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Judgment aggregators and Boolean algebra homomorphisms

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  • Herzberg, Frederik

Abstract

The theory of Boolean algebras can be fruitfully applied to judgment aggregation: assuming universality, systematicity and a sufficiently rich agenda, there is a correspondence between (i) non-trivial deductively closed judgment aggregators and (ii) Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Furthermore, there is a correspondence between (i) consistent complete judgment aggregators and (ii) 2-valued Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Since the shell of such a homomorphism equals the set of winning coalitions and since (ultra)filters are shells of (2-valued) Boolean algebra homomorphisms, we suggest an explanation for the effectiveness of the (ultra)filter method in social choice theory. From the (ultra)filter property of the set of winning coalitions, one obtains two general impossibility theorems for judgment aggregation on finite electorates, even without assuming the Pareto principle.

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  • Herzberg, Frederik, 2010. "Judgment aggregators and Boolean algebra homomorphisms," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 132-140, January.
    • Handle: RePEc:eee:mateco:v:46:y:2010:i:1:p:132-140
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    References listed on IDEAS

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    Cited by:

    1. Mongin, Philippe & Dietrich, Franz, 2011. "An interpretive account of logical aggregation theory," HEC Research Papers Series 941, HEC Paris.
    2. Philippe Mongin, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," Theory and Decision, Springer, vol. 73(3), pages 315-355, September.

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