Judgement aggregation — the series
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The court that contradicts itself
Three judges each answer three questions, and each answers them consistently. Take the majority on each question separately and the answers no longer hang together — the body as a whole endorses a combination no member of it holds, and no rearrangement of the procedure removes the problem.
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No rule escapes the doctrinal paradox
A court whose members each hold a consistent position can reach an inconsistent verdict by majority. One such case is easy to build, which invites the hope that a better rule would avoid it — and every rule that responds to the votes at all fails somewhere.
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Deciding the premises or the conclusion
A body that cannot be both decisive and coherent has to choose which. The two live options are to vote on the reasons and let the verdict follow, or to vote on the verdict and let the reasons look after themselves — and they reach opposite answers on exactly the profiles the impossibility identifies.
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Four ways out, and what each costs
An impossibility theorem lists conditions and says no rule has them all. That leaves exactly as many escapes as there are conditions, each of them a real institution — a dictator, a two-stage procedure, a supermajority, a restricted agenda — and each escape's price can be counted rather than argued about.
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The nearest consistent verdict
When a court's majorities contradict each other, one repair is to announce the consistent verdict that disagrees with the judges least. It treats the premises and the conclusion alike, which neither of the two standard procedures does. On the classic case it returns a three-way tie; on five judges, with every question weighted equally, it never returns a single answer on a troubled profile at all — and what breaks the tie is a decision about which question matters more.
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Agendas that cannot contradict themselves
A court voting on two unconnected questions never contradicts itself, and neither does one voting on a chain of thresholds. A court voting on two premises and their conjunction sometimes does. What separates them is the size of the smallest sets of judgements that cannot all be true: pairs are harmless, because two majorities always share a judge, and triples are not. The same count says exactly how large a supermajority has to be to stay consistent on any agenda.
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What the agenda leaves standing
Ask for a rule that settles each question from the votes on that question, follows a unanimous court and never contradicts itself, and search every such rule for three judges. On a ranking of three options, three survive: one dictator per judge. On two premises and their conjunction, seven survive: every rule in which a fixed set of judges must all agree. On a chain of thresholds, a hundred and twenty-nine, majority among them. The difference is not in the rules. It is in which answers force which, and whether that forcing ever runs back.
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Which vote finds the truth more often
A court that must decide two premises and their conjunction can vote on the premises or on the conclusion, and the two votes sometimes disagree. Suppose there is a right answer and each judge is more often right than wrong. Then the question has an exact answer: for three judges the conclusion vote is more often right when competence is below √½ and the premise vote above it, and in a large court the premise vote wins at any competence — while the conclusion vote turns out to be a built-in standard of proof.
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Every premise makes the verdict vote stricter
A court that must find several things true before it finds liability can vote on each of them or on the verdict. With two elements, judges right seven times in ten are enough for either. With five, the verdict vote needs judges right eighty-seven times in a hundred, or a larger court makes it worse — while the premise vote still needs only better than even. And when any one ground is enough instead of all, the two procedures swap their failures.