Deciding the premises or the conclusion

The previous rung establishes that no aggregation rule is both decisive and coherent, so a court that must reach a verdict has to give up coherence somewhere. Where it gives it up is a choice, and there are exactly two natural places.

Vote on the premises and derive the conclusion. Each judge states findings on the constituent questions; the majority settles each; the verdict follows by the applicable rule of law.

Or vote on the conclusion and let the premises follow. Each judge reaches a verdict; the majority settles it; the findings recorded are whatever supports the majority position.

The two procedures agree most of the time and disagree exactly where the previous rung’s search says they must.

What each preserves #

The two procedures are not symmetric, and the asymmetry is the whole of the practical argument.

The premise-based procedure preserves the reasoning. The findings recorded are what a majority of the body actually found, question by question, and the verdict is a logical consequence of them. Every step is defensible on its own and the record is coherent as a document.

What it gives up is that the verdict may be one no member holds. In the hero’s profile, the court finds for the plaintiff and a majority of judges each individually thinks the plaintiff should lose. That is uncomfortable, and it is not a mistake: each of the two findings has majority support, the law says the conclusion follows from them, and the court has applied the law to its own findings. A verdict nobody wanted is what a body that reasons collectively can produce, and calling it an error would require identifying which step was wrong.

The conclusion-based procedure preserves the verdict. The outcome is what a majority of the body wants, which is the ordinary meaning of a majority decision.

What it gives up is the reasoning. The findings on the premises are not the majority’s findings, so the recorded reasons do not support the recorded verdict, and the document is incoherent as a piece of law even though the outcome is uncontroversial. A later court citing it as authority for a proposition is citing something a majority of the deciding court rejected, which is a defect that compounds rather than staying put.

Which one a body should use #

The argument turns on what the body’s output is for, and that differs sharply between institutions.

A court’s judgment is a precedent. Later courts are bound by the findings, not merely by the outcome, so the reasoning is a substantive product and its coherence matters as much as the verdict’s. That is an argument for the premise-based procedure, and it is the argument Kornhauser and Sager make.

A committee’s recommendation is usually an outcome. A hiring panel, a grant board, a jury deciding guilt — the reasons are internal and the decision is the product. That is an argument for the conclusion-based procedure.

And a legislature is somewhere in between, which is why legislatures use elaborate sequential procedures that are neither, and which a later section is about. A statute’s text is binding and its stated purposes are not, so the output is the conclusion — but amendments are voted on separately, so the process is premise-based in its mechanics and conclusion-based in its product.

So the answer depends on the institution, and the mathematics contributes exactly this: the choice is forced, cannot be avoided by a cleverer rule, and has consequences that can be counted.

The sequential procedure, and its cost #

There is a third thing bodies actually do, and it deserves separating because it is not a rule in the previous rung’s sense.

Take the propositions in some order. Decide each by majority — unless the decision would contradict what has already been decided, in which case take the forced answer.

That always produces a consistent complete set of judgements, so it escapes the impossibility. It escapes it by failing systematicity: the rule applied to a proposition depends on where it comes in the order, so two propositions with identical vote patterns can be decided differently.

And the order decides the outcome. Put the premises first and the sequential procedure is premise-based; put the conclusion first and it is conclusion-based. So the escape is real and the choice reappears as a choice of agenda.

That is the honest reading of every escape from an impossibility theorem, and it is worth stating as a general point. Dropping a condition does not make the difficulty disappear; it moves the difficulty into whatever the dropped condition was preventing. Systematicity prevents the order from mattering; dropping it lets a consistent complete rule exist, and the order now matters. The total quantity of difficulty is conserved, and the choice is about where to keep it.

That is a familiar shape. Whoever sets the order of business controls the outcome, which is agenda manipulation in the voting setting and is the same phenomenon here — and it is the reason parliamentary procedure devotes so much attention to the order in which motions are put.

The frequency deserves a moment because it decides whether the choice is a formality or a live one.

If the two procedures agreed on ninety-nine profiles in a hundred, a body could adopt either and expect the difference to arise once a career. They do not. The share of profiles on which they part company is large enough that any body deciding a stream of cases will meet it, and meet it without noticing — since neither procedure announces that the other would have decided differently.

A body using one procedure never sees the disagreement, which is the practical reason the question was invisible until Kornhauser and Sager pointed at it. The verdict looks like a verdict either way.

What each procedure invites #

Both procedures can be manipulated and they are manipulated differently, which is a practical consideration that the impossibility theorems do not address.

The premise-based procedure invites misreporting the premises. A judge who wants the plaintiff to lose, and who sees that both premises will carry, can misstate a finding on one premise to block the conclusion. That is a strategic report about a question of fact, which is a serious thing for a judge to do and is also invisible.

The conclusion-based procedure invites misreporting the conclusion, which is less useful because the conclusion is what is being voted on directly, and there is nothing to gain by lying about one’s own vote.

There is a second asymmetry in the same direction. Under the premise-based procedure a judge who reports honestly can be outvoted into a conclusion they reject, which gives an honest judge a standing reason to consider misreporting — the incentive is not occasional but structural. Under the conclusion-based procedure an honest judge is only ever outvoted on the thing they were voting on, which is what being outvoted normally means.

So the premise-based procedure is more manipulable and produces better documents. That trade — a more defensible record obtained by a procedure with more room for strategy — is not resolvable and is worth naming, because the usual discussion treats the premise-based procedure as straightforwardly superior on the strength of the record alone.

Deliberation, which is the escape nobody models #

Real bodies talk before they vote, and talking changes what the impossibility is about.

The theorems take the profile of individual judgements as given and ask what a rule can do with it. Deliberation changes the profile: judges hear each other’s reasons, some change their findings, and the profile that reaches the vote is not the one that would have.

If deliberation makes the profile single-plateaued — if there is some proposition on which the body converges — the paradox disappears. That is a domain restriction rather than a rule change, and it is the standard escape from every impossibility of this family, including Arrow’s, where restricting to single-peaked preferences restores majority rule.

Whether deliberation reliably produces such a restriction is an empirical question and the evidence is mixed. What is clear is the shape of the claim: the impossibility is about the worst case over all profiles, and any process narrowing the set of profiles weakens it.

The claim also has a limit worth stating. Deliberation narrows the profiles by persuasion, and persuasion is not neutral: a body that converges because its members defer to one of them has restricted its domain in a way that reintroduces something very like a dictatorship. So “deliberate first” is not a free escape either, and the condition it weakens — universal domain — is weakened by a process with its own politics.

A third property nobody can have either #

There is a further condition worth naming because it sounds innocuous and is not available.

Independence of the agenda’s presentation. A body’s verdict should not depend on how the same question is carved into propositions. Asking “was the contract valid, was it breached” is logically the same as asking “was it a valid contract that was breached, and was it valid” — the same information, differently packaged.

The premise-based procedure fails this outright. Which propositions count as premises is a choice, and different carvings of the same case give different verdicts. So a court adopting it has to fix the carving, which is a substantive legal decision disguised as a procedural one.

The conclusion-based procedure passes it and pays elsewhere, since it consults only the conclusion and is therefore indifferent to how the reasons are described — at the cost of not recording them coherently.

That is a third instance of the same trade, and its shape is worth noting: every property one might want turns out to be purchasable, and to cost one of the others. The impossibility theorems are the systematic statement of that, and the practical literature is a catalogue of which purchases institutions have made.

What is actually done #

The practices are worth recording because they are not uniform and the variation is informative.

Appellate courts mostly vote on the conclusion. A judgment states the outcome and the reasoning of whoever writes the majority opinion, which may not be the majority’s reasoning on each point. Separate concurrences exist precisely to record the discrepancy — a judge who agrees with the outcome and not the reasoning writes one — and their existence is evidence that judges notice the problem and have a procedural device for it rather than a solution.

Some courts vote on issues explicitly, and a few jurisdictions require it. The debate about which is correct is live in legal scholarship and is conducted in the vocabulary of this rung.

And regulatory bodies often use the sequential procedure, deciding a sequence of questions in a fixed statutory order — which fixes the agenda in law and thereby removes the manipulation, at the cost of building the choice of order into the constitution.

The same choice outside courts #

The two procedures are not a legal curiosity, and recognising them elsewhere is most of what makes the rung worth having.

A medical panel deciding treatment. Does the patient have condition A, does the patient have condition B, and does the protocol therefore apply? Deciding the diagnoses and applying the protocol is premise-based; asking each member whether to treat is conclusion-based; they differ, and which is used is usually accidental.

A company’s board deciding on an acquisition. Is the target undervalued, can it be integrated, should it be bought? A board voting on the recommendation is conclusion-based, and a board working through a checklist is premise-based, and the same board can do both on successive items.

An examination board. Has the candidate met each criterion, and do they therefore pass? Criterion-by-criterion marking is premise-based and produces passes nobody voted for; a holistic vote is conclusion-based and produces a record that does not add up.

And a scientific consensus statement, where the propositions are findings and the conclusion is a recommendation. The IPCC’s procedures are explicitly premise-based, which is a deliberate choice with exactly the properties described above and is regularly criticised on exactly those grounds.

In every case the body is unaware there is a choice, and in every case the choice is being made by whoever wrote the procedure. The value of the theorem is not that it settles anything but that it makes the decision visible.

What the pictures cannot show #

A profile is drawn and the procedures are described. The hero shows one arrangement of votes and the two majorities that disagree; what a body would do with it, and what a different body would do, is prose.

The frequency depends on a model. The counts of disagreeing profiles assume every consistent profile is equally likely. Real deliberating bodies produce correlated positions, and the correlation can raise or lower the frequency depending on which axis the body splits along.

The escape figure is from a different anchor’s subject. It shows a domain restriction rescuing majority rule over preferences, which is the analogous escape and is not the same object. The judgement version would need a picture of a set of profiles narrowing, which is a set of tables and not a shape.

And deliberation is entirely outside every figure. The section above is about a process that changes the input, and every figure here takes an input as given. There is no picture of a profile becoming a different profile because people talked.

Where the ladder goes next #

The anchor’s three rungs now cover the paradox, the impossibility and the two procedures. What is left unwritten and named here as debts:

The escape routes in detail. Supermajority rules, which trade decisiveness for consistency in a tunable way; distance-based rules, which choose the nearest consistent judgement set to the majority’s; and quota rules generally, of which majority and unanimity are the extremes.

And the connection to Arrow, made precise. Preference aggregation is judgement aggregation on the agenda of pairwise comparisons, and deriving Arrow’s theorem from the judgement framework is a real if technical exercise.

Also unwritten: what a body should do when it discovers the two procedures disagree on the case in front of it. The theorems say the disagreement must sometimes happen and say nothing about the case where it has, which is the only situation in which anybody consults them.

Sideways, the agenda-setting power is the voting essays’ subject, the domain restriction is the same escape they use, and the impossibility this rung takes as given is the rung below.

Sideways, the same tension appears wherever a rule has to aggregate: four conditions that no voting rule satisfies together is the ancestor of every impossibility on this ladder, a rule that rewards a lie is the strategic cost of choosing one, and the value from both sides is what an aggregation looks like when a genuine agreement is available.

What is worth carrying away #

When a theorem says two desirable properties cannot both hold, the useful work begins afterwards, in deciding which to keep and understanding what the choice costs.

Here the choice is between a coherent record and a majority verdict. The premise-based procedure keeps the first and can reach a conclusion nobody holds; the conclusion-based keeps the second and produces reasons that do not support it. Neither is wrong, both are used, and which is right depends on whether the body’s product is its reasoning or its decision.

The habit worth taking is to ask what an institution’s output is. An impossibility theorem forces a choice; what the output is for decides it; and no amount of further mathematics substitutes for knowing that.

There is a second, quieter reading. Both procedures are defensible, so a body that has not chosen between them will produce whichever answer the order of voting happens to select — and that order is usually set by procedure rather than by principle. An impossibility theorem that forces a choice also forces somebody to make it explicitly, and the alternative is not neutrality but an unexamined default.