@cite{dambrosio-hedden-2024} #
D'Ambrosio, J. & Hedden, B. (2024). Multidimensional Adjectives. Australasian Journal of Philosophy 102(2): 253–277. DOI: 10.1080/00048402.2023.2277923
Key Claims #
Multidimensional adjectives require explicit aggregation functions mapping dimensional assessments to overall assessments (§3).
Arrow's impossibility theorem (adapted): under constraints ONC + WO + U + P + I + D, no aggregation function exists for ≥3 dimensions and ≥3 objects. Multidimensional adjectives would be incoherent (§4.1).
Escape routes determine aggregation type (§4.2–4.3):
- Reject WO (transitivity) → Majority Rule (May 1952)
- Reject WO (completeness) → Strong Pareto Rule (Weymark 1984)
- Reject ONC, accept IUC → Utilitarian / weighted sum (Sen 1970)
- Reject ONC, accept RNC → Cobb-Douglas / weighted product (Tsui-Weymark 1997)
Multiple admissible aggregation functions → comparative vagueness, a source of vagueness specific to multidimensionality (§4.3).
Formalization #
- §1: Arrow's constraints
- §2: athletic example (majority rule and weighted aggregation)
- §3: Comparative vagueness from weight multiplicity
- §4: Connection to @cite{sassoon-2013} (binding types = counting)
Arrow's constraints on dimensional aggregation (§4, adapted from social choice theory).
- unrestrictedDomain : Bool
(U) Defined for all logically possible value profiles.
- weakOrdering : Bool
(WO) Output is a weak ordering (reflexive, transitive, complete).
- weakPareto : Bool
(P) Unanimous dimensional ranking → same overall ranking.
- independence : Bool
(I) Overall ranking of x vs y depends only on their dim values.
- nonDictatorship : Bool
(D) No single dimension dictates the overall ranking.
- ordinalNonComparability : Bool
(ONC) Only ordinal info used (invariant under monotone transforms).
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- Phenomena.Gradability.Studies.DAmbrosioHedden2024.instBEqArrowConstraints.beq x✝¹ x✝ = false
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The full Arrovian constraint set (§4.1).
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fullArrow enables all six constraints. Arrow's impossibility
theorem (adapted, §4.1) says these are jointly unsatisfiable for
≥3 dimensions and ≥3 objects — at least one must be abandoned,
and each escape route yields a different aggregation type.
This theorem only records the constraint specification; the impossibility proof itself would require formalizing aggregation over orderings and is not attempted here.
The adjective athletic has three dimensions: speed (F₁), agility (F₂), and endurance (F₃) (§3). We model two people and show how different aggregation mechanisms yield different verdicts.
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Dimensional profiles for athletic:
- Alice: fast, agile, not enduring
- Bob: not fast, not agile, enduring
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Majority rule: Alice is athletic (2 of 3 dims).
Majority rule: Bob is NOT athletic (1 of 3 dims).
Under speed-heavy weights [3, 1, 1] with θ = 3, Alice IS athletic (score = 3 + 1 + 0 = 4 ≥ 3) but Bob is NOT (score = 0 + 0 + 1 = 1).
Under endurance-heavy weights [1, 1, 3] with θ = 3, Alice is NOT athletic (score = 1 + 1 + 0 = 2 < 3) but Bob IS (0 + 0 + 3 = 3 ≥ 3).
When multiple weight vectors are admissible, the comparative form "x is more athletic than y" is vague: different admissible aggregation functions rank the entities differently.
This is D&H's central prediction (§4.3): multidimensionality
generates comparative vagueness through admissibility multiplicity.
Under speed-heavy weights, Alice outscores Bob.
Under endurance-heavy weights, Bob outscores Alice.
Comparative vagueness: when both weight vectors are admissible, "Alice is more athletic than Bob" is indeterminate — one aggregation function says yes, the other says no.
@cite{sassoon-2013}'s framework classifies binding as conjunctive (∀), disjunctive (∃), or mixed (dimension counting). D&H show all three are counting aggregation — a single escape route from Arrow's theorem. Utilitarian aggregation (weighted sum) is a genuinely different mechanism that Sassoon's typology misses.
All of Sassoon 2013's binding types are counting aggregation.
Utilitarian aggregation is NOT counting — it is a categorically different escape route from Arrow's impossibility.
healthy under conjunctive binding: must satisfy ALL dimensions. A person healthy on musculoskeletal and cardiovascular but with disease present is NOT healthy.
But under counting with k = 2, the same person IS healthy (passes on 2 of 3 dimensions). Counting and conjunctive diverge.