1
Judgments are linked by logic
A judge decides whether p is true, whether q is true, and whether the conclusion p ∧ q is, and a consistent judge accepts the conclusion exactly when they accept both premises. The three questions are one agenda.
Lesson · Justice and Collective Choice · Collective choice
The Discursive Dilemma
Three judges rule on two premises and on a conclusion that is true exactly when both premises are. Each judge is consistent, and the majority accepts each premise, yet the majority rejects the conclusion. A premise-based procedure and a conclusion-based procedure then give opposite verdicts. The wrong model is that if a majority accepts each premise it accepts the conclusion.
a majority can accept p and accept q
and still reject p ∧ q
each judge is consistent, the court is not
premise-based or conclusion-based verdict
A court that contradicts itself
Core ideas
What this lesson establishes
The Condorcet paradox combined consistent rankings into a cycle. The same thing happens to judgments, and the way out of it forces a choice about what the group's reasons are.
2
Two procedures, one set of judgments
The premise-based procedure takes the majority on each premise and derives the conclusion from them. The conclusion-based procedure takes the majority on the conclusion and ignores the premises.
3
A consistent court is the exception
The majority's accepted set can be inconsistent even though every member of the court is consistent. The lab's court shows such a case, and the other presets show a court whose procedures agree and one whose procedures differ the other way.
The material
Three judges, three questions
A court whose majorities contradict one another, a court that agrees with itself, and a disjunction where the premise-based verdict is the one that fails.
A group is often asked for more than a choice. A court gives a verdict and reasons for it; a committee decides a proposal and the facts on which it rests. When the reasons are decided by vote, the question is whether the group's reasons and its verdict agree.
A court of three judges decides whether a defendant is liable. Liability requires two things: p, that there was a valid contract, and q, that the defendant breached it. The conclusion is p ∧ q, that both hold. Each judge is consistent: they accept the conclusion exactly when they accept both premises.
p q p ∧ q
judge 1 yes yes yes
judge 2 yes no no
judge 3 no yes no
Count each column. The premise p is accepted by judges one and two, so it carries 2 to 1. The premise q is accepted by judges one and three, so it carries 2 to 1. The conclusion p ∧ q is accepted only by judge one, so it fails 1 to 2. The court's majority judgments are that p is true, that q is true, and that p ∧ q is false, and those three cannot all be true together.
Definition · The discursive dilemma
The discursive dilemma is the situation in which a group's majority judgments on a set of logically connected propositions are inconsistent, though every member's judgments are consistent.
Two procedures, two verdicts
The court has to say something about liability. The premise-based procedure accepts what the majority accepts on p and on q, and derives that the defendant is liable. The conclusion-based procedure counts the votes on liability itself and finds that the defendant is not. The lab prints both: premise-based true, conclusion-based false.
Each has a cost. The premise-based verdict holds the defendant liable though two of the three judges, judges two and three, would not. The conclusion-based verdict ignores the court's reasons and leaves it saying that a contract existed, that it was breached, and that no liability follows. Choosing between them is a choice about whether a group's verdict should follow its reasons or its votes.
It is the Condorcet paradox again
Three voters, three propositions, each voter consistent, a majority on each proposition that does not cohere. The shape is that of the cycle in “Majority Rule and the Condorcet Paradox”, with truth values in place of rankings. As there, no judge is at fault, and no tie-break repairs it.
The court is Kornhauser and Sager's, who called the pattern the doctrinal paradox; Pettit gave it the name used here when he showed that any group which votes on connected propositions can meet it, courts or not. List and Pettit then proved that, for an agenda like this one, no way of aggregating judgments satisfies all of: accepting any consistent judgments from the judges, returning a consistent and complete set, treating every judge alike, and deciding every proposition by the same rule applied to its own votes. The premise-based procedure gives up the last of these, since it treats the conclusion differently. The theorem is stated, not proved, here.
The direction of the disagreement is not fixed. If the conclusion is the disjunction p ∨ q, take judges who accept p only, q only, and neither. Each premise is accepted by one judge in three, so the premise-based procedure rejects both and rejects the disjunction; but two judges accept the disjunction, and the conclusion-based procedure accepts it. The third preset shows that case.
The lab takes an odd number of judges so that no majority ties, and a conclusion made from the premises with and, or, not and if-then. It takes each judge's verdicts as given. It cannot say whether a premise is true, which is the part the court is there to decide.
Interactive
Collective choice
six voting rules, the majority relation, judgment aggregation and the jury theorem
Method
Aggregating a court's judgments
Five moves for any set of judges. The last one is the choice the lab cannot make for you.
1
Write each judge's verdicts as a row
One column per premise and one for the conclusion. Check that each judge's conclusion follows from their premises, since a judge who is inconsistent is a different problem.
2
Take the majority in every column
Count the judges who accept the proposition. More than half carries it. An odd number of judges means there are no ties.
3
Derive the conclusion from the majority premises
Apply the connective to the majority verdicts on the premises. This is the premise-based verdict.
4
Compare it with the majority on the conclusion
That is the conclusion-based verdict. If the two differ, the court's majority judgments are inconsistent.
5
Decide which procedure to adopt, and say what it costs
The first follows the reasons and may overrule the majority on the verdict. The second follows the verdict and leaves the reasons unexplained. The lab computes both and does not pick.
Reading a court
Score 0 / 0Choose the answer you can justify.
Three columns, three majorities
Judge 1 accepts p, q and p and q; judge 2 accepts p only; judge 3 accepts q only.
p yes, yes, no -> carries 2 to 1
q yes, no, yes -> carries 2 to 1
p and q yes, no, no -> fails 1 to 2
premise-based: p, q hold -> liable
conclusion-based: 1 of 3 -> not liable
Every judge is consistent, and the court is not. The failure is in the combination of the three consistent sets of judgments, not in any one of them.
Common mistakes
1
Believing that if a majority accepts each premise, a majority accepts the conclusionIn the court, p carries 2 to 1 and q carries 2 to 1, and p and q together fails 1 to 2. The majorities are taken column by column and a conjunction is not a column that follows the others. The belief would hold if the majorities were unanimous, and with the court's disagreement it fails.
2
Blaming a judgeEach of the three judges accepts the conclusion exactly when they accept both premises, so each is consistent. The inconsistency appears only when the three sets of verdicts are combined by majority. As with the cycle of rankings, the fault is in the combining.
3
Treating the premise-based procedure as the correct oneIn the disjunction preset it rejects both premises, so it rejects the conclusion, though two of the three judges accept it. The premise-based procedure follows the reasons the majority accepted, and the conclusion-based procedure follows the verdicts. Which to prefer depends on whether the group is answerable for its reasons or its result.
Check your understanding
✓
Finish when you can aggregate a court's judgments on two premises and a conclusion and name the two procedures.
Given three judges' verdicts, compute the majority on each premise and on the conclusion, derive the premise-based verdict, report whether it agrees with the conclusion-based one, and give a profile in which it does not.
Note: Change one judge's row in the court and watch when the contradiction disappears. With three judges and a conjunction it needs one judge who accepts only p, one who accepts only q, and one who accepts both; change any of the three and the two procedures agree.