van Tiel, Franke and Sauerland (2021): Probabilistic Pragmatics Explains Gradience and Focality in Natural Language Quantification

Van Tiel, Franke and Sauerland compare two semantics of quantity words on production data, the frame — of the circles are red over displays of 432 circles, where production is gradient and peaks inside the range in which a word is true. Generalized quantifier theory gives a word a threshold on the intersection set size, a lower bound for a monotone-increasing word and an upper bound for a monotone-decreasing one; prototype theory gives it a degree of truth that falls off with the distance from a prototype. Each semantics is embedded in a literal and a pragmatic speaker, and the pragmatic speaker over thresholds explains the data as well as the prototype models, since it prefers the true word with the smaller extension.

Main definitions

• gq, pt: the threshold and the prototype lexicon.
• speakerLit, listenerLit, speakerPrag: the literal speaker, the literal listener and the pragmatic speaker.
• withConfusion: production under imprecise number representation.
• Word.monotonicity: the monotonicity Experiment 2 assigned a word, which fixes the side of its threshold.

Main results

• speakerLit_apply_singleton_eq_zero, speakerLit_real_eq_of_eq: a literal speaker never produces a false word and is indifferent among true words of equal salience.
• speakerPrag_real_lt_of_card_lt: the pragmatic speaker prefers the true word with the smaller extension, which is where focality comes from.
• gq_no_entailment, some_few_no_entailment: some and few compete without entailment.
• monotone_gq, antitone_gq: a word with a lower bound is monotone in the intersection set size, and a word with an upper bound antitone.
• monotonicity_eq_decreasing_iff: on the words the English fragment shares with the study, Experiment 2's coding agrees with the scope antitonicity of the fragment's readings.
• pt_lt_pt_of_abs_lt: the prototype semantics is gradient by itself.

Implementation notes

States are the intersection set sizes Fin (n + 1) and speakers are kernels from states to words built with Kernel.ofWeights; the literal listener has a uniform prior over states. Thresholds, prototypes, spreads and saliences are parameters, not the fitted posterior values, and the Weber-fraction confusion kernel of Experiment 3 is an arbitrary kernel on states. The words are the paper's seventeen, and the side of a word's threshold is the monotonicity Experiment 2 assigned it, read from Data.Experiments.VanTielEtAl2021. The model comparison of Table 1 and the ratings of Experiment 4 are not formalized. The examples are the rows of Data.Examples.VanTielEtAl2021.

References

• [vTFS21]
• [BC81]
• [FG12]
• [Gri75]
• [Sau04]

@[reducible, inline]

abbrev VanTielEtAl2021.State (n : ℕ) : Type

The states are the intersection set sizes of a display of n circles.

Equations

• VanTielEtAl2021.State n = Fin (n + 1)

@[reducible, inline]

abbrev VanTielEtAl2021.Lexicon (n : ℕ) (M : Type u_2) : Type u_2

A lexical meaning function gives the truth value of a word at a state.

Equations

• VanTielEtAl2021.Lexicon n M = (M → VanTielEtAl2021.State n → ℝ)

def VanTielEtAl2021.gq {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) : Lexicon n M

In the generalized-quantifier lexicon a word is true at the states on its side of its threshold, above it for a monotone-increasing word and below it for a monotone-decreasing one.

Equations

• VanTielEtAl2021.gq mono θ m t = match mono m with | VanTielEtAl2021.Monotonicity.increasing => if θ m ≤ ↑t then 1 else 0 | VanTielEtAl2021.Monotonicity.decreasing => if ↑t ≤ θ m then 1 else 0

noncomputable def VanTielEtAl2021.pt {n : ℕ} {M : Type u_1} (p d : M → ℝ) : Lexicon n M

In the prototype lexicon the degree of truth falls off with the distance from the prototype p, scaled by the spread d.

Equations

• VanTielEtAl2021.pt p d m t = Real.exp (-((↑↑t - p m) / d m) ^ 2)

theorem VanTielEtAl2021.gq_eq_one_or_zero {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) (m : M) (t : State n) : gq mono θ m t = 1 ∨ gq mono θ m t = 0

theorem VanTielEtAl2021.gq_no_entailment {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) {m m' : M} (hm : mono m = Monotonicity.increasing) (hθm : 1 ≤ θ m) (hθm' : θ m ≤ n) (hm' : mono m' = Monotonicity.decreasing) (hθ : θ m' < n) : gq mono θ m' 0 = 1 ∧ gq mono θ m 0 = 0 ∧ gq mono θ m (Fin.last n) = 1 ∧ gq mono θ m' (Fin.last n) = 0

Some and few stand in no entailment relation. With a lower bound for some and an upper bound for few inside the range, few is true and some false of an empty intersection, and the other way round of a full one.

theorem VanTielEtAl2021.monotone_gq {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) {m : M} (h : mono m = Monotonicity.increasing) : Monotone (gq mono θ m)

A word with a lower bound is true at more states as the intersection grows.

theorem VanTielEtAl2021.antitone_gq {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) {m : M} (h : mono m = Monotonicity.decreasing) : Antitone (gq mono θ m)

A word with an upper bound is true at fewer states as the intersection grows.

theorem VanTielEtAl2021.pt_pos {n : ℕ} {M : Type u_1} (p d : M → ℝ) (m : M) (t : State n) : 0 < pt p d m t

theorem VanTielEtAl2021.pt_le_one {n : ℕ} {M : Type u_1} (p d : M → ℝ) (m : M) (t : State n) : pt p d m t ≤ 1

theorem VanTielEtAl2021.pt_lt_pt_of_abs_lt {n : ℕ} {M : Type u_1} (p d : M → ℝ) (m : M) {t t' : State n} (hd : 0 < d m) (h : |↑↑t - p m| < |↑↑t' - p m|) : pt p d m t' < pt p d m t

The prototype semantics is gradient by itself, a state closer to the prototype being truer.

noncomputable def VanTielEtAl2021.extension {M : Type u_1} (n : ℕ) (mono : M → Monotonicity) (θ : M → ℕ) (m : M) : Finset (State n)

The extension of a word is the set of states where it is true.

Equations

• VanTielEtAl2021.extension n mono θ m = {t : VanTielEtAl2021.State n | VanTielEtAl2021.gq mono θ m t = 1}

theorem VanTielEtAl2021.gq_nonneg {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) (m : M) (t : State n) : 0 ≤ gq mono θ m t

theorem VanTielEtAl2021.sum_gq {n : ℕ} {M : Type u_1} (mono : M → Monotonicity) (θ : M → ℕ) (m : M) : ∑ t : State n, gq mono θ m t = ↑(extension n mono θ m).card

Speakers

noncomputable def VanTielEtAl2021.withConfusion {n : ℕ} {M : Type u_1} [MeasurableSpace M] (S : ProbabilityTheory.Kernel (State n) M) (cf : ProbabilityTheory.Kernel (State n) (State n)) : ProbabilityTheory.Kernel (State n) M

Under imprecise number representation a speaker applies its rule at the state the confusion kernel cf represents the true state as.

Equations

• VanTielEtAl2021.withConfusion S cf = S.comp cf

theorem VanTielEtAl2021.withConfusion_id {n : ℕ} {M : Type u_1} [MeasurableSpace M] (S : ProbabilityTheory.Kernel (State n) M) : withConfusion S ProbabilityTheory.Kernel.id = S

Exact number representation changes nothing.

noncomputable def VanTielEtAl2021.speakerLit {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] (L : Lexicon n M) (sal : M → ℝ) : ProbabilityTheory.Kernel (State n) M

The literal speaker produces a word in proportion to its salience and its truth value.

Equations

• VanTielEtAl2021.speakerLit L sal = ProbabilityTheory.Kernel.ofWeights fun (t : VanTielEtAl2021.State n) (m : M) => ENNReal.ofReal (sal m * L m t)

noncomputable def VanTielEtAl2021.listenerLit {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] [MeasurableSingletonClass M] (L : Lexicon n M) : ProbabilityTheory.Kernel M (State n)

The literal listener, with a uniform prior over states, infers a state in proportion to the truth value of the word there.

Equations

• VanTielEtAl2021.listenerLit L = ProbabilityTheory.Kernel.ofWeights fun (m : M) (t : VanTielEtAl2021.State n) => ENNReal.ofReal (L m t)

noncomputable def VanTielEtAl2021.speakerPrag {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] [MeasurableSingletonClass M] (L : Lexicon n M) (sal : M → ℝ) (α : ℝ) : ProbabilityTheory.Kernel (State n) M

The pragmatic speaker with rationality α produces a word in proportion to its salience and the probability that the literal listener recovers the state from it.

Equations

• VanTielEtAl2021.speakerPrag L sal α = ProbabilityTheory.Kernel.ofWeights fun (t : VanTielEtAl2021.State n) (m : M) => ENNReal.ofReal (sal m * ((VanTielEtAl2021.listenerLit L) m).real {t} ^ α)

theorem VanTielEtAl2021.speakerLit_apply_singleton_eq_zero {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] [MeasurableSingletonClass M] (L : Lexicon n M) (sal : M → ℝ) {m : M} {t : State n} (h : L m t = 0) : ((speakerLit L sal) t) {m} = 0

A literal speaker never produces a false word.

theorem VanTielEtAl2021.speakerLit_real_eq_of_eq {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] [MeasurableSingletonClass M] (L : Lexicon n M) (sal : M → ℝ) {m m' : M} {t : State n} (hL : L m t = L m' t) (hs : sal m = sal m') : ((speakerLit L sal) t).real {m} = ((speakerLit L sal) t).real {m'}

A literal speaker is indifferent among words of equal salience and truth value, so that with a threshold semantics its productions are step functions.

theorem VanTielEtAl2021.listenerLit_real_gq {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] [MeasurableSingletonClass M] (mono : M → Monotonicity) (θ : M → ℕ) {m : M} {t : State n} (h : gq mono θ m t = 1) : ((listenerLit (gq mono θ)) m).real {t} = 1 / ↑(extension n mono θ m).card

Under the threshold semantics the literal listener recovers a state from a word true there with the reciprocal of the size of the word's extension.

theorem VanTielEtAl2021.speakerPrag_real_lt_of_card_lt {n : ℕ} {M : Type u_1} [Fintype M] [MeasurableSpace M] [MeasurableSingletonClass M] (mono : M → Monotonicity) (θ : M → ℕ) {sal : M → ℝ} (hsal : ∀ (m : M), 0 < sal m) {α : ℝ} (hα : 0 < α) {m m' : M} {t : State n} (hm : gq mono θ m t = 1) (hm' : gq mono θ m' t = 1) (hs : sal m = sal m') (hcard : (extension n mono θ m).card < (extension n mono θ m').card) : ((speakerPrag (gq mono θ) sal α) t).real {m'} < ((speakerPrag (gq mono θ) sal α) t).real {m}

Among words true at a state and equally salient, the pragmatic speaker prefers the one with the smaller extension, the more informative one, which is focality.

The quantity words of the study

def VanTielEtAl2021.Word.monotonicity (w : Word) : Monotonicity

The monotonicity of a word is the one Experiment 2 assigned it.

Equations

• w.monotonicity = (VanTielEtAl2021.classification w).monotonicity

theorem VanTielEtAl2021.some_few_no_entailment {n : ℕ} (θ : Word → ℕ) (hs : 1 ≤ θ Word.some) (hs' : θ Word.some ≤ n) (hf : θ Word.few < n) : gq Word.monotonicity θ Word.few 0 = 1 ∧ gq Word.monotonicity θ Word.some 0 = 0 ∧ gq Word.monotonicity θ Word.some (Fin.last n) = 1 ∧ gq Word.monotonicity θ Word.few (Fin.last n) = 0

Some and few compete without entailment for any thresholds inside the range.

The English quantity words

def VanTielEtAl2021.Word.toQuantityWord? : Word → Option English.Determiners.QuantityWord

A word of the study is a quantity word of Fragments/English/Determiners.lean when the fragment has it.

Equations

• VanTielEtAl2021.Word.none.toQuantityWord? = some English.Determiners.QuantityWord.none_
• VanTielEtAl2021.Word.few.toQuantityWord? = some English.Determiners.QuantityWord.few
• VanTielEtAl2021.Word.some.toQuantityWord? = some English.Determiners.QuantityWord.some_
• VanTielEtAl2021.Word.half.toQuantityWord? = some English.Determiners.QuantityWord.half
• VanTielEtAl2021.Word.many.toQuantityWord? = some English.Determiners.QuantityWord.many
• VanTielEtAl2021.Word.most.toQuantityWord? = some English.Determiners.QuantityWord.most
• VanTielEtAl2021.Word.all.toQuantityWord? = some English.Determiners.QuantityWord.all
• x✝.toQuantityWord? = none

theorem VanTielEtAl2021.monotonicity_eq_decreasing_iff {w : Word} {q : English.Determiners.QuantityWord} (h : w.toQuantityWord? = some q) : w.monotonicity = Monotonicity.decreasing ↔ ∃ d ∈ Semantics.Denotes.denote q, ∀ (α : Type) [inst : Fintype α], (d α).ScopeAntitone

Experiment 2 coded a word the fragment shares monotone decreasing exactly when the word has a reading antitone in its scope on every finite domain. Half, whose reading is neither monotone nor antitone, was coded increasing.