Representing voting rules in Łukasiewicz’s three-valued logic

We show how voting rules like the simple and the absolute majority rules, unanimity, consensus, etc. can be represented as logical operators in Łukasiewicz’s three-valued logic. First, we prove that the binary simple majority operator can be extended to express the majority choice of groups formed of n voters. Consequently, decisions in larger groups can be represented as more complex decisions in groups formed of pairs of voters. This property is not shared, however, by the consensus and the absolute majority operators. Secondly, we prove that the simple majority operator can be defined in terms of the consensus operator, but that the converse does not hold. Finally, we construct the classes of logical operators definable in terms of the simple majority and of the consensus and point to some more general implications of these results.