Source
- Sen (1970), The Impossibility of a Paretian Liberal, LIT-285, pp. 152–157, as read in NOTE-255. The reader checked the proof.
- Sen (1998), The Possibility of Social Choice, LIT-282, §XI, pp. 198–200, as read in NOTE-263.
What was actually shown
A social decision function need only yield a best element in every subset. Its social relation need not be transitive, and independence of irrelevant alternatives is not imposed. Minimal liberalism (L*) asks only that two individuals each be decisive, both ways, over one pair of states. Theorem II (p. 154) says no such function satisfies U, weak Pareto and L*.
The proof splits on how the two decisive pairs overlap:
- Same pair. Opposed preferences give a direct contradiction.
- One shared alternative. L* and Pareto force a strict 3-cycle.
- All four distinct. They force a strict 4-cycle.
In the cycle cases no element of the subset is unbeaten. The reader in NOTE-255 checked every case, and also checked the footnote-3 strengthening to one-way decisiveness, which Sen asserts without proof. Lady Chatterley (p. 155) is a correct instance of the shared-alternative case.
Because the proof uses only pairwise conditions and best-element existence, the two usual escapes from Arrow are shut. Dropping transitivity does nothing, since only a choice function is required. Relaxing independence does nothing, since it is never used (C4).
The 1998 lecture adds the contrast that makes this more than a restatement. Arrow’s impossibility is broken even by ordinal interpersonal comparability (p. 189, cited results). The liberal paradox “cannot be satisfactorily resolved through the use of interpersonal comparisons”. The reason given is that “neither liberty claims nor Pareto depend on the intensities of preferences” (pp. 199–200).
The record’s own check agrees. U, P and L* are stated on individual orderings alone, and the cycle is forced by those orderings. Supplying comparability information changes none of the premises. So the only exits are to weaken P, weaken L*, or restrict the domain, which is what Sen’s “priorities between personal liberty and overall desire fulfillment” amounts to (p. 200).
What this does not say
- It does not say that liberty is pairwise decisiveness over complete social states. That is the premise the later literature disputes: Nozick, Gärdenfors, Sugden, and Gaertner et al. The record holds those critics only as Sen’s citations and his reply. His 1998 reply is that the process view is right that opportunity is not sufficient, but does not by itself resolve the paradox (fn. 46). That reply is his, not a result.
- It does not say that Pareto should give way. Sen’s moral (“deeply illiberal”) is an interpretation of the theorem. His positive suggestion, mutually respectful individual values (pp. 155–156) or a multistage choice (1998, fn. 49), is not developed.
- The Arrow half is held by citation, not proof. That comparability breaks Arrow rests on Sen 1970a and other works the record does not hold. This document relies on it only as the contrast Sen draws. The curation entry of 2026-09-28 declined a THEORY on the pair for this reason. This one is restricted to the liberal half, which is proved in the record.
- Full U is far stronger than needed. The proof uses a handful of profiles, and the paper does not ask which domain restrictions escape. Theorem I, stated with L, tacitly needs n ≥ 2.
Connections
- LIT-289 (Sen 1979) cites this result for the claim that libertarian non-utility information may require rejecting Pareto (p. 212).
- THEORY-025, on conversion and adaptation, takes up the other half of Sen’s 1998 lecture: once comparisons are admitted, “comparison of what?”.