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

A general model of judgment aggregation is presented in which judgments on propositions are not binary but come in degrees. The primitives of the model 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. ∗We are extremely grateful to Franz Dietrich, Christian Hellwig, Maurice Salles and two referees for their helpful comments and suggestions. An earlier version of this paper was presented at the 10th International Meeting of the Society for Social Choice and Welfare, Moscow, July 2010, and in seminars at the Paris School of Economics and the Université de Caen. We are grateful to the participants for their comments. Financial support from the Spanish Ministry of Science and Innovation through MICINN/FEDER grant ECO2010-21624, the NUI Galway Millennium Fund and the Irish Research Council for the Humanities and Social Sciences co-funding from the European Commission is gratefully acknowledged. †Department of Philosophy, Logic and Scientific Method, London School of Economics, U.K. and J.E. Cairnes School of Business and Economics, National University of Ireland Galway, Ireland. Email: [email protected] ‡J.E. Cairnes School of Business and Economics, National University of Ireland Galway, Ireland. Email: [email protected]. Tel.: +353 91 492300; fax: +353 91 524130.