[Coh99]: 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. The
conditional probability P(Q | P) is prevalenceOn (the proportion of
restrictor-satisfying elements where the scope holds — the analogue of
Rel.edgeDensity), and cohenGEN is prevalenceOn > 1/2. The
division-free, kernel-decidable cross-multiplied form is
thresholdGtOn _ _ _ 1 2; the two agree (cohen_iff_thresholdGt).
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 ([Nic09]) #
[Nic09] 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 Studies/Nickel2009.lean, which states the
divergence as a theorem against cohenGEN.
Cohen's Probability-Based GEN #
The operators are polymorphic over the domain carrier (prevalenceOn is), so
the same cohenGEN applies to situation-based models (here) and entity-based
models ([Nic09]).
Cohen's GEN: a generic "Ps are Q" is true iff the conditional probability
P(Q | P) exceeds 0.5. prevalenceOn domain restrictor scope is that
conditional probability, so Cohen's GEN is prevalenceOn > 1/2.
Equations
- Cohen1999.cohenGEN domain restrictor scope = (Quantification.prevalenceOn domain restrictor scope > 1 / 2)
Instances For
Equations
- Cohen1999.instDecidableCohenGEN domain restrictor scope = id inferInstance
Cohen's GEN agrees with the division-free, kernel-decidable threshold form
thresholdGtOn … 1 2 whenever the restrictor is satisfied somewhere.
Cohen's GEN is proportional majority quantification #
Cohen's θ = 1/2 GEN is, over a non-empty restrictor domain, exactly the
canonical majority quantifier mostOn — the θ = 1/2 threshold is the cutpoint
at the 1 : 1 cell ratio. It therefore inherits Proportional from the GQ
substrate. This is the precise content of "generics as majority quantification"
— and exactly the claim the genericity literature rejects: real generics are
not majority statements (cohen_wrong_on_mosquitoes; [Nic09];
[Les08]; [TG19]). The theorems below state what is true
of Cohen's operator, not of generics in general.
Cohen's GEN over a non-empty restrictor domain is exactly the canonical
majority quantifier mostOn (strictly more R∧S than R∧¬S): θ = 1/2 is
the 1 : 1 cell-ratio cutpoint (thresholdGtOn_one_two_iff_mostOn).
Cohen's majority GEN over the whole (finite) carrier is a proportional
quantifier ([PW06]): its truth depends only on the ratio
|R∩S| : |R∖S|. Inherited from mostOn_univ_proportional via cohen_iff_mostOn,
not re-proved.
This proportionality is a property of Cohen's majority operator, not of
generics in general — the prevalence asymmetry ([Les08];
TesslerGoodman2019.same_prevalence_opposite_endorsement) shows real generic
endorsement is not ratio-determined.
Generic-quantifier interface #
cohenGEN over the whole carrier IS the majority generalized quantifier
Quantification.most_sem, slotting Cohen's absolute reading into the GQ
framework (Quantification.Generic) alongside genNormalcy / genWays.
Cohen's absolute-reading GEN over the whole carrier is exactly the majority
generalized quantifier most_sem — the GQ-interface form of cohenGEN.
The bare P(Q | P) > ½ truth condition only: the homogeneity presupposition
(homogeneous) and the relative reading are not part of it.
Cohen's "always": no exceptions — every restrictor-element satisfies the scope.
For a non-empty restrictor domain this is exactly P(Q | P) = 1; stated as the
decidable universal everyOn rather than the (non-kernel-decidable) ℚ equality.
Equations
- Cohen1999.cohenAlways domain restrictor scope = Quantification.everyOn domain restrictor scope
Instances For
Equations
- Cohen1999.instDecidableCohenAlways domain restrictor scope = id inferInstance
Homogeneity Constraint #
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 — i.e. 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.
Instances For
Examples #
Ten situations: 8 with a barking dog, 2 with a sleeping dog.
Equations
- Cohen1999.dogSituations = (List.map (fun (n : ℕ) => { id := n }) (List.range 10)).toFinset
Instances For
Ten cases: 8 flying, 2 non-flying (penguins, ostriches).
Equations
- Cohen1999.birdSituations = (List.map (fun (n : ℕ) => { id := n }) (List.range 10)).toFinset
Instances For
Cohen GEN vs the relativized universal #
When the relativized universal is true (all normal restrictor-cases satisfy scope) and the restrictor cases are a majority, Cohen's GEN is also true. Agreement in the typical case.
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:
Rare property generics: "Mosquitoes carry malaria" is judged true despite prevalence well below 50%.
Conjunctive generics: [Nic09]'s "Elephants live in Africa and Asia" shows the majority-based view predicts the wrong truth conditions for conjoined habitat claims.
Striking property generics: "Sharks attack swimmers" — low prevalence but judged true. [TG19]'s RSA account handles this via pragmatic reasoning over priors, not a fixed threshold.
Ten cases: only 1 carries malaria (a low-prevalence property).
Equations
- Cohen1999.mosquitoSituations = (List.map (fun (n : ℕ) => { id := n }) (List.range 10)).toFinset
Instances For
Cohen's prediction conflicts with empirical judgments ([Les08]): "Mosquitos carry malaria" has prevalence ~1/100 but judgment ~85/100 (clearly true). Cohen predicts false (1/100 < 1/2).
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
- Cohen1999.mixedDogSituations = (List.map (fun (n : ℕ) => { id := n }) (List.range 10)).toFinset
Instances For
...but homogeneity FAILS: the urban rate (5/5) differs from the overall rate
(6/10), witnessed by cross-multiplication 5·10 ≠ 6·5.