Domain Vagueness #
@cite{kadmon-landman-1993}
Kadmon & Landman (1993) "Any". Linguistics and Philosophy 16: 353–422.
Core Distinction #
K&L distinguish two kinds of quantificational expressions:
- Domain precise: the contextual restriction determines a unique precise domain of quantification (e.g., every owl, no owl).
- Domain vague: the contextual restriction is inherently vague — different precisifications yield different domains (e.g., an owl in generic use).
This distinction explains why generics allow exceptions while universal quantifiers do not: the vague restricting set leaves room for objects that might or might not fall under the generalization.
Formal Apparatus #
A vague restriction ⟨v₀, V⟩ consists of:
v₀: the precise part — properties known to hold of the relevant entitiesV: the precisifications — consistent ways to complete the restriction
Each precisification v ∈ V extends v₀ (i.e., v₀ ⊆ v) and determines
a domain of entities: those satisfying all properties in v.
Widening and Dimensional Universality #
K&L's analysis of FC any relies on widening along a contextual dimension {P, ¬P}. When widening removes both P and ¬P from all precisifications, the quantifier becomes universal with respect to that dimension: objects are not excluded merely for having or lacking property P.
Any CN is dimensionally universal — universal wrt its dimension of widening in every context. This is why almost can modify any owl (dimensionally universal) but not an owl (not dimensionally universal).
A vague restriction on a type of properties.
precise is the set of properties known to hold (v₀ in K&L's notation).
precisifications are the consistent complete extensions of precise.
Every precisification must extend the precise part, and the precise
part itself is always a valid (minimal) precisification.
- precise : Set Property
The precise part: properties definitely in the restriction
- precisifications : Set (Set Property)
The set of precisifications: consistent ways to complete the restriction
- extends_precise (v : Set Property) : v ∈ self.precisifications → self.precise ⊆ v
Every precisification extends the precise part
- precise_mem : self.precise ∈ self.precisifications
The precise part is itself a (minimal) precisification. K&L: v₀ trivially extends itself, so it is always among V.
Instances For
The domain of entities induced by a set of properties and an application function. An entity is in the domain iff it satisfies every property.
Equations
- Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.domainOf props apply = {e : Entity | ∀ P ∈ props, e ∈ apply P}
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A vague restriction is domain precise on a base set iff every precisification determines the same domain of entities.
Every owl and no owl are domain precise: the contextual restriction on "owl" determines a unique set of owls to quantify over.
K&L (164): ⟨v₀, V⟩ is domain precise on B iff for every v ∈ V, D_{v,B} = D_{v₀,B}.
Equations
- One or more equations did not get rendered due to their size.
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A vague restriction is domain vague iff it is not domain precise: some precisifications yield different domains.
Generic NPs like an owl are domain vague: different precisifications of "normalcy for owls" yield different sets of relevant owls. This is why generics tolerate exceptions — it is always possible that an apparent counterexample falls outside the domain under the "right" precisification.
Equations
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A trivially precise restriction: when every precisification equals the precise part, the restriction is domain precise. This is the case for standard universal quantifiers like every and no.
Remove a property and its negation from a vague restriction.
K&L (174): ⟦X_A ∼ {P, ¬P}⟧_c is the result of removing every property S except for P and ¬P from ⟦X_A⟧_c and from all its precisifications.
Here we model "removing a dimension" by filtering out a specific
property from the precise part and all precisifications. In the
full K&L treatment, both P and ¬P are removed; we parameterize
by a predicate onDimension that identifies properties on the
dimension to be removed.
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- One or more equations did not get rendered due to their size.
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Widening weakens the restriction: the precise part of the widened restriction is a subset of the original precise part.
Widening expands the domain: if we remove properties from the restriction, fewer constraints means more entities qualify.
A quantificational expression Q ↾ X_A(A) is universal with respect to a pair of properties {P, ¬P} in context c iff removing P and ¬P from the restriction leaves the quantifier ranging over all objects in ⟦A⟧_c.
K&L (175): Q ↾ X_A(A) is universal wrt {P, ¬P} in c iff ⟦Q ↾ X_A ∼ {P, ¬P}(A)⟧_c = ⟦∀(A)⟧_c or ⟦¬∃(A)⟧_c.
We formalize the positive case: after removing properties on the dimension, every entity in the base denotation is in the domain.
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- One or more equations did not get rendered due to their size.
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A pair of properties {P, ¬P} is non-trivial on a base set iff some entity in the base satisfies P and some entity satisfies ¬P.
K&L (176): {P, ¬P} is non-trivial on A iff both P and ¬P contain some object in A. This rules out vacuous dimensions where universality is trivially achieved.
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- One or more equations did not get rendered due to their size.
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An NP is dimensionally universal if there exists some non-trivial dimension wrt which it is universal.
K&L (177): Q ↾ X_A(A) is dimensionally universal iff for every context c, there is a pair {P, ¬P} which is non-trivial on ⟦A⟧_c such that Q ↾ X_A(A) is universal wrt {P, ¬P} in c.
any CN is dimensionally universal (universal wrt its dimension of widening), while generic a CN is not. This explains why almost can modify any owl but not an owl: almost requires a quantifier that is (dimensionally) universal.
Equations
- One or more equations did not get rendered due to their size.
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Theorem (K&L §4.3): Widening creates universality wrt its dimension.
After widening a vague restriction along a dimension, the widened restriction is universal wrt that dimension. This is because widening removes all properties on the dimension from both the precise part and all precisifications — so no entity is excluded on the basis of that dimension.
This is the core of K&L's explanation for why any CN is dimensionally universal: any widens along a contextual dimension, and widening along a dimension automatically creates universality wrt that dimension.
Corollary: any CN is dimensionally universal (given a non-trivial dimension of widening).
If a vague restriction X has a non-trivial dimension along which widening is applied, and the base denotation satisfies the non-dimensional properties, then the widened restriction is dimensionally universal.
DE on a constant parameter: a multi-place function is DE in one argument when another argument is held constant.
K&L use this pattern twice:
- Adversative predicates are "DE on a constant perspective" (§3.3): sorry(x, p, A) is DE in A when perspective p is held constant
- Conditionals are "DE on a constant restriction" (§3.5): if_R(A, C) is DE in A when the implicit restriction R is held constant
This is the same structure as Kratzer's modal base: necessity(f, g, p) is DE in p when the modal base f and ordering source g are held constant.
Equations
- Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.IsDE_OnConstant f param = Antitone (f param)
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Connecting Domain Restriction to Domain Vagueness #
DomainRestriction.lean formalizes domain-precise quantifiers: each
DomainRestrictor E is a single predicate Set E that determines a
unique domain. K&L's VagueRestriction generalizes this: domain-precise
quantifiers are the degenerate case where all precisifications agree.
The bridge theorem below shows that any DomainRestrictor-based quantifier
is trivially domain precise in K&L's sense: the restriction has only one
precisification (the restrictor itself), so all precisifications yield the
same domain. This connects @cite{ritchie-schiller-2024}'s DDRP apparatus
to K&L's domain vagueness distinction.
Lift a single DomainRestrictor into a trivially precise
VagueRestriction. The precise part is the singleton set containing
the restrictor, and the only precisification equals the precise part.
This embedding shows that standard domain-restricted quantifiers (every, no) are the degenerate case of K&L's vague restriction framework — they have exactly one precisification.
Equations
- Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.VagueRestriction.ofRestrictor C = { precise := {C}, precisifications := {{C}}, extends_precise := ⋯, precise_mem := ⋯ }
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A DomainRestrictor-based vague restriction is domain precise.
This is the formal bridge: standard quantifiers (which use a single
DomainRestrictor) are always domain precise in K&L's sense, because
their restriction has only one precisification. This is why every owl
and no owl do not tolerate exceptions — there is a unique domain to
quantify over, and counterexamples cannot be excluded by shifting to a
different precisification.
Generic Quantification (K&L §4.1.1) #
K&L propose that generic statements involve a universal quantifier (∀) restricted by a vague set of properties. For example, "An owl hunts mice" means roughly:
∀ ↾ X_owl(Owl)(Hunts mice)
where X_owl is a vague restriction defining "normality" for owls. The
quantifier ranges over all entities satisfying a precisification of X_owl
that is compatible with the CN denotation.
This explains two key properties of generics:
Exception tolerance: Because the restriction is vague, apparent counterexamples can always be excluded under some precisification. This is why "A poodle gives live birth" is true despite male poodles.
Incompatibility with almost: Generic NPs (an owl) are domain vague, so there is no unique precise domain to be "almost all of." Almost requires a domain-precise or dimensionally-universal NP.
The generic operator as a vague universal: a generic sentence
GEN(CN)(VP) is true under precisification v iff every entity
in the v-induced domain satisfies the VP scope.
K&L (p. 407, eq. (159)): ∀ ↾ X_owl(Owl)(Hunts mice)
Equations
- Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.genericTrue _X apply scope v = ∀ e ∈ Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.domainOf v apply, scope e
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A generic is true on its standard (supervaluationist) semantics iff it is true under EVERY precisification.
K&L (p. 411): The generic quantifier ∀ ↾ X_A(A) is domain precise iff for every context, the restriction determines a unique domain. For domain-vague generics, truth requires all precisifications to agree.
Equations
- Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.genericSuperTrue X apply scope = ∀ v ∈ X.precisifications, Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.genericTrue X apply scope v
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A generic is true on its subvaluationist semantics iff it is true under SOME precisification.
This captures the exception-tolerant reading: the generic holds as long as there exists a "way of being normal" under which all entities in the domain satisfy the scope.
Equations
- Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.genericSubTrue X apply scope = ∃ v ∈ X.precisifications, Phenomena.Polarity.Studies.KadmonLandman1993.Apparatus.genericTrue X apply scope v
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Exception tolerance from domain vagueness.
If a restriction is domain vague, then for any entity e, there exists a precisification under which e is outside the domain. This means apparent counterexamples to the generic are always "excusable" — they can be excluded by choosing the right precisification.
K&L (p. 409): "When we encounter objects that do not fall under the generalization, there is always the possibility that they are not among the objects that the generalization is supposed to apply to."
Formalized: if the restriction is domain vague, there exist two precisifications with different domains, so some entity is in one domain but not the other.
Domain-precise quantifiers have unique domains per precisification.
When a restriction is domain precise, all precisifications yield the same domain as the precise part. This means the generic degenerates into a standard universal: there is one domain, and truth is checked against it. No exceptions are tolerated.
K&L (p. 412): "Every owl and no owl are domain precise universal quantifiers. Every owl hunts mice and no owl hunts mice express in every context a generalization about all objects in D_{X_owl,c}."
Total widening creates true universality (K&L p. 419).
If widening is total — it eliminates ALL properties from the restriction (both precise part and precisifications) — then the widened restriction imposes no constraints at all. The quantifier becomes a standard universal over the entire base denotation.
K&L: "if in a context widening is total and eliminates all properties from the restriction, then any CN becomes not only universal wrt some pair {P, ¬P}, but also truly universal."