@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

1. Multidimensional adjectives require explicit aggregation functions mapping dimensional assessments to overall assessments (§3).

2. 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).

3. 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)

4. 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)

structure DAmbrosioHedden2024.ArrowConstraints : Type

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).

def DAmbrosioHedden2024.instReprArrowConstraints.repr : ArrowConstraints → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance DAmbrosioHedden2024.instReprArrowConstraints : Repr ArrowConstraints

Equations

• DAmbrosioHedden2024.instReprArrowConstraints = { reprPrec := DAmbrosioHedden2024.instReprArrowConstraints.repr }

@[implicit_reducible]

instance DAmbrosioHedden2024.instBEqArrowConstraints : BEq ArrowConstraints

Equations

• DAmbrosioHedden2024.instBEqArrowConstraints = { beq := DAmbrosioHedden2024.instBEqArrowConstraints.beq }

def DAmbrosioHedden2024.instBEqArrowConstraints.beq : ArrowConstraints → ArrowConstraints → Bool

Equations

• One or more equations did not get rendered due to their size.
• DAmbrosioHedden2024.instBEqArrowConstraints.beq x✝¹ x✝ = false

def DAmbrosioHedden2024.fullArrow : ArrowConstraints

The full Arrovian constraint set (§4.1).

Equations

• DAmbrosioHedden2024.fullArrow = { unrestrictedDomain := true, weakOrdering := true, weakPareto := true, independence := true, nonDictatorship := true, ordinalNonComparability := true }

theorem DAmbrosioHedden2024.fullArrow_all_enabled : fullArrow.unrestrictedDomain = true ∧ fullArrow.weakOrdering = true ∧ fullArrow.weakPareto = true ∧ fullArrow.independence = true ∧ fullArrow.nonDictatorship = true ∧ fullArrow.ordinalNonComparability = true

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.

inductive DAmbrosioHedden2024.Person : Type

• alice : Person
• bob : Person

def DAmbrosioHedden2024.instReprPerson.repr : Person → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance DAmbrosioHedden2024.instReprPerson : Repr Person

Equations

• DAmbrosioHedden2024.instReprPerson = { reprPrec := DAmbrosioHedden2024.instReprPerson.repr }

@[implicit_reducible]

instance DAmbrosioHedden2024.instDecidableEqPerson : DecidableEq Person

Equations

• DAmbrosioHedden2024.instDecidableEqPerson x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def DAmbrosioHedden2024.athleticDims : List (Person → Bool)

Dimensional profiles for athletic:

• Alice: fast, agile, not enduring
• Bob: not fast, not agile, enduring

Equations

• One or more equations did not get rendered due to their size.

theorem DAmbrosioHedden2024.alice_athletic_majority : Semantics.Gradability.Aggregation.majorityBinding athleticDims Person.alice = true

Majority rule: Alice is athletic (2 of 3 dims).

theorem DAmbrosioHedden2024.bob_not_athletic_majority : Semantics.Gradability.Aggregation.majorityBinding athleticDims Person.bob = false

Majority rule: Bob is NOT athletic (1 of 3 dims).

theorem DAmbrosioHedden2024.alice_athletic_speed_heavy : Semantics.Gradability.Aggregation.weightedBinding [3, 1, 1] 3 athleticDims Person.alice = true

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).

theorem DAmbrosioHedden2024.bob_not_athletic_speed_heavy : Semantics.Gradability.Aggregation.weightedBinding [3, 1, 1] 3 athleticDims Person.bob = false

theorem DAmbrosioHedden2024.alice_not_athletic_endurance_heavy : Semantics.Gradability.Aggregation.weightedBinding [1, 1, 3] 3 athleticDims Person.alice = false

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).

theorem DAmbrosioHedden2024.bob_athletic_endurance_heavy : Semantics.Gradability.Aggregation.weightedBinding [1, 1, 3] 3 athleticDims Person.bob = true

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. 

theorem DAmbrosioHedden2024.speed_heavy_alice_wins : Semantics.Gradability.Aggregation.weightedScore [3, 1, 1] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.alice > Semantics.Gradability.Aggregation.weightedScore [3, 1, 1] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.bob

Under speed-heavy weights, Alice outscores Bob.

theorem DAmbrosioHedden2024.endurance_heavy_bob_wins : Semantics.Gradability.Aggregation.weightedScore [1, 1, 3] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.bob > Semantics.Gradability.Aggregation.weightedScore [1, 1, 3] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.alice

Under endurance-heavy weights, Bob outscores Alice.

theorem DAmbrosioHedden2024.comparative_vagueness : Semantics.Gradability.Aggregation.weightedScore [3, 1, 1] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.alice > Semantics.Gradability.Aggregation.weightedScore [3, 1, 1] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.bob ∧ Semantics.Gradability.Aggregation.weightedScore [1, 1, 3] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.bob > Semantics.Gradability.Aggregation.weightedScore [1, 1, 3] (Semantics.Gradability.Aggregation.boolMeasures athleticDims) Person.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.

theorem DAmbrosioHedden2024.sassoon_framework_is_counting (b : Semantics.Gradability.DimensionBindingType) : Semantics.Gradability.Aggregation.toAggregationType b = Semantics.Gradability.Aggregation.AggregationType.counting

All of Sassoon 2013's binding types are counting aggregation.

theorem DAmbrosioHedden2024.utilitarian_not_counting : Semantics.Gradability.Aggregation.AggregationType.utilitarian ≠ Semantics.Gradability.Aggregation.AggregationType.counting

Utilitarian aggregation is NOT counting — it is a categorically different escape route from Arrow's impossibility.

theorem DAmbrosioHedden2024.healthy_conjunctive_rejects_partial : Semantics.Gradability.conjunctiveBinding [fun (x : Unit) => true, fun (x : Unit) => true, fun (x : Unit) => false] () = false

healthy under conjunctive binding: must satisfy ALL dimensions. A person healthy on musculoskeletal and cardiovascular but with disease present is NOT healthy.

theorem DAmbrosioHedden2024.counting_accepts_partial : Semantics.Gradability.Aggregation.countBinding 2 [fun (x : Unit) => true, fun (x : Unit) => true, fun (x : Unit) => false] () = true

But under counting with k = 2, the same person IS healthy (passes on 2 of 3 dimensions). Counting and conjunctive diverge.