a majority of the court rejects each of the plaintiff's arguments J1 and J2 in the same way that Table 2 suggests. Would it, nevertheless, be possible for the plaintiff to argue that she should win the judgement J in column (3)? After all, a majority of the judges, A and B, do think that the plaintiff should win J, albeit for different reasons. A lawyer is tempted to reply that the plaintiff's position is inadequate because in law she has an obligation to frame her claim against the defendant as an argument, that is, under some sort of conceptual structure. It will not do for the plaintiff to show only that the defendant owes (or, even, probably owes) her damages for some reason. Instead, she must show (more probably than not) that there is a reason (that is, at least one particular reason, woven out of some particular account of transactional wrong) for the claim. It is in this respect that some claims, even if they are right (or right more probably than not), may not be right for the right reasons. We might reasonably wonder, therefore, whether they are both right and rational. On this, compare the point made above, at n. 4.