Denić et al. (2021): The influence of polarity items on inferential judgments

Participants rated how far a conclusion follows from a premise that differs from it in a superset and a subset verb phrase (saw birds, saw doves), inside ten environments the paper classes as upward entailing, downward entailing, non-monotone (exactly 12, only 12) or doubly negative, a downward-entailing operator inside another. A negative polarity item in the premise lowered the directional rating of the non-monotone environments in all four experiments and the meta-analysis, and the positive polarity item some raised that of the doubly negative ones in Experiment 3 and the meta-analysis (§8.1); the printed results are in Data/Experiments/DenicEtAl2021.json. Here each environment denotes a function of its verb phrase built from the substrate's quantifiers, from which the paper's classes and valid inferences follow. The meaning route of §10.2 combines Kadmon and Landman's domain widening for any with Chater and Oaksford's probabilistic semantics of quantified sentences.

Main statements

• Environment.classifies_iff: the class the paper gives each environment is the one its denotation has, the doubly negative ones being upward entailing with the item in a downward-entailing constituent (§3.1.2, §5).
• monotone_wideSome: on its wide-scope reading some sits in an upward-entailing position whatever the environment, the confound of (9) and (27).
• cond_eq_one_of_widen, exists_cond_lt_one_of_widen: widening the domain of the object, (30), weakens the bridge premise of the probabilistic downward inference (33), and strictly so for some believer, (34).

Implementation notes

• The English fragment gives many no reading; a model fixes one of Partee's cardinal and proportional readings, and every result holds for both. Few is the fragment's proportional reading.
• Only 12 is its presupposition and assertion together, at least and at most twelve, which is exactly 12 (Environment.denotation_only12); the licensing context is still focus only, downward entailing modulo the presupposition. Under exactly 12 the substrate licenses no polarity item, although the paper finds them acceptable there to a degree (§3).
• No alien spent a year without seeing birds is no over the aliens and the year-spenders who did not see birds.

References

• [DHRC21]
• [Par89]
• [KL93]
• [CO99]

The environments

def DenicEtAl2021.manyReadings {ι : Type u_1} : Set (Quantifier.GQ ι)

The readings of many put at least a contextual number, or at least a contextual proportion, of the restrictor in the scope.

Equations

• DenicEtAl2021.manyReadings = Set.range Quantifier.GQ.atLeast ∪ Set.range fun (p : ℕ × ℕ) => (Quantifier.NumberTree.threshold p.1 p.2).toGQ

theorem DenicEtAl2021.scopeMonotone_of_mem_manyReadings {ι : Type u_1} [Finite ι] {q : Quantifier.GQ ι} (h : q ∈ manyReadings) : q.ScopeMonotone

structure DenicEtAl2021.Model (ι : Type u_1) : Type u_1

A model fixes what the stimuli's nouns and adjectives are true of and the reading of many.

• alien : ι → Prop

  The aliens.

• purpleAlien : ι → Prop

  The purple alien, the restrictor of the purple alien.

• hairy : ι → Prop

  The hairy individuals.

• spentYear : ι → Prop

  The individuals who spent a year on Earth.

• many : Quantifier.GQ ι

  The reading of many.

• many_mem : self.many ∈ manyReadings

  The reading of many is one the context leaves open.

def DenicEtAl2021.Environment.host {ι : Type u_1} : Environment → (ι → Prop) → ι → Prop

The constituent hosting the item, as a function of its verb phrase, is the negated verb phrase in the negative and doubly negative environments and the verb phrase itself elsewhere.

Equations

• DenicEtAl2021.Environment.negative.host = compl
• DenicEtAl2021.Environment.everyNot.host = compl
• DenicEtAl2021.Environment.noWithout.host = compl
• x✝.host = id

def DenicEtAl2021.Environment.outer {ι : Type u_1} (m : Model ι) : Environment → (ι → Prop) → Prop

The rest of the sentence, as a function of the host constituent.

Equations

• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.positive = Quantifier.GQ.the m.purpleAlien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.negative = Quantifier.GQ.the m.purpleAlien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.every = Quantifier.GQ.every m.alien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.many = m.many m.alien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.no = Quantifier.GQ.no m.alien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.few = Quantifier.GQ.few m.alien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.exactly12 = Quantifier.GQ.exactly 12 m.alien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.only12 = (Quantifier.GQ.atLeast 12 ⊓ Quantifier.GQ.atMost 12) m.alien
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.everyNot = fun (V : ι → Prop) => Quantifier.GQ.every (m.alien ⊓ V) m.hairy
• DenicEtAl2021.Environment.outer m DenicEtAl2021.Environment.noWithout = fun (V : ι → Prop) => Quantifier.GQ.no m.alien (m.spentYear ⊓ V)

def DenicEtAl2021.Environment.denotation {ι : Type u_1} (e : Environment) (m : Model ι) : (ι → Prop) → Prop

The truth of an environment's sentence as a function of its verb phrase.

Equations

• e.denotation m = DenicEtAl2021.Environment.outer m e ∘ e.host

theorem DenicEtAl2021.Environment.denotation_only12 {ι : Type u_1} (m : Model ι) : only12.denotation m = exactly12.denotation m

def DenicEtAl2021.Environment.Valid (e : Environment) : Direction → Prop

An inference is valid in an environment when it holds in every finite model, so the subset-to-superset inference is valid when the denotation is always monotone and the converse when it is always antitone.

Equations

• e.Valid DenicEtAl2021.Direction.subsetToSuperset = ∀ (ι : Type) [Fintype ι] (m : DenicEtAl2021.Model ι), Monotone (e.denotation m)
• e.Valid DenicEtAl2021.Direction.supersetToSubset = ∀ (ι : Type) [Fintype ι] (m : DenicEtAl2021.Model ι), Antitone (e.denotation m)

def DenicEtAl2021.Environment.Classifies (e : Environment) : Monotonicity → Prop

An environment's denotation classes it as upward entailing when the subset-to-superset inference is valid and the item's host is monotone, as doubly negative when that inference is valid and the host is antitone, as downward entailing when the converse is valid, and as non-monotone when neither is.

Equations

• e.Classifies DenicEtAl2021.Monotonicity.ue = (e.Valid DenicEtAl2021.Direction.subsetToSuperset ∧ ∀ (ι : Type), Monotone e.host)
• e.Classifies DenicEtAl2021.Monotonicity.dn = (e.Valid DenicEtAl2021.Direction.subsetToSuperset ∧ ∀ (ι : Type), Antitone e.host)
• e.Classifies DenicEtAl2021.Monotonicity.de = e.Valid DenicEtAl2021.Direction.supersetToSubset
• e.Classifies DenicEtAl2021.Monotonicity.nm = (¬e.Valid DenicEtAl2021.Direction.subsetToSuperset ∧ ¬e.Valid DenicEtAl2021.Direction.supersetToSubset)

theorem DenicEtAl2021.Environment.valid_subsetToSuperset {e : Environment} (he : (environments e).monotonicity = Monotonicity.ue ∨ (environments e).monotonicity = Monotonicity.dn) : e.Valid Direction.subsetToSuperset

theorem DenicEtAl2021.Environment.valid_supersetToSubset {e : Environment} (he : (environments e).monotonicity = Monotonicity.de) : e.Valid Direction.supersetToSubset

def DenicEtAl2021.Environment.thirteen : Model (Fin 13)

The countermodel has thirteen aliens, none hairy, all on Earth for a year, the first one purple, and reads many as at least one.

Equations

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

theorem DenicEtAl2021.Environment.not_valid_subsetToSuperset {e : Environment} (he : (environments e).monotonicity = Monotonicity.de ∨ (environments e).monotonicity = Monotonicity.nm) : ¬e.Valid Direction.subsetToSuperset

theorem DenicEtAl2021.Environment.not_valid_supersetToSubset {e : Environment} (he : (environments e).monotonicity ≠ Monotonicity.de) : ¬e.Valid Direction.supersetToSubset

theorem DenicEtAl2021.Environment.classifies_iff (e : Environment) (k : Monotonicity) : e.Classifies k ↔ (environments e).monotonicity = k

The paper's classification of the ten environments, §3.1.2 and §5, is the one their denotations give. Only the subset-to-superset inference is valid in the upward-entailing and doubly negative ones, only the converse in the downward-entailing ones, neither in the non-monotone ones, and a doubly negative environment hosts the item in an antitone constituent.

Licensing

def DenicEtAl2021.Environment.licensingContext : Environment → Option PolarityItem.LicensingContext

The licensing context of the narrowest operator over the item that the substrate counts as downward entailing is negation, no, few, without, or focus only, downward entailing modulo its presupposition; the upward-entailing environments and exactly 12 have none.

Equations

• DenicEtAl2021.Environment.negative.licensingContext = some PolarityItem.LicensingContext.negation
• DenicEtAl2021.Environment.everyNot.licensingContext = some PolarityItem.LicensingContext.negation
• DenicEtAl2021.Environment.no.licensingContext = some PolarityItem.LicensingContext.nobody
• DenicEtAl2021.Environment.few.licensingContext = some PolarityItem.LicensingContext.few
• DenicEtAl2021.Environment.noWithout.licensingContext = some PolarityItem.LicensingContext.withoutClause
• DenicEtAl2021.Environment.only12.licensingContext = some PolarityItem.LicensingContext.onlyFocus
• DenicEtAl2021.Environment.positive.licensingContext = none
• DenicEtAl2021.Environment.every.licensingContext = none
• DenicEtAl2021.Environment.many.licensingContext = none
• DenicEtAl2021.Environment.exactly12.licensingContext = none

theorem DenicEtAl2021.Environment.licenses_of_mem_licensingContext {e : Environment} {c : PolarityItem.LicensingContext} (h : c ∈ e.licensingContext) : c.licenser.Licenses English.PolarityItems.any ∧ c.licenser.Licenses English.PolarityItems.ever ∧ c.licenser.Licenses English.PolarityItems.atAll

Any, ever and at all are licensed in the downward-entailing environments, by the inner operator in the doubly negative ones (§1, §5), and under only 12.

Wide scope of some

def DenicEtAl2021.wideSome {ι : Type u_1} {ε : Type u_2} (f : (ι → Prop) → Prop) (saw : ε → ι → Prop) (N : Set ε) : Prop

On the wide-scope reading of some N in an environment f, (9b) and (27), some member of N is such that f holds of seeing it.

Equations

• DenicEtAl2021.wideSome f saw N = ∃ y ∈ N, f (saw y)

theorem DenicEtAl2021.monotone_wideSome {ι : Type u_1} {ε : Type u_2} (f : (ι → Prop) → Prop) (saw : ε → ι → Prop) : Monotone (wideSome f saw)

On its wide-scope reading some sits in an upward-entailing position, whatever the environment, (9).

theorem DenicEtAl2021.narrow_of_wideSome {ι : Type u_1} {ε : Type u_2} {f : (ι → Prop) → Prop} (hf : Monotone f) {saw : ε → ι → Prop} {N N' : Set ε} (hN : N ⊆ N') (h : wideSome f saw N) : f (KadmonLandman1993.existsInDomain N' saw)

In an upward-entailing environment the wide-scope reading of some doves entails the narrow-scope sentence about birds, so the inference from (26a) to (26b) goes through on reading (27).

Directional ratings

def DenicEtAl2021.Direction.sign : Direction → SignType

The sign with which a direction enters the directional rating.

Equations

• DenicEtAl2021.Direction.subsetToSuperset.sign = 1
• DenicEtAl2021.Direction.supersetToSubset.sign = -1

def DenicEtAl2021.directional : Direction → ℝ → ℝ

The directional rating keeps a subset-to-superset rating and reverses a superset-to-subset one, in percent.

Equations

• DenicEtAl2021.directional DenicEtAl2021.Direction.subsetToSuperset x✝ = x✝
• DenicEtAl2021.directional DenicEtAl2021.Direction.supersetToSubset x✝ = 100 - x✝

theorem DenicEtAl2021.directional_add (d : Direction) (r b : ℝ) : directional d (r + b) = directional d r + ↑d.sign * b

A response bias enters the directional rating with the sign of the direction, so a yes-bias common to both directions cancels in their sum (§3.2).

The meaning route

On the probabilistic semantics, No aliens saw birds says that a random alien saw birds with probability zero, (33), and the downward inference needs seeing doves to make seeing birds certain.

theorem DenicEtAl2021.cond_eq_zero_of_cond_eq_one {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s e u : Set Ω} (hs : MeasurableSet s) (he : MeasurableSet e) (hu : MeasurableSet u) (h₁ : μ[e | s] = 0) (h₂ : μ[e | u] = 1) : μ[u | s] = 0

In the probabilistic downward inference (33), if e has probability zero given s and probability one given u, then u has probability zero given s.

theorem DenicEtAl2021.cond_eq_one_of_widen {Ω : Type u_1} {ε : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {saw : ε → Set Ω} {birds anyBirds : Set ε} {D : Set Ω} (h : birds ⊆ anyBirds) (hD : μ[KadmonLandman1993.existsInDomain birds saw | D] = 1) : μ[KadmonLandman1993.existsInDomain anyBirds saw | D] = 1

Since any widens the domain of the object, (30), seeing doves makes seeing any birds certain whenever it makes seeing birds certain, (34). So the bridge premise of the probabilistic downward inference (33) from No aliens saw birds (cond_eq_zero_of_cond_eq_one) gives that of the inference from No aliens saw any birds.

theorem DenicEtAl2021.exists_cond_lt_one_of_widen : ∃ (saw : Fin 2 → Set (Fin 2)) (doves : Set (Fin 2)) (birds : Set (Fin 2)) (anyBirds : Set (Fin 2)), birds ⊆ anyBirds ∧ MeasureTheory.Measure.count[KadmonLandman1993.existsInDomain birds saw | KadmonLandman1993.existsInDomain doves saw] < 1 ∧ MeasureTheory.Measure.count[KadmonLandman1993.existsInDomain anyBirds saw | KadmonLandman1993.existsInDomain doves saw] = 1

A believer may take an individual to have seen a dove outside the plain domain of birds, so that seeing doves makes seeing any birds certain but not seeing birds, the room (34) leaves for any to help.