Source
List & Pettit (2006), LIT-705, pp. 9–17, read in full in NOTE-550. Its claims table: Proposition 1 has a sketch proof, Propositions 4 and 5 short arguments, and the possibility result is “constructive”. Propositions 2 and 3 are cited, not proved, and are not part of this account.
What was actually shown
The setting. The agenda is P, Q, R, B = (R ↔ (P ∧ Q)) and their negations. An individual judgment set is complete, consistent and deductively closed on it. A constitution has universal domain if it accepts every profile of such sets. The group is robustly rational if its judgment set is complete, consistent and deductively closed for every such profile (p. 9).
Proposition 1: majority voting fails. Take three members.
| member | P | Q | B | R |
|---|---|---|---|---|
| J1 | yes | yes | yes | yes |
| J2 | yes | no | yes | no |
| J3 | no | yes | yes | no |
Each row is complete and consistent, with R forced by B and the premises. The majorities accept P, Q and B and reject R. But P ∧ Q together with B entail R, so the majority set is inconsistent. One profile in the domain is enough to defeat robust rationality.
Proposition 4: the premise-based procedure succeeds. With an odd number of members, take the majority on each of P, Q and B, so each majority exists. Then fix R from them: if B is accepted, R iff P ∧ Q; if B is rejected, R iff ¬(P ∧ Q). Any combination of verdicts on P, Q and B is consistent, and R chosen this way makes the set complete, consistent and deductively closed, for every profile. The group’s set is a function of the members’ whole sets (set-wise supervenience). It is not proposition-wise, because the group’s verdict on R does not depend on the members’ verdicts on R.
Weak discontinuity. In the profile above, the members’ verdicts on R are (yes, no, no), and the group accepts R. Now take J1 = (yes, yes, yes, yes), J2 = (no, yes, yes, no) and J3 = (no, no, yes, no), in the column order P, Q, B, R. Each row is consistent, and the members’ verdicts on R are again (yes, no, no). But the premise majorities are ¬P, Q and B, so the group rejects R. The same judgments on the conclusion go with opposite group verdicts, so they do not fix it.
Strong discontinuity (Proposition 5). Under the distributed premise-based procedure, each premise is settled by its own subgroup, and R is derived as above. No member’s judgment on R enters the procedure at all, and the group’s set is still wholly fixed by the members’ judgments (pp. 16–17).
What this does not say
- Anything about proposition-wise rules in general. That no proposition-wise rule depending on more than one individual and respecting unanimity is robustly rational (Propositions 2–3) is cited in the source, not proved. It is THEORY-127’s, still Proposed.
- Anything for other agendas. The source names the wider classes (n. 3) without stating their conditions.
- That the premise-based procedure is the right one. It is one illustration, and the authors do not recommend it.