@cite{cohen-1999a}: Probability-Based Generic Quantification

Ariel Cohen, Think Generic! The Meaning and Use of Generic Sentences, 1999.

Core Proposal

Cohen proposes that the generic quantifier GEN is a probability operator:

GEN(P, Q) is true iff P(Q | P) > 0.5

That is, a generic "Ps are Q" is true iff the conditional probability of an object having property Q, given that it has property P, exceeds 0.5.

This contrasts with the frequency adverb always, which requires P(Q | P) = 1.

Connection to Threshold Semantics

Cohen's GEN is a special case of threshold semantics with θ = 1/2. In linglib's infrastructure, cohenGEN is definitionally equal to thresholdGeneric situations restrictor scope (1/2).

Homogeneity Constraint

Cohen introduces a homogeneity presupposition: the conditional probability P(Q | P) must be uniform across all suitable partitions of the domain. If the domain splits into subgroups with different rates, the generic presupposition fails and the sentence is neither true nor false.

This constraint is discussed in the introduction to Genericity (Mari, Beyssade, Del Prete, OUP 2013):

"A homogeneity requirement is introduced as a presupposition of generics and frequency statements, according to which the relative probability in every part of a suitable partition of any admissible history H must be the same as the probability in the whole H."

Nickel's Critique (@cite{nickel-2009})

@cite{nickel-2009} shows that even with homogeneity, the majority-based view cannot handle conjunctive generics like "Elephants live in Africa and Asia." If this is equivalent to the conjunction "Elephants live in Africa AND Elephants live in Asia," then both conjuncts would need to hold with probability > 0.5, which is impossible if the populations are disjoint. See Phenomena/Generics/Studies/Nickel2009.lean.

def Phenomena.Generics.Studies.Cohen1999.cohenGEN (situations : List Semantics.Lexical.Noun.Kind.Generics.Situation) (restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor) (scope : Semantics.Lexical.Noun.Kind.Generics.Scope) : Bool

Cohen's GEN: a generic "Ps are Q" is true iff the conditional probability P(Q | P) exceeds 0.5.

prevalence situations restrictor scope computes exactly this conditional probability: the proportion of restrictor-satisfying situations where scope holds. So Cohen's GEN is prevalence > 1/2.

Equations

• Phenomena.Generics.Studies.Cohen1999.cohenGEN situations restrictor scope = decide (Semantics.Lexical.Noun.Kind.Generics.prevalence situations restrictor scope > 1 / 2)

theorem Phenomena.Generics.Studies.Cohen1999.cohen_is_threshold (situations : List Semantics.Lexical.Noun.Kind.Generics.Situation) (restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor) (scope : Semantics.Lexical.Noun.Kind.Generics.Scope) : cohenGEN situations restrictor scope = Semantics.Lexical.Noun.Kind.Generics.thresholdGeneric situations restrictor scope (1 / 2)

Cohen's GEN is a special case of threshold semantics with θ = 1/2.

def Phenomena.Generics.Studies.Cohen1999.cohenAlways (situations : List Semantics.Lexical.Noun.Kind.Generics.Situation) (restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor) (scope : Semantics.Lexical.Noun.Kind.Generics.Scope) : Bool

Cohen's "always" requires conditional probability = 1 (no exceptions).

Equations

• Phenomena.Generics.Studies.Cohen1999.cohenAlways situations restrictor scope = (Semantics.Lexical.Noun.Kind.Generics.prevalence situations restrictor scope == 1)

def Phenomena.Generics.Studies.Cohen1999.homogeneous (situations : List Semantics.Lexical.Noun.Kind.Generics.Situation) (restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor) (scope : Semantics.Lexical.Noun.Kind.Generics.Scope) : Prop

Cohen's homogeneity constraint: the conditional probability P(scope | restrictor) must be the same in every non-empty sub-partition of the domain.

Formally: for any sub-predicate part, if there are restrictor-satisfying elements in that partition, the proportion of scope-satisfying elements among restrictor ∧ part elements equals the overall proportion among restrictor elements.

This is P(Q | P ∧ Pᵢ) = P(Q | P) for all partition cells Pᵢ.

When homogeneity fails, the generic presupposition fails and the sentence is neither true nor false.

Equations

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

structure Phenomena.Generics.Studies.Cohen1999.CohenGenericJudgment : Type

A generic assertion according to Cohen: the prevalence exceeds 0.5 AND the homogeneity presupposition is satisfied.

• situations : List Semantics.Lexical.Noun.Kind.Generics.Situation
• restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor
• scope : Semantics.Lexical.Noun.Kind.Generics.Scope
• truth : Bool
• presupposition : Prop

def Phenomena.Generics.Studies.Cohen1999.dogSituations : List Semantics.Lexical.Noun.Kind.Generics.Situation

Ten situations: 8 with a barking dog, 2 with a sleeping dog.

Equations

• Phenomena.Generics.Studies.Cohen1999.dogSituations = List.map (fun (n : ℕ) => { id := n }) (List.range 10)

def Phenomena.Generics.Studies.Cohen1999.isDog : Semantics.Lexical.Noun.Kind.Generics.Restrictor

Equations

• Phenomena.Generics.Studies.Cohen1999.isDog x✝ = true

def Phenomena.Generics.Studies.Cohen1999.barks : Semantics.Lexical.Noun.Kind.Generics.Scope

Equations

• Phenomena.Generics.Studies.Cohen1999.barks s = decide (s.id < 8)

def Phenomena.Generics.Studies.Cohen1999.birdSituations : List Semantics.Lexical.Noun.Kind.Generics.Situation

Birds: 80 flying, 20 non-flying (penguins, ostriches).

Equations

• Phenomena.Generics.Studies.Cohen1999.birdSituations = List.map (fun (n : ℕ) => { id := n }) (List.range 100)

def Phenomena.Generics.Studies.Cohen1999.isBird : Semantics.Lexical.Noun.Kind.Generics.Restrictor

Equations

• Phenomena.Generics.Studies.Cohen1999.isBird x✝ = true

def Phenomena.Generics.Studies.Cohen1999.flies : Semantics.Lexical.Noun.Kind.Generics.Scope

Equations

• Phenomena.Generics.Studies.Cohen1999.flies s = decide (s.id < 80)

theorem Phenomena.Generics.Studies.Cohen1999.traditional_true_majority_implies_cohen (situations : List Semantics.Lexical.Noun.Kind.Generics.Situation) (normal : Semantics.Lexical.Noun.Kind.Generics.NormalcyPredicate) (restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor) (scope : Semantics.Lexical.Noun.Kind.Generics.Scope) (_hTrad : Semantics.Lexical.Noun.Kind.Generics.traditionalGEN situations normal restrictor scope = true) (hMajority : Semantics.Lexical.Noun.Kind.Generics.prevalence situations restrictor scope > 1 / 2) : cohenGEN situations restrictor scope = true

When traditional GEN is true (all normal restrictor-cases satisfy scope), and the normal+restrictor cases are a majority, Cohen's GEN is also true.

This shows agreement in the typical case: the traditional analysis with a well-chosen normalcy predicate yields the same truth value as the probability-based analysis.

Cohen's Advantage over Traditional GEN

Traditional GEN has a hidden normalcy parameter that does all the explanatory work. Cohen's probability-based GEN eliminates this parameter: the threshold 0.5 is fixed, and the truth value is determined by observable prevalence.

However, Cohen's approach faces its own challenges:

1. Rare property generics: "Mosquitoes carry malaria" is judged true despite prevalence well below 50%. Cohen must either deny these are true generics or invoke the homogeneity constraint.

2. Conjunctive generics: @cite{nickel-2009}'s "Elephants live in Africa and Asia" shows the majority-based view predicts the wrong truth conditions for conjoined habitat claims.

3. Striking property generics: "Sharks attack swimmers" — low prevalence but judged true. @cite{tessler-goodman-2019}'s RSA account handles this via pragmatic reasoning over priors, not a fixed threshold.

def Phenomena.Generics.Studies.Cohen1999.mosquitoSituations : List Semantics.Lexical.Noun.Kind.Generics.Situation

Mosquitoes: only ~1% carry malaria.

Equations

• Phenomena.Generics.Studies.Cohen1999.mosquitoSituations = List.map (fun (n : ℕ) => { id := n }) (List.range 100)

def Phenomena.Generics.Studies.Cohen1999.isMosquito : Semantics.Lexical.Noun.Kind.Generics.Restrictor

Equations

• Phenomena.Generics.Studies.Cohen1999.isMosquito x✝ = true

def Phenomena.Generics.Studies.Cohen1999.carriesMalaria : Semantics.Lexical.Noun.Kind.Generics.Scope

Equations

• Phenomena.Generics.Studies.Cohen1999.carriesMalaria s = (s.id == 0)

theorem Phenomena.Generics.Studies.Cohen1999.cohen_wrong_on_mosquitoes : 1 / 100 < 1 / 2 ∧ 85 / 100 > 1 / 2

Cohen's prediction conflicts with empirical judgments (@cite{leslie-2008}): "Mosquitos carry malaria" has prevalence ~1/100 but judgment ~85/100 (clearly true). Cohen predicts false (1/100 < 1/2).

def Phenomena.Generics.Studies.Cohen1999.mixedDogSituations : List Semantics.Lexical.Noun.Kind.Generics.Situation

A domain that VIOLATES homogeneity: urban vs rural dogs. Urban dogs bark more (all 5 bark), rural dogs bark less (1 of 5 barks). Overall prevalence = 6/10, but the partition into urban/rural shows different rates (5/5 vs 1/5).

Equations

• Phenomena.Generics.Studies.Cohen1999.mixedDogSituations = List.map (fun (n : ℕ) => { id := n }) (List.range 10)

def Phenomena.Generics.Studies.Cohen1999.allDogs : Semantics.Lexical.Noun.Kind.Generics.Restrictor

Equations

• Phenomena.Generics.Studies.Cohen1999.allDogs x✝ = true

def Phenomena.Generics.Studies.Cohen1999.mixedBarks : Semantics.Lexical.Noun.Kind.Generics.Scope

Equations

• Phenomena.Generics.Studies.Cohen1999.mixedBarks s = if s.id < 5 then true else s.id == 5

def Phenomena.Generics.Studies.Cohen1999.urbanPartition : Semantics.Lexical.Noun.Kind.Generics.Situation → Bool

Equations

• Phenomena.Generics.Studies.Cohen1999.urbanPartition s = decide (s.id < 5)

theorem Phenomena.Generics.Studies.Cohen1999.cohen_fixes_threshold (situations : List Semantics.Lexical.Noun.Kind.Generics.Situation) (restrictor : Semantics.Lexical.Noun.Kind.Generics.Restrictor) (scope : Semantics.Lexical.Noun.Kind.Generics.Scope) : cohenGEN situations restrictor scope = Semantics.Lexical.Noun.Kind.Generics.thresholdGeneric situations restrictor scope (1 / 2)

Cohen's account is itself a special case of the general GEN eliminability result in Comparisons/GenericSemantics.lean. While that theorem shows ANY GEN configuration can be matched by SOME threshold, Cohen's contribution is fixing the threshold at 0.5 and adding the homogeneity presupposition.