@cite{nickel-2009}: Generics and the Ways of Normality

Bernhard Nickel, "Generics and the Ways of Normality", Linguistics and Philosophy 31 (2009), 629–648.

The Problem: Conjunctive Generics

Nickel criticizes majority-based views of generics (including @cite{cohen-1999a}'s probability-based GEN) by showing they cannot handle conjunctive generics like:

(1) Elephants live in Africa and Asia.

If (1) is equivalent to the sentential conjunction:

(2) Elephants live in Africa AND Elephants live in Asia.

then a majority-based view would require both conjuncts to hold with prevalence > 0.5 over the same domain. But African elephants and Asian elephants are disjoint populations — most elephants can't live in BOTH places. So the majority view predicts (1) is false, contrary to speaker judgments.

Nickel's Solution: Ways of Being Normal

Nickel proposes that normality is not a single binary predicate but comes in multiple ways. For the elephant case:

• Way w₁: normal w.r.t. habitat → lives in Africa
• Way w₂: normal w.r.t. habitat → lives in Asia

GEN existentially quantifies over ways of being normal, then universally quantifies over entities that are normal in that way:

GEN[A][F] is true iff
∃w (way of being normal for As w.r.t. F).
  ∀x. normalIn(x, w) → F(x)

Conjunctive generics can then use different normality ways for each conjunct:

(1) is true iff
(∃w₁. ∀x. normalIn(x, w₁) → livesInAfrica(x)) ∧
(∃w₂. ∀x. normalIn(x, w₂) → livesInAsia(x))

This is discussed in the introduction to Genericity (OUP 2013).

structure Nickel2009.NormalcyWay : Type

A way of being normal — an index that selects which entities count as "normal" for a given generalization. Different generic claims can appeal to different normality ways.

• id : ℕ

def Nickel2009.instDecidableEqNormalcyWay.decEq (x✝ x✝¹ : NormalcyWay) : Decidable (x✝ = x✝¹)

Equations

• Nickel2009.instDecidableEqNormalcyWay.decEq { id := a } { id := b } = if h : a = b then h ▸ isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Nickel2009.instDecidableEqNormalcyWay : DecidableEq NormalcyWay

Equations

• Nickel2009.instDecidableEqNormalcyWay = Nickel2009.instDecidableEqNormalcyWay.decEq

def Nickel2009.instReprNormalcyWay.repr : NormalcyWay → ℕ → Std.Format

Equations

• Nickel2009.instReprNormalcyWay.repr x✝ prec✝ = Std.Format.bracket "{ " (Std.Format.nil ++ Std.Format.text "id" ++ Std.Format.text " := " ++ (Std.Format.nest 6 (repr x✝.id)).group) " }"

@[implicit_reducible]

instance Nickel2009.instReprNormalcyWay : Repr NormalcyWay

Equations

• Nickel2009.instReprNormalcyWay = { reprPrec := Nickel2009.instReprNormalcyWay.repr }

structure Nickel2009.Entity : Type

An entity in the domain of a generic.

• id : ℕ

def Nickel2009.instDecidableEqEntity.decEq (x✝ x✝¹ : Entity) : Decidable (x✝ = x✝¹)

Equations

• Nickel2009.instDecidableEqEntity.decEq { id := a } { id := b } = if h : a = b then h ▸ isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Nickel2009.instDecidableEqEntity : DecidableEq Entity

Equations

• Nickel2009.instDecidableEqEntity = Nickel2009.instDecidableEqEntity.decEq

@[implicit_reducible]

instance Nickel2009.instReprEntity : Repr Entity

Equations

• Nickel2009.instReprEntity = { reprPrec := Nickel2009.instReprEntity.repr }

def Nickel2009.instReprEntity.repr : Entity → ℕ → Std.Format

Equations

• Nickel2009.instReprEntity.repr x✝ prec✝ = Std.Format.bracket "{ " (Std.Format.nil ++ Std.Format.text "id" ++ Std.Format.text " := " ++ (Std.Format.nest 6 (repr x✝.id)).group) " }"

@[reducible, inline]

abbrev Nickel2009.NormalIn : Type

Whether an entity is normal in a given way.

Equations

• Nickel2009.NormalIn = (Nickel2009.Entity → Nickel2009.NormalcyWay → Bool)

@[reducible, inline]

abbrev Nickel2009.Property : Type

A property of entities.

Equations

• Nickel2009.Property = (Nickel2009.Entity → Bool)

def Nickel2009.nickelGEN (entities : List Entity) (normalIn : NormalIn) (ways : List NormalcyWay) (restrictor scope : Property) : Bool

Nickel's GEN with way-indexed normality:

GEN[restrictor][scope] is true iff there exists a way of being normal such that all entities that are normal in that way AND satisfy the restrictor also satisfy the scope.

The key innovation: the existential quantification over normality ways allows different conjuncts of a conjunctive generic to use different ways.

Equations

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

def Nickel2009.nickelConjunctiveGEN (entities : List Entity) (normalIn : NormalIn) (ways : List NormalcyWay) (restrictor scope1 scope2 : Property) : Bool

Conjunctive generic: both GEN[A][F₁] and GEN[A][F₂] hold, potentially via different normality ways.

Equations

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

def Nickel2009.majorityGEN (entities : List Entity) (restrictor scope : Property) : Bool

Majority-based GEN (@cite{cohen-1999a}'s view): generic is true iff prevalence exceeds 1/2. Structurally identical to cohenGEN in Cohen1999.lean — both are thresholdGeneric with θ = 1/2, just instantiated at different domain types (Entity here, Situation there).

Equations

• Nickel2009.majorityGEN entities restrictor scope = Semantics.Genericity.Generics.thresholdGeneric entities restrictor scope (1 / 2)

def Nickel2009.elephants : List Entity

10 elephants: 6 African (ids 0-5), 4 Asian (ids 6-9).

Equations

• Nickel2009.elephants = List.map (fun (n : ℕ) => { id := n }) (List.range 10)

def Nickel2009.isElephant : Property

Equations

• Nickel2009.isElephant x✝ = true

def Nickel2009.livesInAfrica : Property

Equations

• Nickel2009.livesInAfrica e = decide (e.id < 6)

def Nickel2009.livesInAsia : Property

Equations

• Nickel2009.livesInAsia e = decide (e.id ≥ 6)

def Nickel2009.africanWay : NormalcyWay

Equations

• Nickel2009.africanWay = { id := 1 }

def Nickel2009.asianWay : NormalcyWay

Equations

• Nickel2009.asianWay = { id := 2 }

def Nickel2009.ways : List NormalcyWay

Equations

• Nickel2009.ways = [Nickel2009.africanWay, Nickel2009.asianWay]

def Nickel2009.elephantNormalIn : NormalIn

Normal in the "African way" = African elephants; Normal in the "Asian way" = Asian elephants.

Equations

• Nickel2009.elephantNormalIn e w = match w.id with | 1 => decide (e.id < 6) | 2 => decide (e.id ≥ 6) | x => false

def Nickel2009.bears : List Entity

20 bears across 4 continents: North America (0-4), South America (5-9), Europe (10-14), Asia (15-19). The majority view fails for ALL four habitat conjuncts since each subpopulation is only 5/20 = 25%.

Equations

• Nickel2009.bears = List.map (fun (n : ℕ) => { id := n }) (List.range 20)

def Nickel2009.isBear : Property

Equations

• Nickel2009.isBear x✝ = true

def Nickel2009.bearNA : Property

Equations

• Nickel2009.bearNA e = decide (e.id < 5)

def Nickel2009.bearSA : Property

Equations

• Nickel2009.bearSA e = (decide (e.id ≥ 5) && decide (e.id < 10))

def Nickel2009.bearEU : Property

Equations

• Nickel2009.bearEU e = (decide (e.id ≥ 10) && decide (e.id < 15))

def Nickel2009.bearAS : Property

Equations

• Nickel2009.bearAS e = decide (e.id ≥ 15)

def Nickel2009.naWay : NormalcyWay

Equations

• Nickel2009.naWay = { id := 1 }

def Nickel2009.saWay : NormalcyWay

Equations

• Nickel2009.saWay = { id := 2 }

def Nickel2009.euWay : NormalcyWay

Equations

• Nickel2009.euWay = { id := 3 }

def Nickel2009.asWay : NormalcyWay

Equations

• Nickel2009.asWay = { id := 4 }

def Nickel2009.bearWays : List NormalcyWay

Equations

• Nickel2009.bearWays = [Nickel2009.naWay, Nickel2009.saWay, Nickel2009.euWay, Nickel2009.asWay]

def Nickel2009.bearNormalIn : NormalIn

Equations

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

def Nickel2009.waysIncompatible (entities : List Entity) (normalIn : NormalIn) (ways : List NormalcyWay) : Bool

Normality ways are pairwise incompatible: no entity in the domain is normal in two distinct ways simultaneously. The paper (p.643) states: "Being F-normal in one way is (perhaps always) incompatible with being F-normal in any other way."

Equations

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

theorem Nickel2009.majority_fails_elephant_asia : majorityGEN elephants isElephant livesInAsia = false

The majority view fails for the elephant example: "Elephants live in Asia" is false under majority semantics because only 4/10 < 1/2 are Asian.

theorem Nickel2009.majority_fails_conjunction : (majorityGEN elephants isElephant livesInAfrica && majorityGEN elephants isElephant livesInAsia) = false

The conjunctive generic fails under the majority view.

theorem Nickel2009.nickel_handles_elephant_conjunction : nickelConjunctiveGEN elephants elephantNormalIn ways isElephant livesInAfrica livesInAsia = true

Nickel's view succeeds for the elephant conjunction.

theorem Nickel2009.bears_majority_fails_nickel_succeeds : majorityGEN bears isBear bearNA = false ∧ majorityGEN bears isBear bearSA = false ∧ majorityGEN bears isBear bearEU = false ∧ majorityGEN bears isBear bearAS = false ∧ nickelGEN bears bearNormalIn bearWays isBear bearNA = true ∧ nickelGEN bears bearNormalIn bearWays isBear bearSA = true ∧ nickelGEN bears bearNormalIn bearWays isBear bearEU = true ∧ nickelGEN bears bearNormalIn bearWays isBear bearAS = true

The bears example (paper's ex. 2a): majority view fails for ALL four habitat conjuncts (each is 25%), while Nickel's view succeeds.

theorem Nickel2009.elephant_ways_incompatible : waysIncompatible elephants elephantNormalIn ways = true ∧ waysIncompatible bears bearNormalIn bearWays = true

Normality ways are pairwise incompatible in both examples.

theorem Nickel2009.nickel_single_way_is_traditional (entities : List Entity) (normalIn : NormalIn) (w : NormalcyWay) (restrictor scope : Property) : nickelGEN entities normalIn [w] restrictor scope = entities.all fun (e : Entity) => !(restrictor e && normalIn e w) || scope e

Nickel's GEN with a single normality way reduces to traditional GEN: if there is only one way of being normal, the existential quantification is trivial and we get back ∀x. normal(x) ∧ restrictor(x) → scope(x).

Summary: Three Views of Normality

View	Normality	GEN formula	Handles elephants?
@cite{cohen-1999a}	Probability > 0.5	P(Q|P) > 0.5	No
@cite{asher-pelletier-2012}	Modal ordering	∀w ≤ w₀. P(x,w) → Q(x,w)	Partially
@cite{nickel-2009}	Ways of being normal	∃w. ∀x. normal(x,w) → Q(x)	Yes

The three views are formalized in:

• Cohen1999.lean (probability)
• Comparisons/GenericModality.lean (modal)
• This file (way-indexed normality)