The doctrinal paradox, the discursive dilemma, and logical aggregation theory

Mongin, Philippe (2012): The doctrinal paradox, the discursive dilemma, and logical aggregation theory.

Abstract

Judgment aggregation theory, or rather, as we conceive of it here, logical aggregation theory generalizes social choice theory by having the aggregation rule bear on judgments of all kinds instead of merely preference judgments. It derives from Kornhauser and Sager’s doctrinal paradox and List and Pettit’s discursive dilemma, two problems that we distinguish emphatically here. The current theory has developed from the discursive dilemma, rather than the doctrinal paradox, and the final objective of the paper is to give the latter its own theoretical development along the line of recent work by Dietrich and Mongin. However, the paper also aims at reviewing logical aggregation theory as such, and it covers impossibility theorems by Dietrich, Dietrich and List, Dokow and Holzman, List and Pettit, Mongin, Nehring and Puppe, Pauly and van Hees, providing a uniform logical framework in which they can be compared with each other. The review goes through three historical stages: the initial paradox and dilemma, the scattered early results on the independence axiom, and the so-called canonical theorem, a collective achievement that provided the theory with its specific method of analysis. The paper goes some way towards philosophical logic, first by briefly connecting the modern philosophy of judgment, and second by thoroughly discussing and axiomatizing the "general logic" built in this framework.

• Item Type: MPRA Paper
• Language: English
• Keywords: Judgment Aggregation, Logical Aggregation, Doctrinal Paradox, Judgment Aggregation; Logical Aggregation; Doctrinal Paradox; Discursive Dilemma; General Logic; Premiss-Based vs Conclusion-Based Approach; Social Choice Theory; Impossibility Theorems
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D70 - General; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D79 - Other

References

Arrow, K.J. (1951), Social Choice and Individual Values, New York, Cowles Foundation and Wiley, 2nd ed., 1963.

d’Aspremont, C. (1985), Axioms for Social Welfare Orderings, in Social Goals and Social Organization, ed. by Hurwicz, L., Schmeidler, D., Sonnenschein, H., Cambridge, Cambridge University Press.

d’Aspremont, C., Gevers, L. (2002), Social Welfare Functionals and Interpersonal Comparability, in Handbook of Social Choice and Welfare, ed. by Arrow, K.J., Sen, A.K., Suzumura, K., Elsevier, I, 449-541.

Barwise, J. (1977), An Introduction to First-Order Logic, in Handbook of Mathematical Logic, ed. by Barwise, J., Amsterdam, North Holland, 5-46.

Bovens, L., Rabinowicz, W. (2006), Democratic Answers to Complex Questions. An Epistemic Perspective, Synthese, 150, 131-153.

Brennan, G. (2001), Collective Coherence?, International Review of Law and Economics, 21, 197-211.

Cariani, F., Pauly, M., Snyder, J. (2008), Decision Framing in Judgment Aggregation, Synthese, 163, 1-24.

Chang, C.C., Keisler, H.J. (1973), Model Theory, Amsterdam, Elsevier.

Chapman, B. (2002), Rational Aggregation, Politics, Philosophy and Economics, 1, 337-354.

Condorcet (Nicolas de Caritat, Marquis de) (1785), Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Paris, Imprimerie Royale. Also in Oeuvres, ed. by Condorcet-O’Connor, A., Arago, M.F., Paris, Firmin Didot, 1847-1849. Partial English transl. by McLean, I., Urkenas, A.B., as An Essay on the Application of Analysis to the Probability of Decisions Rendered by a Plurality of Votes, Classics of Social Choice, Ann Arbor, University of Michigan Press, 1995, 91-112.

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

Dietrich, F. (2007a), A Generalized Model of Judgment Aggregation, Social Choice and Welfare, 28, 529-565.

Dietrich, F. (2007b), Aggregation and the Relevance of Some Issues for Others, Research Memorandum 002, Maastricht Research School of Economics, Technology and Organization.

Dietrich, F (2010), The Possibility of Judgment Aggregation on Agendas with Subjunctive Implications, Journal of Economic Theory, 145, Symposium on Judgment Aggregation, 603-638.

Dietrich, F., List, C. (2007a), Judgment Aggregation by Quota Rules: Majority Voting Generalized, Journal of Theoretical Politics, 19, 391-424.

Dietrich, F., List, C. (2007b), Arrow’s Theorem in Judgment Aggregation, Social Choice and Welfare, 29, 19-33.

Dietrich, F., List, C. (2008a), Judgment Aggregation Without Full Rationality, Social Choice and Welfare, 31, 15-39.

Dietrich, F., List, C. (2008b), A Liberal Paradox for Judgment Aggregation, Social Choice and Welfare, 31, 59—78

Dietrich, F., List, C. (2010a), Majority Voting on Restricted Domains, Journal of Economic Theory, 145, Symposium on Judgment Aggregation, 512-543.

Dietrich, F. and List, C. (2010b), The Impossibility of Unbiased Judgment Aggregation, Theory and Decision, 68, 281-299.

Dietrich, F., Mongin, P. (2007), The Premiss-Based Approach to Logical Aggregation, Cahier de recherche HEC 886/2008, Paris.

Dietrich, F., Mongin, P. (2010), The Premiss-Based Approach to Judgment Aggregation, Journal of Economic Theory, 145, Symposium on Judgment Aggregation, 562-582.

Dokow, E., Holzman, R. (2009), Aggregation of Binary Evaluations for Truth-functional Agendas, Social Choice and Welfare, 32, 221—241.

Dokow, E., Holzman, R. (2010a), Aggregation of Binary Evaluations, Journal of Economic Theory, Symposium on Judgment Aggregation, 145, 495-511.

Dokow, E., Holzman, R. (2010b), Aggregation of Binary Evaluations with Abstentions, Journal of Economic Theory, Symposium on Judgment Aggregation, 544-561

Duddy, C., Piggins, A. (2009), Many-valued Judgment Aggregation: Characterizing the Possibility/impossibility Boundary for an Important Class of Agendas, Department of EconomicsWP 0154, National University of Ireland, Galway.

Eckert, D., Monjardet, B. (2009), Guilbaud’s Theorem: An Early Contribution to Judgment Aggregation, Document de travail 2009.47, Centre d’Economie de la Sorbonne, Paris.

Edgington, D., Conditionals, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), ed. by Zalta, E.N., online (http://plato.stanford.edu/archives/win2008/entries

Fishburn, P. C., Rubinstein, A. (1986), Algebraic Aggregation Theory, Journal of Economic Theory, 38, 63-77.

Fleurbaey, M., Mongin, P. (2005), The News of the Death of Welfare Economics is Greatly Exaggerated, Social Choice and Welfare, 25, 381-418.

Frege, G. (1918-1919-1923), Logische Untersuchungen, Beiträge zur Philosophie der deutschen Idealismus, Hefte I/2, II/3-4, III/1. English transl. in Gottlob Frege: Collected Papers on Mathematics, Logic, and Philosophy, ed. by McGuinness, B., Oxford, Blackwell, 1984.

Gaertner,W. (2006), Domain Conditions in Social Choice Theory, Cambridge, Cambridge University Press

Gärdenfors, P. (2006), An Arrow-like Theorem for Voting with Logical Consequences, Economics and Philosophy, 22, 181-190.

Genest, C., Zidek, J.V. (1986), Combining Probability Distributions: a Critique and an Annotated Bibliography, Statistical Science, 1, 114-148.

Gibbard, A. (1969), Social Choice and the Arrow Conditions, Unpublished, Department of Philosophy, University of Michigan (http://wwwpersonal.umich.edu/∼gibbard/).

Guilbaud, G.T. (1952), Les théories de l’intérêt général et le problème logique de l’agrégation, Economie appliquée, 5, 501-551. To be republished in Revue économique, 2012.

Hartmann, S., Pigozzi, G., Sprenger, J. (2010), Reliable Methods of Judgment Aggregation, Journal of Logic and Computation, 20, 603-617.

Hartmann, S., Sprenger, J. (forthcoming), Judgment Aggregation and the Problem of Tracking the Truth, Synthese.

Herzberg, F.S. (2010), Judgment Aggregators and Boolean Algebra Homomorphisms, Journal of Mathematical Economics, 46, 132-140.

Herzberg, F.S., Eckert, D. (2010), Impossibility Results for Infinite Electorate Abstract Aggregation Rules, IMW WP 427, Universität Bielefeld.

Heifetz, A., Mongin, P. (2001), Probability Logic for Type Spaces, Games and Economic Behavior, 35, 31-53.

Kant, E. (1800), Logik, ein Handbuch für Vorlesungen, herausgegegen by Jäsche, G.B., Königsberg. English transl. in Kant Lectures on Logic, The Cambridge Edition of the Works of Immanuel Kant, Cambridge, Cambridge University Press, 1992, Part IV, The Jäsche Logic, 521-642.

Kirman, A.P., Sondermann, D. (1972), Arrow’s Theorem, Many Agents, and Invisible Dictators, Journal of Economic Theory, 267—277.

Konieczny, S., Pino-Perez, R. (2002), Merging Information Under Constraints: a Logical Framework, Journal of Logic and Computation, 12, 773-808.

Kornhauser, L.A. (1992), Modelling Collegial Courts II: Legal Doctrine, Journal of Law, Economics and Organization, 8, 441-470.

Kornhauser, L.A., Sager L.G. (1986), Unpacking the Court, The Yale Law Journal, 96, 82-117.

Kornhauser, L.A., Sager L.G. (1993), The One and the Many: Adjudication in Collegial Courts, California Law Review, 81, 1-59.

Kornhauser, L.A, Sager, L.G. (2004), The Many as One: Integrity and Group Choice in Paradoxical Cases, Philosophy and Public Affairs, 32, 249-276.

Lang, J., Pigozzi, G., Slavkovik, M., van der Torre, L. (2011), Judgment Aggregation Rules Based on Minimization, in Proceedings of the 13th Conference on the Theoretical Aspects of Rationality and Knowledge (TARK XIII), 238-246. Extended version, LAMSADE WP, Université Paris Dauphine, 2011 (www.pigozzi.org/Publications.html).

Le Breton, M., Weymark, J. (2011), Arrovian Social Choice Theory on Economic Domains, in Handbook of Social Choice and Welfare, ed. by Arrow, K.J., Sen, A.K., Suzumura, K., Amsterdam, North Holland, II, 191-299.

Lehrer, K., Wagner, C. (1981), Rational Consensus in Science and Society, Dordrecht, Reidel.

Lehtinen, A. (2011), A Welfarist Critique of Social Choice Theory, Journal of Theoretical Politics, 23, 359-381.

Lewis, D. (1973), Counterfactuals, Oxford, Blackwell

Lismont, L., Mongin, P. (2003), Strong Completeness for Weak Logics of Common Belief, Journal of Philosophical Logic, 32, 115-137.

List, C. (2003), A Possibility Theorem on Aggregation over Multiple Interconnected Propositions, Mathematical Social Sciences, 45, 1-13. (Corrigendum, Mathematical Social Sciences 52, 109-110.)

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

List, C. (2005), The Probability of Inconsistencies in Complex Collective Decisions, Social Choice and Welfare, 24, 3-32.

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

List, C., Petit, P. (2005), On the Many as One: A Reply to Kornhauser and Sager, Philosophy and Public Affairs, 33, 377-390.

List, C., Polak, B. (2010), Introduction to Judgment Aggregation, Journal of Economic Theory, 145, Symposium on Judgment Aggregation, 441—466.

List, C., Puppe, C. (2009), Judgment Aggregation: A Survey, in Oxford Handbook of Rational and Social Choice, ed. by Anand, P., Puppe, C., Pattanaik, P., Oxford, Oxford University Press.

McConway, K.J. (1981), Marginalization and Linear Opinion Pools, Journal of the American Statistical Association, 76, 410-414.

Makinson, D. (2005), Bridges from Classical Logic to Nonmonotonic Logic, Texts in Computer Science Series, King’s College London.

Miller, M.K., Osherson, D. (2008), Methods for Distance-based Judgment Aggregation, Social Choice and Welfare, 32, 575-601.

Mongin, P. (1995), Consistent Bayesian aggregation, Journal of Economic Theory, 66, 313-351.

Mongin, P. (1997), Spurious Unanimity and the Pareto Principle, THEMA WP, Université de Cergy-Pontoise.

Mongin, P. (2003), L’axiomatisation et les théories économiques, Revue économique, 54, 99-138.

Mongin, P. (2008), Factoring Out the Impossibility of Logical Aggregation, Journal of Economic Theory, 141, 100-113. 48

Mongin, P. (2012), Un précurseur méconnu de la théorie de l’agrégation logique, Revue économique, forthcoming.

Monjardet, B. (2003), De Condorcet à Arrow via Guilbaud, Nakamura et les "jeux simples", Mathématiques et sciences humaines / Mathematics and Social Sciences, n◦ 163, 5-32.

Nash, J.R. (2003), A Context-Sensitive Voting Protocol Paradigm for Multimembers Courts, Stanford Law Review, 56, 75-159.

Nehring, K. (2003), Arrow’s Theorem as a Corollary, Economics Letters, 80, 379-382.

Nehring, K. (2005), The (Im)Possibility of a Paretian Rational, Economics WP 0068, Institute for Advanced Study, School of Social Science.

Nehring, K., Puppe, C. (2002), Strategy-Proof Social Choice on Single-Peaked Domains: Possibility, Impossibility and the Space Between, Department of Economics WP, University of California, Davies.

Nehring, K., Puppe, C. (2005), The Structure of Strategy-Proof Social Choice. Part II: Non-dictatorship, Anonymity, and Neutrality, Department of Economics WP, University of Karlsruhe.

Nehring, K., Puppe, C. (2008), Consistent Judgement Aggregation: The Truth-Functional Case, Social Choice and Welfare, 31, 41-57.

Nehring, K., Puppe, C. (2010a), Abstract Arrowian Aggregation, Journal of Economic Theory, 145, Symposium on Judgment Aggregation, 467-494.

Nehring, K., Puppe, C. (2010b), Justifiable Group Choice, Journal of Economic Theory, 145, Symposium on Judgment Aggregation, 583-602.

Nute, D., Cross, C.B. (2001), Conditional Logic, in Handbook of Philosophical Logic, ed. by Gabbay, D., Guenthner, F., Dordrecht, Kluwer, IV.

Pauly, M. (2007), Axiomatizing Collective Judgment Sets in a Minimal Logical Language, Synthese, 158, 233-250.

Pauly, M. (2008), On the Role of Language in Social Choice Theory, Synthese, 163, 227-243.

Pauly, M., van Hees, M. (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.

Pigozzi, G., Hartmann, S. (2007), Judgment Aggregation and the Problem of Truth-tracking, in Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK XI), 248-252.

Priest, G. (2002), Paraconsistent Logic, in Handbook of Philosophical Logic (2nd ed.) ed. by Gabbay, D., Guenthner, F., Dordrecht, Kluwer, VI, 287-393.

Priest, G., Routley, R., Norman, J. (ed. by) (1989), Paraconsistent Logic: Essays on the Inconsistent, München, Philosophia Verlag. 49

Sen, A. (1970), Collective Choice and Social Welfare, Amsterdam, North Holland.

Sen, A. (1986), Social Choice Theory, in Handbook of Mathematical Economics, ed. by Arrow, K.J., Intriligator, M.D., Amsterdam, North Holland, III, 63-77.

Stalnaker, R. (1968), A Theory of Conditionals, in Studies in Logical Theory, American Philosophical Quarterly Monograph Series, 2, 98-112.

van Hees, M. (2007), The Limits of Epistemic Democracy, Social Choice and Welfare, 28, 649-666.

Vickers, J. (1989), An Agenda for Subjectivism, Erkenntnis, 31, 397-416.

Wilson, R. (1972), Social Choice Theory without the Pareto Principle,Journal of Economic Theory, 5, 478-486.

Wilson, R. (1975), On the Theory of Aggregation, Journal of Economic Theory, 10, 89-99.