Majority rule in the absence of a majority

What is the meaning of "majoritarianism" as a principle of democratic group decision-making in a judgement aggregation problem, when the propositionwise majority view is logically inconsistent? We argue that the majoritarian ideal is best embodied by the principle of "supermajority efficiency" (SME). SME reflects the idea that smaller supermajorities must yield to larger supermajorities. We show that in a well-demarcated class of judgement spaces, the SME outcome is generically unique. But in most spaces, it is not unique; we must make trade-offs between the different supermajorities. We axiomatically characterize the class of "additive majority rules", which specify how such trade-offs are made. This requires, in general, a hyperreal-valued representation.