From degrees of belief to beliefs: Lessons from judgment-aggregation theory

Dietrich, Franz and List, Christian (2014): From degrees of belief to beliefs: Lessons from judgment-aggregation theory.

Abstract

What is the relationship between degrees of belief and (all-or-nothing) beliefs? Can the latter be expressed as a function of the former, without running into paradoxes? We reassess this “belief-binarization” problem from the perspective of judgmentaggregation theory. Although some similarities between belief binarization and judgment aggregation have been noted before, the literature contains no general study of the implications of aggregation-theoretic impossibility and possibility results for belief binarization. We seek to fill this gap. At the centre of this paper is an impossibility theorem showing that, except in simple cases, there exists no belief-binarization rule satisfying four baseline desiderata (“universal domain”, “belief consistency and completeness”, “propositionwise independence”, “certainty preservation”). We show that this result is a corollary of the judgment-aggregation variant of Arrow’s impossibility theorem and explore several escape routes from it.

• Item Type: MPRA Paper
• Language: English
• Keywords: subjective probability, yes/no belif, impossibility theorem on binarization, analytic philosophy, judgment aggregation
• Subjects: C - Mathematical and Quantitative Methods > C0 - General; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; D - Microeconomics > D0 - General; D - Microeconomics > D0 - General > D01 - Microeconomic Behavior: Underlying Principles; D - Microeconomics > D7 - Analysis of Collective Decision-Making; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D70 - General; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D80 - General; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D89 - Other

References

Arrow, K. (1951/1963) Social Choice and Individual Values. New York: Wiley.

Bovens, L., and W. Rabinowicz (2006) “Democratic answers to complex questions—an epistemic perspective.” Synthese 150(1): 131–153.

Chapman, B. ( 2002) “Rational Aggregation.” Politics, Philosophy and Economics 1: 337–354.

Dietrich, F. (2006) “Judgment Aggregation: (Im)Possibility Theorems.” Journal of Economic Theory 126: 286–298.

Dietrich, F. (2007) “A generalised model of judgment aggregation.” Social Choice and Welfare 28: 529–565.

Dietrich, F., and C. List (2007a) “Judgment aggregation by quota rules: majority voting generalized.” Journal of Theoretical Politics 19: 391–424.

Dietrich, F., and C. List (2007b) “Arrow’s theorem in judgment aggregation.” Social Choice and Welfare 29: 19–33.

Dietrich, F., and C. List (2013) “Propositionwise judgment aggregation: the general case.” Social Choice and Welfare 40(4): 1067–1095.

Dietrich, F. and P. Mongin (2010) “The premise-based approach to judgment aggregation.” Journal of Economic Theory 145: 562–582.

Dokow, E., and R. Holzman (2010a) “Aggregation of binary evaluations.” Journal of Economic Theory 145: 495–511.

Dokow, E., and R. Holzman (2010b) “Aggregation of binary evaluations with abstentions.” Journal of Economic Theory 145: 544–561.

Douven, I., and J.-W. Romeijn (2007) “The Discursive Dilemma as a Lottery Paradox.” Economics and Philosophy 23(3): 301–319.

Douven, I., and T. Williamson (2006) “Generalizing the Lottery Paradox.” British Journal for the Philosophy of Science 57(4): 755–779.

Easwaran, K., and B. Fitelson (forthcoming) “Accuracy, Coherence and Evidence.” In T. Szab´o Gendler and J. Hawthorne (eds.), Oxford Studies in Epistemology 5.

Hawthorne, J., and L. Bovens (1999) “The preface, the lottery, and the logic of belief.” Mind 108(430): 241–264.

Kelly, K. T., and H. Lin (2011) “Judgment aggregation: A geometrical impossibility proof.” Presentation at the 2011 Episteme Conference, Carnegie Mellon University, June 2011.

Konieczny, S., and R. Pino P´erez (2002) “Merging Information Under Constraints: A Logical Framework.” Journal of Logic and Computation 12: 773–808.

Kornhauser, L. A. (1992) “Modeling collegial courts. II. Legal doctrine.” Journal of Law, Economics and Organization 8: 441–470.

Kornhauser, L. A., and L. G. Sager (1986) “Unpacking the Court.” Yale Law Journal 96: 82–117.

Kyburg, H. E. (1961) Probability and the Logic of Rational Belief. Middletown: Wesleyan University Press (1961)

Leitgeb, H. (2014) “The Stability Theory of Belief.” Philosophical Review Volume 123(2): 131–171.

Levi, I. (2004) “List and Pettit.” Synthese 140(1/2): 237–242.

Lin, H., and K. T. Kelly (2012a) “Propositional Reasoning that Tracks Probabilistic Reasoning.” Journal of Philosophical Logic 41(6): 957–981.

Lin, H., and K. T. Kelly (2012b) “A Geo-logical Solution to the Lottery Paradox.” Synthese 186: 531–575.

List, C. (2004) “A Model of Path-Dependence in Decisions over Multiple Propositions.” American Political Science Review 98: 495–513.

List, C. (2006) “The Discursive Dilemma and Public Reason.” Ethics 116: 362–402.

List, C. (2012) “The Theory of Judgment Aggregation: An Introductory Review.” Synthese 187: 179–207.

List, C. (2014) “When to defer to supermajority testimony – and when not.” In J. Lackey (ed.), Essays in Collective Epistemology. Oxford: Oxford University Press (in press).

List, C., and P. Pettit (2002) “Aggregating Sets of Judgments: An Impossibility Result.” Economics and Philosophy 18(1): 89–110.

List, C., and P. Pettit (2004) “Aggregating Sets of Judgments: Two Impossibility Results Compared.” Synthese 140(1/2): 207–235.

Nehring, K., and C. Puppe (2010) “Abstract Arrovian Aggregation.” Journal of Economic Theory 145: 467–494.

Pauly, M., and M. van Hees (2006) “Logical Constraints on Judgment Aggregation.” Journal of Philosophical Logic 35: 569–585.

Pettit, P. (2001) “Deliberative Democracy and the Discursive Dilemma.” Philosophical Issues 11: 268–299.

Pigozzi, G. (2006) “Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation.” Synthese 152: 285–298.