Agendas that cannot contradict themselves

The doctrinal paradox needs a particular kind of agenda. A court asked whether a contract was valid, whether it was breached, and whether the defendant is liable — where liability means both — can have majorities saying yes, yes and no. A court asked only whether it will rain tomorrow and whether the bus will be late, two questions with no logical link, can take the majority on each and never say anything impossible, because nothing it says is impossible.

Between those extremes is every agenda a body might face, and the question is which ones are safe: on which agendas does the majority on each question, taken separately, always produce a consistent set of answers, whatever the judges think?

The table was made by brute force. For each agenda it lists every consistent judgement set, gives every judge every one of them in every combination, takes the majority on each question, and checks whether the result is one of the consistent sets. Two agendas never fail: the unconnected pair and the chain of thresholds. Five fail on some profiles — from 6 of 64 profiles of three judges for the conjunction agenda, to 42 of 512 for a conjunction of three premises. And one column predicts the split exactly.

The smallest things that cannot all be true #

The column is about minimally inconsistent sets: collections of judgements that cannot all be true together, but any one of which could be dropped to leave something that can.

Every agenda has some minimally inconsistent pairs: a judgement and its own denial, at the very least. What distinguishes agendas is whether they have larger ones. The conjunction agenda has exactly one set of three: accepting both premises and denying the conclusion. Each two of those three are compatible — a judge can accept both premises and the conclusion, or accept one premise and deny the conclusion — but all three together are not. The chain of thresholds has none larger than two. Every contradiction on it is a pair: accepting that a sentence should exceed two years while denying it should exceed one.

The sizes grow with the agenda’s logic. A conjunction of three premises has a minimally inconsistent set of four — accept each of the three premises and deny that all three hold — and no smaller set captures that contradiction, since any three of the four can be held together. In general a conjunction of $m$ premises has one of size $m + 1$, and the larger that set, the more majorities it takes to assemble it and the more room there is for them to miss one another.

Finding these sets is a search, not an inspection. The figure tries every collection of judgements of up to four members, one judgement per question, checks whether any consistent set satisfies all of them, and keeps the collections that fail while every smaller part succeeds. On a three-question agenda that is a few dozen checks; on an agenda of thirty questions it would be millions, which is one reason real bodies rarely know which kind of agenda they are voting on.

The size of the largest minimally inconsistent set is the number in the second column of the first table, and majority is safe exactly on the agendas where that number is two. This is the median property, identified in its general form by Klaus Nehring and Clemens Puppe in 2007, and the table checks it on seven agendas and every profile of up to five judges.

Why two is safe and three is not #

The reason is a fact about majorities that has nothing to do with logic. Any two majorities of the same body share a member. If more than half the judges accept one judgement, and more than half accept another, then together they account for more than all the judges, so at least one judge is counted twice.

Now suppose the majority accepted both members of a minimally inconsistent pair. Some judge is in both majorities and so accepts both judgements. But the pair cannot both be true, and every judge is consistent. So this cannot happen, and since every contradiction on the agenda contains a minimally inconsistent set, a majority verdict containing no bad pair contains no contradiction at all.

Three majorities need not share a member, and the doctrinal paradox is three majorities that do not.

Each of the three judgements is held by two judges of three. The pairs of majorities overlap, as they must, but the three together have no judge in common: judge 1 accepts the conclusion, judge 2 rejects the second premise, judge 3 rejects the first. A minimally inconsistent set of three can be accepted by three majorities of two because no single judge is asked to hold all three.

So the argument is complete in both directions. With only pairs, every accepted contradiction would need a judge holding a contradiction; with a triple, a profile like this one lets three majorities assemble the triple from pieces no judge holds together. The converse needs a little more care in general — a triple has to be realisable by some profile — but on every agenda in the table it is, and the search finds the profile.

The chain, and the judge in the middle #

The chain of thresholds is safe for a reason that is older than the theory of judgement aggregation, and more visible.

On a chain, a judge’s position is a single level: accept every threshold up to some point and none beyond it. Line the judges up by level. A threshold is accepted by the judges above some point in the line, so it is accepted by a majority exactly when the middle judge accepts it. The majority’s verdicts are the middle judge’s verdicts, and the middle judge is consistent.

This is Duncan Black’s median voter theorem of 1948 in judgement form. Black showed that when voters’ preferences are single-peaked along one dimension, pairwise majority always has a winner, the median voter’s favourite. The chain is one dimension, and its consistent positions are the points along it. The general median property is what remains of Black’s theorem when there is no line to arrange the judges on: every contradiction is a pair, so every set of majority verdicts is held, piece by piece, by judges who overlap enough to force consistency.

The chain’s safety has a second consequence, about incentives. On a chain, a judge who wants the verdict closer to their own position cannot gain by misreporting: moving one’s reported level only matters if it moves the middle, and it can only move the middle away from where the judge wanted it. Nehring and Puppe showed that this is general. The agendas on which some reasonable rule resists the lie that pays are, once again, the ones with the median property, so a single structural fact about an agenda decides both whether majority is consistent and whether it can be gamed.

The same structure explains the fourth escape of the essay on ways out of the impossibility, where majority was safe on profiles whose judges could be lined up. There the restriction was on the judges; here it is on the questions. Both work because the majority collapses onto one member.

A conditional is as dangerous as a conjunction #

The third row of the first table is an agenda that looks harmless: is $p$ true, is it true that $p$ implies $q$, and is $q$ true? It is the shape of most legal reasoning — the statute applies if the facts are established, the facts are established, so the statute applies — and it fails on exactly as many profiles as the conjunction agenda, 6 of 64 for three judges and 150 of 1,024 for five.

It has one set of three, $\{p,\ p \to q,\ \text{not } q\}$: accept the fact, accept the rule, and deny the consequence. That is modus ponens refused, and three majorities can refuse it with no judge refusing it. One judge accepts the fact and the rule and so the consequence; one accepts the fact, denies the rule and denies the consequence; one denies the fact, accepts the rule and denies the consequence. Fact and rule are each carried two to one, and so is the denial of the consequence.

The count of failures is the same as for the conjunction because the two agendas are the same agenda in disguise. Each consistent judgement set is a corner of a cube — four corners of eight — and a relabelling carries one set of four corners onto the other. Replace the second premise of the conjunction agenda by its denial, so that the conjunction reads “$p$ and not $q$”, and then replace the conjunction by its own denial, which is “$p$ implies $q$”. Each judgement set on one agenda becomes a judgement set on the other, consistent sets go to consistent sets, and the minimally inconsistent sets go to minimally inconsistent sets of the same size. The search does not know about the relabelling; it simply finds the same counts, 6 and 150, twice. What matters to a majority is not the connective but the combinatorial shape of the consistent corners, and a conditional and a conjunction have the same shape.

Rankings are unsafe for the same reason #

The agenda of a ranking — is $a$ preferred to $b$, $b$ to $c$, $a$ to $c$ — sits in the table with minimal inconsistencies of size three.

The two sets of three on the ranking agenda are the two cycles. So majority on a ranking agenda can fail, and when it fails it produces a cycle: Condorcet’s paradox is the median property failing on the agenda of pairwise comparisons. The table measures how often. For three voters, 12 of the 216 profiles produce a cycle, which is the classical figure of one in eighteen; for five voters, 540 of 7,776, about seven in a hundred.

The “exactly one” agenda is worse still, failing on 24 of 64 profiles of three judges, because it has four triples rather than one. The count of profiles that fail is not decided by the largest triple alone — it depends on how many inconsistent sets there are and how they overlap — but whether any profile fails is decided by it exactly. That is the difference between the median property, which is a yes-or-no condition, and the frequency of trouble, which is a count.

This is also how Arrow’s theorem for preferences becomes a theorem about judgements. Franz Dietrich and Christian List showed in 2007 that Arrow’s four conditions follow from the judgement-aggregation impossibility applied to the ranking agenda, precisely because that agenda has triples. An impossibility needs an agenda with enough interlocking; rankings of three or more candidates have it.

Raising the bar, by exactly the right amount #

The median property decides when plain majority is safe. On an unsafe agenda, a stricter rule can still be safe, and the triple count says how strict.

A quota rule accepts a judgement when at least $q$ of the $n$ judges hold it. With $q$ above half, it takes no view on a question where neither side reaches the quota, so it may be incomplete. The question is when it can accept a contradiction.

The argument generalises the two-majority argument. Suppose the rule accepts all $k$ members of a minimally inconsistent set. Each is held by at least $q$ judges, so each is rejected — or at least not held — by at most $n - q$. If $k(n - q) < n$, the judges failing to hold some member of the set are fewer than all the judges, so some judge holds every member — a contradiction. So the rule is safe whenever $q > n(k - 1)/k$. Dietrich and List proved in 2007 that this is also necessary: at any lower quota, some profile makes the rule accept an inconsistent set.

For six judges the thresholds are $3$ on the chain, $4$ on the conjunction agenda and $4.5$ on the conjunction of three premises. The table has zeros exactly from the next whole quota on: from $q = 4$ on the chain, $q = 5$ on the other two. At $q = 4$ the three-premise agenda still fails on 6,390 profiles; at $q = 5$ it fails on none. The required supermajority rises with the size of the largest contradiction, from a bare majority for pairs, to two thirds for triples, to three quarters for sets of four.

What the search cannot show #

Every table here is exhaustive over its agendas and over every profile of up to six judges, and the general theorems are about every agenda and every number of judges. The median property is proved by the two-majority argument in one direction; its converse, that any agenda with a larger minimal inconsistency fails on some profile, needs an argument that such a set can always be split among majorities, which the search confirms for seven agendas and does not establish for others.

The seven agendas are small, and real agendas are not. A legal case, a committee’s budget or a panel’s findings may involve dozens of interlinked propositions, and computing their minimal inconsistent sets is itself a hard problem — the number of candidate sets grows exponentially with the agenda. Deciding whether an agenda has the median property is known to be computationally hard in general, so for a large agenda the safe-or-unsafe question may have no quick answer.

And the tables count profiles as though every combination of opinions were equally likely. A body whose members agree about most things may never meet its unsafe profiles, and one riven along a single axis may behave like a chain even on an agenda that is not one. The median property is a guarantee over every possible profile; it says nothing about the profiles a particular body tends to produce.

The question it leaves: what else the agenda decides #

The size of an agenda’s contradictions decides whether majority is safe and how high a quota must go. It also decides the shape of the impossibility itself. Nehring and Puppe showed that some agendas force more than inconsistency: on a totally blocked agenda — one where any consistent position can be reached from any other through a chain of minimal inconsistencies — every rule satisfying the other conditions is a dictatorship. On agendas that are unsafe but less interlocked, the same conditions allow non-dictatorial rules.

So the agenda’s logic is not a background detail. It determines whether there is a problem, how strict a fix must be, and whether the only fix is a dictator. The questions that remain on this subject concern agendas with more structure than the ones here — agendas whose propositions are linked by conditionals rather than conjunctions, or where some judgements are fixed by law in advance — and in each case the answer comes back to the same object: the smallest collections of judgements that cannot all be held.

The practical upshot for a body with an unsafe agenda is the choice already on the table. It can decide the premises or the conclusion, accepting the impossibility and choosing which condition to give up; it can take the nearest consistent verdict and accept ties; or it can restructure its agenda so that its contradictions come in pairs — asking a sequence of threshold questions instead of a conjunction, for instance. The last option changes the questions rather than the rule, and it is the only one that makes the problem disappear rather than moving it.

Two judges in common #

The whole theory rests on a counting fact about majorities. Two majorities of one body always share a member, and three need not. An agenda whose contradictions all involve two judgements can therefore be voted on question by question with complete safety, because the majorities that would have to assemble a contradiction are forced to overlap in someone who would have to hold it.

The doctrinal paradox, Condorcet’s cycle and the failures of every unsafe agenda in the table are the same event: three or more majorities, each large enough, with nobody in all of them. Raising the quota forces them to overlap again, and the quota needed is set by how many of them there are. What looked like a paradox about reasoning is a fact about how large sets can overlap, and the logic of the agenda only decides how many sets have to.

That reframing also explains why the problem is so general. Five voting rules returning five winners, a court contradicting itself, a committee’s budget adding up to more than its total: each is a collection of majorities too large to be held by any one member, on an agenda whose contradictions are too large to be caught by pairwise overlap. The agendas where that cannot happen are rare and recognisable, and the table on this page is a list of the smallest ones.