van Tiel, van Miltenburg, Zevakhina and Geurts (2016): Scalar Diversity

This file formalizes [VTVMZG16], which tests the uniformity assumption of the literature on scalar inferences, that observations about one lexical scale, typically ⟨some, all⟩ or ⟨or, and⟩, generalize to the whole family of scales, Uniform. Two experiments over 43 scales, the rows of Data.Examples.VanTielEtAl2016, find that the rate at which the weaker term gives rise to the upper-bounding inference that the stronger term does not hold ranges from almost never, ⟨content, happy⟩, to always, ⟨cheap, free⟩, with ⟨some, all⟩ an extreme case. A scale is a pair of thresholds on a dimension, the terms denoting the degrees from their threshold up, LexicalScale, and the scalar inference is the weak term strengthened by the negation of the strong one, upperBounded. The paper explains the diversity by distinctness rather than availability: none of four measures of how available the stronger scalemate is predicts the rates, whereas the two measures of how distinct the scalemates are do, the rated semantic distance between the terms and boundedness, whether the stronger term denotes the end point of the dimension, IsBounded, in the sense of [RW04b] and [KMcN05]. On a bounded scale the terms are told apart on formal grounds alone, one denoting an interval and the other an end point, upperBounded_eq_of_isBounded, which is why bounded scales license the inference far more often; in the sample boundedness subsumes the closed grammatical classes, though not in general, ⟨some, most⟩ being open. The two-stage account of the inference of [Sau04] and [Geu10], from the primary inference that the speaker does not believe the stronger alternative to the scalar inference by the speaker's competence, two_stage_inference, is considered as a source of the remaining variance and set aside, since a speaker of She is pretty is plausibly opinionated about beautiful and the inference is nonetheless rare.

Implementation notes

The rates of Experiments 1 and 2, the cloze, frequency, relatedness and distance measures of Experiments 3 and 4, and the mixed model of Table 5, which explains about half of the variance with boundedness the largest factor, are reported in the paper and not formalized; the rows record the scales, their grammatical class and their boundedness. The emotional valence of the stronger term, Appendix B, does not predict the rates either.

References

• [VTVMZG16]
• [Hor72]
• [Hir85]
• [RW04b]
• [KMcN05]
• [Sau04]
• [Geu10]

Lexical scales

structure VanTielEtAl2016.LexicalScale (D : Type u_1) [Preorder D] : Type u_1

A lexical scale over a dimension of degrees: the weaker and the stronger term denote the degrees from a threshold up, the stronger threshold at or above the weaker.

• weakThreshold : D
• strongThreshold : D
• le : self.weakThreshold ≤ self.strongThreshold

def VanTielEtAl2016.LexicalScale.weak {D : Type u_1} [Preorder D] (s : LexicalScale D) : Set D

The denotation of the weaker term.

Equations

• s.weak = Set.Ici s.weakThreshold

def VanTielEtAl2016.LexicalScale.strong {D : Type u_1} [Preorder D] (s : LexicalScale D) : Set D

The denotation of the stronger term.

Equations

• s.strong = Set.Ici s.strongThreshold

theorem VanTielEtAl2016.LexicalScale.strong_subset_weak {D : Type u_1} [Preorder D] (s : LexicalScale D) : s.strong ⊆ s.weak

The stronger term entails the weaker.

def VanTielEtAl2016.LexicalScale.upperBounded {D : Type u_1} [Preorder D] (s : LexicalScale D) : Set D

The upper-bounding inference: the weaker term strengthened by the negation of the stronger, the degrees from the weak threshold up to the strong one.

Equations

• s.upperBounded = s.weak \ s.strong

def VanTielEtAl2016.LexicalScale.IsBounded {D : Type u_1} [Preorder D] (s : LexicalScale D) [OrderTop D] : Prop

A bounded scale: the stronger term denotes the end point of the dimension.

Equations

• s.IsBounded = (s.strongThreshold = ⊤)

theorem VanTielEtAl2016.LexicalScale.upperBounded_eq_Ico {L : Type u_2} [LinearOrder L] (s : LexicalScale L) : s.upperBounded = Set.Ico s.weakThreshold s.strongThreshold

On a linear dimension the inference is the interval from the weak threshold up to the strong one.

theorem VanTielEtAl2016.LexicalScale.strong_eq_singleton_top {E : Type u_2} [PartialOrder E] [OrderTop E] (s : LexicalScale E) (h : s.IsBounded) : s.strong = {⊤}

On a bounded scale the stronger term denotes the end point alone.

theorem VanTielEtAl2016.LexicalScale.upperBounded_eq_of_isBounded {E : Type u_2} [PartialOrder E] [OrderTop E] (s : LexicalScale E) (h : s.IsBounded) : s.upperBounded = s.weak \ {⊤}

On a bounded scale the scalemates are told apart on formal grounds alone: the inference excludes the end point, without inspecting the reach of the stronger term.

Uniformity and the two-stage inference

def VanTielEtAl2016.Uniform {ι : Type u_2} (rate : ι → ℚ) : Prop

The uniformity assumption: every scale gives rise to the inference at the same rate.

Equations

• VanTielEtAl2016.Uniform rate = ∀ (s t : ι), rate s = rate t

theorem VanTielEtAl2016.not_uniform_of_ne {ι : Type u_2} {rate : ι → ℚ} {s t : ι} (h : rate s ≠ rate t) : ¬Uniform rate

Two scales with different rates refute uniformity.

theorem VanTielEtAl2016.two_stage_inference {believesStrong believesNotStrong : Prop} (primary : ¬believesStrong) (competence : believesStrong ∨ believesNotStrong) : believesNotStrong

The two-stage account: from the primary inference, that the speaker does not believe the stronger alternative, the competence assumption, that the speaker is opinionated about it, yields the scalar inference that the speaker believes it false.