Documentation

Linglib.Phenomena.Nonliteral.Hyperbole.KaoEtAl2014

@cite{kao-etal-2014-hyperbole} @cite{kao-etal-2014-hyperbole} #

"Nonliteral understanding of number words" PNAS 111(33): 12002-12007

Experimental Design #

Participants heard price utterances ("It cost $X") about items with known typical prices (electric kettles, laptops, watches). Three experiments measured: (3a) price priors, (3b) affect priors conditional on price, (3c) listener interpretations of literal, hyperbolic, and round/sharp utterances.

The Model #

Speakers may use literally false utterances to convey affective information when the listener is uncertain about the speaker's communicative goal.

S1(u|s,a,g) ∝ exp(α · [ln L0(g(s,a)|u) - C(u)]) [Eq. 5,7]

where g composes precision (exact vs approximate) with relevance (price, affect, or both), yielding 5 distinct goals.

L1 marginalizes over goals:

L1(s,a|u) ∝ P_S(s) · P_A(a|s) · Σ_g P_G(g) · S1(u|s,a,g) [Eq. 10]

Grounding #

Price semantics is grounded in HasDegree. The literal meaning of "fifty dollars" is numeralExact 50 price — the price equals $50.

Qualitative Findings #

Any model of hyperbole should account for these 6 findings:

#	Finding	Theorem
1	At the inferred price, the listener infers notable affect	`hyperbole_affect_at_modal`
2	Marginal notable affect > no affect for hyperbolic utterances	`hyperbole_affect`
3	Listener infers speaker's QUD is valence, not price	`hyperbole_qud`
4	Literal utterance → listener infers correct price	`literal_correct`
5	Literal utterance is not interpreted hyperbolically	`literal_not_hyperbolic`
6	Sharp numbers interpreted more precisely than round	`halo_sharp_500`

inductive Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Finding :

The 6 qualitative findings from @cite{kao-etal-2014-hyperbole} Experiments 3a–3c. Each model of hyperbole should formalize and prove all 6 findings.

affect_at_modal : Finding
Hearing "$10,000" for a kettle, the listener infers notable affect at the modal price (the most likely actual price under the posterior).
affect_marginal : Finding
Marginalizing over all prices, notable affect dominates for "$10,000": Σ_s P(s, notable | "$10K") > Σ_s P(s, none | "$10K").
qud_valence : Finding
The listener infers the speaker's communicative goal (QUD) is to express valence/affect rather than to communicate exact price.
literal_correct : Finding
Hearing "$50" for a $50 kettle, the listener infers the correct price: P($50 | "$50") > P($500 | "$50").
literal_not_hyperbolic : Finding
Literal utterances are not interpreted hyperbolically: P($50 | "$50") > P($10K | "$50").
halo_sharp_precise : Finding
Sharp numbers are interpreted more precisely than round numbers: P(exact match | "$501") > P(exact match | "$500").

Instances For

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqFinding :

DecidableEq Finding

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqFinding x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqFinding :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqFinding = { beq := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqFinding.beq }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqFinding.beq :

Finding → Finding → Bool

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqFinding.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprFinding :

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Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprFinding = { reprPrec := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprFinding.repr }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprFinding.repr :

Finding → ℕ → Std.Format

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.findings :

All findings from the paper.

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inductive Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Item :

Item types (Experiments 3a/3b): electric kettles, laptops, watches.

electricKettle : Item
laptop : Item
watch : Item

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprItem :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprItem = { reprPrec := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprItem.repr }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprItem.repr :

Item → ℕ → Std.Format

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqItem :

DecidableEq Item

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqItem x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqItem.beq :

Item → Item → Bool

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqItem.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqItem :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqItem = { beq := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqItem.beq }

inductive Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.PriceState :

Price states: round/sharp pairs {50,51,500,501,...,10000,10001}.

s50 : PriceState
s51 : PriceState
s500 : PriceState
s501 : PriceState
s1000 : PriceState
s1001 : PriceState
s5000 : PriceState
s5001 : PriceState
s10000 : PriceState
s10001 : PriceState

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprPriceState :

Repr PriceState

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprPriceState = { reprPrec := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprPriceState.repr }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprPriceState.repr :

PriceState → ℕ → Std.Format

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqPriceState :

DecidableEq PriceState

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqPriceState x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqPriceState :

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Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqPriceState = { beq := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqPriceState.beq }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqPriceState.beq :

PriceState → PriceState → Bool

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqPriceState.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instHasDegreePriceState :

Core.Scale.HasDegree PriceState

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instHasDegreePriceState = { degree := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.PriceState.value }

inductive Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Affect :

Affect: no affect vs notable affect.

none : Affect
notable : Affect

Instances For

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprAffect :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprAffect = { reprPrec := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprAffect.repr }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprAffect.repr :

Affect → ℕ → Std.Format

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqAffect :

DecidableEq Affect

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqAffect x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqAffect :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqAffect = { beq := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqAffect.beq }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqAffect.beq :

Affect → Affect → Bool

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqAffect.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

Instances For

@[reducible, inline]

abbrev Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.World :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.World = (Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.PriceState × Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Affect)

Instances For

inductive Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Goal :

Communicative goals: 5 = (3 relevance × 2 precision) minus 1 collapse.

price : Goal
valence : Goal
priceValence : Goal
approxPrice : Goal
approxPriceValence : Goal

Instances For

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprGoal :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprGoal = { reprPrec := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprGoal.repr }

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instReprGoal.repr :

Goal → ℕ → Std.Format

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqGoal :

DecidableEq Goal

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instDecidableEqGoal x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqGoal.beq :

Goal → Goal → Bool

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqGoal.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqGoal :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqGoal = { beq := Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instBEqGoal.beq }

instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instFintypePriceState :

Fintype PriceState

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instFintypeAffect :

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instFintypeGoal :

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instance Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.instFintypeItem :

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.pricePrior (item : Item) :

PriceState → ℝ

Price prior P_S (Experiment 3a). Unnormalized.

Equations

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.affectPrior :

PriceState → Affect → ℝ

Affect prior P_A(a|s) (Experiment 3b). Unnormalized (pairs sum to 10000).

Equations

Instances For

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.PriceState.isRound :

PriceState → Bool

Equations

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noncomputable def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.cost :

PriceState → ℝ

C(round) = 0, C(sharp) = 1.

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.cost x✝ = if x✝.isRound = true then 0 else 1

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.costPredicate :

Semantics.Lexical.Numeral.MeasurePredicate PriceState

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.costPredicate = Semantics.Lexical.Numeral.MeasurePredicate.fromHasDegree Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.PriceState "price (USD)"

Instances For

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literalCompositional (u world : PriceState) :

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literal (u world : PriceState) :

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literal u world = Semantics.Lexical.Numeral.numeralExact u.value world

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theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literal_eq_compositional (u world : PriceState) :

literal u world = literalCompositional u world

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literal_uses_degree (u world : PriceState) :

literal u world = (Core.Scale.HasDegree.degree world == u.value)

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.costPredicate_is_hasDegree :

costPredicate.μ = Core.Scale.HasDegree.degree

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.meaning (_q : Goal) (u : PriceState) (w : World) :

L0(s,a|u) = P_A(a|s) if s = u, 0 otherwise. [Eq. 9]

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.meaning _q u w = if (u == w.1) = true then Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.affectPrior w.1 w.2 else 0

Instances For

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.PriceState.round :

PriceState → PriceState

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Affect.toIdx :

Equations

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def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.project (w : World) (q : Goal) :

QUD projection function [Eq. 6]: maps a world to its equivalence-class tag. Two worlds are QUD-equivalent iff project w₁ q = project w₂ q.

Equations

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noncomputable def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.qudProject (q : Goal) (f : World → ℝ) (w : World) :

Sum L0 over the QUD equivalence class of w under goal q. The class is derived from project: {w' | project w' q = project w q}.

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noncomputable def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.cfg (item : Item) :

RSA.RSAConfig PriceState World

@cite{kao-etal-2014-hyperbole} hyperbole model, parametric in item.

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@[reducible, inline]

noncomputable abbrev Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.kettleCfg :

RSA.RSAConfig PriceState World

Equations

Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.kettleCfg = Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.cfg Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.Item.electricKettle

Instances For

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.hyperbole_affect_at_modal :

kettleCfg.L1 PriceState.s10000 (PriceState.s10000 , Affect.notable ) > kettleCfg.L1 PriceState.s10000 (PriceState.s10000 , Affect.none )

At the modal price ($10K), notable affect > no affect. The speaker saying "$10K" about a kettle signals frustration.

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.hyperbole_affect :

(kettleCfg.L1_marginal PriceState.s10000 fun (w : World) => w.2 == Affect.notable) > kettleCfg.L1_marginal PriceState.s10000 fun (w : World) => w.2 == Affect.none

Marginal: notable affect dominates overall for "$10K". Σ_s L1(s, notable | "$10K") > Σ_s L1(s, none | "$10K").

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.hyperbole_qud :

kettleCfg.L1_latent PriceState.s10000 Goal.valence > kettleCfg.L1_latent PriceState.s10000 Goal.price

The listener infers valence QUD over price QUD.

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literal_correct :

kettleCfg.L1 PriceState.s50 (PriceState.s50 , Affect.none ) > kettleCfg.L1 PriceState.s50 (PriceState.s500 , Affect.none )

Hearing "$50", the listener infers $50 > $500.

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.literal_not_hyperbolic :

kettleCfg.L1 PriceState.s50 (PriceState.s50 , Affect.none ) > kettleCfg.L1 PriceState.s50 (PriceState.s10000 , Affect.none )

Literal utterances are not interpreted hyperbolically.

theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.halo_sharp_500 :

(kettleCfg.L1_marginal PriceState.s501 fun (w : World) => w.1 == PriceState.s501) > kettleCfg.L1_marginal PriceState.s500 fun (w : World) => w.1 == PriceState.s500

Sharp "$501" is interpreted more precisely than round "$500".

def Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.formalize :

Finding → Prop

Map each empirical finding to the RSA model prediction that accounts for it.

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theorem Phenomena.Nonliteral.Hyperbole.KaoEtAl2014.all_findings_verified (f : Finding) :

The RSA model accounts for all 6 empirical findings from @cite{kao-etal-2014-hyperbole}.