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I present an example in which the individuals' preferences are strictorderings, and under the majority rule, a transitive social ordering can beobtained and thus a non-empty choice set can also be obtained. However, theindividuals' preferences in that example do not satisfy any conditions(restrictions) of which at least one is required by Inada (1969) for socialpreference transitivity under the majority rule. Moreover, the consideredindividuals' preferences satisfy none of the conditions of value restriction(VR), extremal restriction (ER) or limited agreement (LA), some of which isrequired by Sen and Pattanaik (1969) for the existence of a non-empty socialchoice set. Therefore, the example is an exception to a number of theorems ofsocial preference transitivity and social choice set existence under themajority rule. This observation indicates that the collection of the conditionslisted by Inada (1969) is not as complete as might be supposed. This is alsothe case for the collection of conditions VR, ER and LA considered by Sen andPattanaik (1969). This observation is a challenge to some necessary conditionsin the current social choice theory. In addition to seeking new conditions, onepossible way to deal with this challenge may be, from a theoreticalprospective, to represent the identified conditions (such as the VR, ER and LA)in terms of a common mathematical tool, and then, people may find more.

A counter example to the theorems of social preference transitivity and social choice set existence under the majority rule