Documentation

Studies.KursatDegen2021

Kursat and Degen (2021): Perceptual difficulty differences predict asymmetry in redundant modification with color and material adjectives #

Kursat and Degen's Perceptual Difficulty Hypothesis. In the continuous-semantics model of Degen and colleagues an adjective is noisy to a degree, and the hypothesis is that the noise reflects how hard it is to see whether an object has the property, so the semantic value of an adjective falls as its property gets harder to verify. The model then predicts that a harder property is mentioned redundantly less often. This file derives the prediction for the critical display of the paper's production experiment, which shows a target, a competitor sharing the redundant property with it, and two distractors sharing the sufficient property with the competitor, and reads the paper's findings off its printed statistics.

Main statements #

redundant_eq_sigmoid: the probability of the redundant expression is a logistic function of the informativeness the redundant adjective adds, less its extra cost.
harder_property_less_redundant: under the hypothesis, of two properties that trade the sufficient and the redundant role the harder is mentioned redundantly less, at every positive rationality and every cost. Material was found harder to verify than colour (material_harder) and was mentioned redundantly less (color_more_redundant).
harder_item_less_redundant: under the hypothesis the same holds of two redundant properties beside one sufficient property, the strong version of the hypothesis, for which the paper found no evidence (no_within_property_effect).

Implementation notes #

The display is reduced to the roles of its two properties, so a colour-redundant trial and a material-redundant trial are one model with the semantic values swapped, and the cost of an expression depends only on how many adjectives it has.
The hypothesis is rendered as a strictly decreasing map from difficulty to semantic value. The paper says only that the noise reflects the difficulty.
An effect counts as found when its printed p-value is below .05.

References #

The critical display of the production experiment #

inductive KursatDegen2021.Object :

A critical trial shows four objects of one kind, the target, a competitor that shares the redundant property with it, and two distractors that share the sufficient property with the competitor.

target : Object
competitor : Object
distractor₁ : Object
distractor₂ : Object

Instances For

@[instance_reducible]

instance KursatDegen2021.instDecidableEqObject :

DecidableEq Object

Equations

KursatDegen2021.instDecidableEqObject x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def KursatDegen2021.instReprObject.repr :

Object → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance KursatDegen2021.instReprObject :

Equations

KursatDegen2021.instReprObject = { reprPrec := KursatDegen2021.instReprObject.repr }

@[instance_reducible]

instance KursatDegen2021.instInhabitedObject :

Inhabited Object

Equations

KursatDegen2021.instInhabitedObject = { default := KursatDegen2021.instInhabitedObject.default }

@[instance_reducible]

instance KursatDegen2021.instFintypeObject :

Fintype Object

Equations

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

@[instance_reducible]

instance KursatDegen2021.instMeasurableSpaceObject :

MeasurableSpace Object

Equations

KursatDegen2021.instMeasurableSpaceObject = ⊤

instance KursatDegen2021.instDiscreteMeasurableSpaceObject :

DiscreteMeasurableSpace Object

inductive KursatDegen2021.Utterance :

A referring expression has the sufficient adjective alone or both adjectives.

sufficient : Utterance
redundant : Utterance

Instances For

@[instance_reducible]

instance KursatDegen2021.instDecidableEqUtterance :

DecidableEq Utterance

Equations

KursatDegen2021.instDecidableEqUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def KursatDegen2021.instReprUtterance.repr :

Utterance → ℕ → Std.Format

Equations

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

Instances For

@[instance_reducible]

instance KursatDegen2021.instReprUtterance :

Equations

KursatDegen2021.instReprUtterance = { reprPrec := KursatDegen2021.instReprUtterance.repr }

@[instance_reducible]

instance KursatDegen2021.instInhabitedUtterance :

Inhabited Utterance

Equations

KursatDegen2021.instInhabitedUtterance = { default := KursatDegen2021.instInhabitedUtterance.default }

@[instance_reducible]

instance KursatDegen2021.instFintypeUtterance :

Fintype Utterance

Equations

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

@[instance_reducible]

instance KursatDegen2021.instMeasurableSpaceUtterance :

MeasurableSpace Utterance

Equations

KursatDegen2021.instMeasurableSpaceUtterance = ⊤

instance KursatDegen2021.instDiscreteMeasurableSpaceUtterance :

DiscreteMeasurableSpace Utterance

def KursatDegen2021.meaning (xs xr : ℝ) :

Utterance → Object → ℝ

The continuous meaning gives the sufficient adjective the semantic value xs and the redundant one xr. The Boolean extension of the sufficient adjective is the target alone, and that of the redundant adjective is the target and the competitor.

Equations

One or more equations did not get rendered due to their size.
KursatDegen2021.meaning xs xr KursatDegen2021.Utterance.sufficient x✝ = DegenEtAl2020.noisy xs (x✝ = KursatDegen2021.Object.target)

Instances For

theorem KursatDegen2021.meaning_nonneg {xs xr : ℝ} (hs0 : 0 ≤ xs) (hs1 : xs ≤ 1) (hr0 : 0 ≤ xr) (hr1 : xr ≤ 1) (u : Utterance) (o : Object) :

0 ≤ meaning xs xr u o

theorem KursatDegen2021.redundant_eq_sigmoid {xs xr α : ℝ} {C : Utterance → ℝ} (hα : 0 < α) (hs0 : 0 < xs) (hs1 : xs ≤ 1) (hr0 : 0 < xr) (hr1 : xr ≤ 1) :

((DegenEtAl2020.S1 α C (meaning xs xr)) Object.target).real {Utterance.redundant} = (α * (Real.log (DegenEtAl2020.gain 2 xs xr) - (C Utterance.redundant - C Utterance.sufficient))).sigmoid

The speaker's probability of the redundant expression for the target is a logistic function of the log of the gain of the redundant adjective, less the extra cost.

theorem KursatDegen2021.redundant_of_boolean_sufficient {xr α : ℝ} {C : Utterance → ℝ} (hα : 0 < α) (hr0 : 0 < xr) (hr1 : xr ≤ 1) :

((DegenEtAl2020.S1 α C (meaning 1 xr)) Object.target).real {Utterance.redundant} = (-(α * (C Utterance.redundant - C Utterance.sufficient))).sigmoid

Beside a Boolean sufficient adjective the redundant one adds no information, and the probability of the redundant expression does not depend on its noise.

theorem KursatDegen2021.redundant_lt_iff {xs xr xr' α : ℝ} {C : Utterance → ℝ} (hα : 0 < α) (hs0 : 0 < xs) (hs1 : xs < 1) (hr0 : 0 < xr) (hr1 : xr ≤ 1) (hr0' : 0 < xr') (hr1' : xr' ≤ 1) :

((DegenEtAl2020.S1 α C (meaning xs xr)) Object.target).real {Utterance.redundant} < ((DegenEtAl2020.S1 α C (meaning xs xr')) Object.target).real {Utterance.redundant} ↔ xr < xr'

Beside one sufficient adjective short of Boolean, the redundant expression is the likelier the less noisy the redundant adjective.

theorem KursatDegen2021.redundant_swap_lt_iff {α : ℝ} {C : Utterance → ℝ} {x₁ x₂ : ℝ} (hα : 0 < α) (h₁0 : 0 < x₁) (h₁1 : x₁ ≤ 1) (h₂0 : 0 < x₂) (h₂1 : x₂ ≤ 1) :

((DegenEtAl2020.S1 α C (meaning x₂ x₁)) Object.target).real {Utterance.redundant} < ((DegenEtAl2020.S1 α C (meaning x₁ x₂)) Object.target).real {Utterance.redundant} ↔ x₁ < x₂

When two adjectives trade the sufficient and the redundant role, the redundant expression is the likelier on the trial whose redundant adjective is the less noisy of the two.

The hypothesis #

The semantic value of an adjective is a strictly decreasing function x of the difficulty of verifying its property, with values strictly between zero and one.

theorem KursatDegen2021.harder_property_less_redundant {D : Type u_1} [LinearOrder D] {x : D → ℝ} {α : ℝ} {C : Utterance → ℝ} (hx : StrictAnti x) (hx01 : ∀ (d : D), x d ∈ Set.Ioo 0 1) (hα : 0 < α) (d d' : D) :

((DegenEtAl2020.S1 α C (meaning (x d) (x d'))) Object.target).real {Utterance.redundant} < ((DegenEtAl2020.S1 α C (meaning (x d') (x d))) Object.target).real {Utterance.redundant} ↔ d < d'

Of two properties that trade the sufficient and the redundant role, the harder to verify is mentioned redundantly less.

theorem KursatDegen2021.harder_item_less_redundant {D : Type u_1} [LinearOrder D] {x : D → ℝ} {α : ℝ} {C : Utterance → ℝ} (hx : StrictAnti x) (hx01 : ∀ (d : D), x d ∈ Set.Ioo 0 1) (hα : 0 < α) (s d d' : D) :

((DegenEtAl2020.S1 α C (meaning (x s) (x d'))) Object.target).real {Utterance.redundant} < ((DegenEtAl2020.S1 α C (meaning (x s) (x d))) Object.target).real {Utterance.redundant} ↔ d < d'

Beside one sufficient property, the harder of two redundant properties is mentioned redundantly less. This is the strong version of the hypothesis, on which difficulty predicts redundancy within a property type.

The findings #

def KursatDegen2021.Bound.Significant :

Bound → Data.Experiments.Decimal → Prop

A printed p-value is below .05 when it is an upper bound of at most .05 or a value under it.

Equations

KursatDegen2021.Bound.below.Significant x✝ = (x✝.toRat ≤ 5 / 100)
KursatDegen2021.Bound.exact.Significant x✝ = (x✝.toRat < 5 / 100)

Instances For

@[instance_reducible]

instance KursatDegen2021.instDecidableSignificant (b : Bound) (p : Data.Experiments.Decimal) :

Decidable (b.Significant p)

Equations

theorem KursatDegen2021.material_harder (e : Norming) (m : DifficultyMeasure) :

0 < (verification e m).beta.toRat ∧ (verification e m).bound.Significant (verification e m).p

Material was harder to verify than colour. In both norming experiments it raised the error rate and the response time.

theorem KursatDegen2021.color_more_redundant :

0 < (redundantUse Predictor.redundantProperty).beta.toRat ∧ (redundantUse Predictor.redundantProperty).bound.Significant (redundantUse Predictor.redundantProperty).p

Colour was mentioned redundantly more often than material.

theorem KursatDegen2021.no_within_property_effect (r : Predictor) :

r ∈ [Predictor.difficulty, Predictor.interaction, Predictor.logRatio] → ¬(redundantUse r).bound.Significant (redundantUse r).p

No measure of difficulty within a property type predicted redundant mention, so the strong version of the hypothesis found no support.