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Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary

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  • Duddy, Conal
  • Piggins, Ashley

Abstract

A model of judgment aggregation is presented in which judgments on propositions are not binary but come in degrees. The primitives are a set of propositions, an entailment relation, and a “triangular norm” which establishes a lower bound on the degree to which a proposition is true whenever it is entailed by a set of propositions. Under standard assumptions, we identify a necessary and sufficient condition for the collective judgments to be both deductively closed and free from veto power. This condition says that the triangular norm used to establish the lower bound must contain a zero divisor.

Suggested Citation

  • Duddy, Conal & Piggins, Ashley, 2013. "Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary," Journal of Economic Theory, Elsevier, vol. 148(2), pages 793-805.
    • Handle: RePEc:eee:jetheo:v:148:y:2013:i:2:p:793-805
    DOI: 10.1016/j.jet.2012.07.005
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    References listed on IDEAS

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    1. Franz Dietrich, 2007. "A generalised model of judgment aggregation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(4), pages 529-565, June.
    2. Mongin, Philippe, 2008. "Factoring out the impossibility of logical aggregation," Journal of Economic Theory, Elsevier, vol. 141(1), pages 100-113, July.
    3. Martin Hees, 2007. "The limits of epistemic democracy," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(4), pages 649-666, June.
    4. Richard Barrett & Maurice Salles, 2006. "Social Choice With Fuzzy Preferences," Economics Working Paper Archive (University of Rennes & University of Caen) 200615, Center for Research in Economics and Management (CREM), University of Rennes, University of Caen and CNRS.
    5. List, Christian & Pettit, Philip, 2002. "Aggregating Sets of Judgments: An Impossibility Result," Economics and Philosophy, Cambridge University Press, vol. 18(1), pages 89-110, April.
    6. Franz Dietrich & Christian List, 2007. "Arrow’s theorem in judgment aggregation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 29(1), pages 19-33, July.
    7. Conal Duddy & Juan Perote-Peña & Ashley Piggins, 2011. "Arrow’s theorem and max-star transitivity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(1), pages 25-34, January.
    8. List, Christian & Polak, Ben, 2010. "Introduction to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 441-466, March.
    9. K. J. Arrow & A. K. Sen & K. Suzumura (ed.), 2011. "Handbook of Social Choice and Welfare," Handbook of Social Choice and Welfare, Elsevier, edition 1, volume 2, number 2.
    10. Franz Dietrich & Christian List, 2008. "Judgment aggregation without full rationality," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 31(1), pages 15-39, June.
    11. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations with abstentions," Journal of Economic Theory, Elsevier, vol. 145(2), pages 544-561, March.
    12. Rubinstein, Ariel & Fishburn, Peter C., 1986. "Algebraic aggregation theory," Journal of Economic Theory, Elsevier, vol. 38(1), pages 63-77, February.
    13. Wilson, Robert, 1975. "On the theory of aggregation," Journal of Economic Theory, Elsevier, vol. 10(1), pages 89-99, February.
    14. Nehring, Klaus & Puppe, Clemens, 2010. "Abstract Arrowian aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 467-494, March.
    15. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations," Journal of Economic Theory, Elsevier, vol. 145(2), pages 495-511, March.
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    Citations

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    Cited by:

    1. Conal Duddy & Ashley Piggins, 2012. "A measure of distance between judgment sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 855-867, October.
    2. Dietrich, Franz, 2015. "Aggregation theory and the relevance of some issues to others," Journal of Economic Theory, Elsevier, vol. 160(C), pages 463-493.
    3. Dietrich, Franz & List, Christian, 2014. "Probabilistic Opinion Pooling," MPRA Paper 54806, University Library of Munich, Germany.
    4. List, Christian & Polak, Ben, 2010. "Introduction to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 441-466, March.
    5. Dietrich, Franz, 2016. "Judgment aggregation and agenda manipulation," Games and Economic Behavior, Elsevier, vol. 95(C), pages 113-136.
    6. repec:hal:pseose:halshs-01249513 is not listed on IDEAS
    7. Philippe Mongin, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," Theory and Decision, Springer, vol. 73(3), pages 315-355, September.
    8. Piggins, Ashley & Duddy, Conal, 2016. "Oligarchy and soft incompleteness," MPRA Paper 72392, University Library of Munich, Germany.
    9. Conal Duddy & Ashley Piggins, 2018. "On some oligarchy results when social preference is fuzzy," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(4), pages 717-735, December.
    10. Danilov, V., 2024. "Judgment aggregation: A critical view," Journal of the New Economic Association, New Economic Association, vol. 65(4), pages 30-53.
    11. Conal Duddy & Juan Perote-Peña & Ashley Piggins, 2011. "Arrow’s theorem and max-star transitivity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(1), pages 25-34, January.

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    More about this item

    Keywords

    Judgment aggregation; Deductive closure; Many-valued logic; Triangular norm; Zero divisor;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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