A formal result stating that when there are at least three alternatives, no social welfare function that maps every profile of individual strict preference orderings to a complete and transitive collective ordering can simultaneously satisfy the axioms of unrestricted domain, Pareto (unanimity), independence of irrelevant alternatives, and non‑dictatorship.

Principle

Under the specified ordinal, deterministic aggregation setting, the conjunction of the listed axioms is inconsistent: any aggregation rule satisfying unrestricted domain, Pareto and independence must be dictatorial for three or more alternatives.

Demonstration

Situation: Three voters provide strict rankings over three options. Recognition: Assume an aggregation rule purports to satisfy unanimity, independence, unrestricted domain and non‑dictatorship. Action: Construct preference profiles that, by independence and unanimity, force contradictory social orderings or single out one voter's preferences as decisive. Consequence: The construction shows that no non‑dictatorial rule can satisfy all axioms, establishing the impossibility claim for ordinal aggregations of three or more alternatives.

Misapplication

Interpreting the theorem as a claim that 'democracy is impossible' in all senses; or applying it to contexts with cardinal utilities, restricted preference domains, randomized rules, or procedures that produce incomplete or intransitive social judgments, where the theorem's assumptions do not hold.

Consequence

Forces explicit choice about which axioms to relax or which alternative frameworks to adopt (e.g., restrict preference domain, allow cardinal utility, permit randomized social choice), thereby shaping the theoretical and institutional design of aggregation rules.

Reversal

If one relaxes unrestricted domain (e.g., single‑peaked preferences), drops independence of irrelevant alternatives, permits interpersonal utility comparisons or cardinal utilities, or allows social preferences to be incomplete or probabilistic, the impossibility conclusion no longer applies.

Boundary

Applies to: deterministic social welfare functions from complete strict ordinal preference profiles to a complete transitive social ranking. Does not apply to: scoring rules that use cardinal information, probabilistic social choice, judgment aggregation producing non‑rank outputs, or settings with restricted preference domains.

Semantic Tension

Fairness axioms (unanimity, independence) ↔ feasibility of consistent collective orderings, revealing that commitment to certain fairness constraints precludes other desiderata like non‑dictatorship.

Synthesis

Arrow's theorem formalizes the trade‑offs inherent in ordinal aggregation: it does not prohibit collective decision‑making but compels explicit specification of which axioms to abandon or which alternative representation to use.