@cite{kao-etal-2014-hyperbole} — Metaphor @cite{kao-etal-2014-metaphor} #

"Formalizing the Pragmatics of Metaphor Understanding" Proceedings of the Annual Meeting of the Cognitive Science Society, 36, 719-724

The Model #

Domain: "He is a whale" metaphor. 2 categories (whale, person) × 2³ feature combinations (large, graceful, majestic) = 16 world states. 2 utterances (= categories). 3 QUDs (= features).

L0: L0(w|u) = P(features|category) if category = u, 0 otherwise
S1: QUD-projected rpow: S1(u|g,w) ∝ [Σ_{w': g(w')=g(w)} L0(w'|u)]^α
L1: L1(w|u) ∝ P(cat) · P(features|cat) · Σ_g P(g) · S1(u|g,w)

Parameters: α = 3, P(whale) = 1/100, P(person) = 99/100

Qualitative Findings #

#	Finding	Theorem
1	P(person \| "whale") > P(whale \| "whale")	`nonliteral`
2	P(large=T \| "whale") > P(large=F \| "whale")	`feature_large`
3	P(graceful=T \| "whale") > P(graceful=F \| "whale")	`feature_graceful`
4	P(majestic=T \| "whale") > P(majestic=F \| "whale")	`feature_majestic`
5	Specific QUD → higher P(large=T) than vague QUD	`context_sensitivity`
6	P(person \| "person") > P(whale \| "person")	`literal_correct`

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inductive Phenomena.Nonliteral.Metaphor.KaoEtAl2014.Finding :

Type

The 6 qualitative findings from @cite{kao-etal-2014-hyperbole}. Each model of metaphor should formalize and prove all 6 findings.

nonliteral : Finding
Hearing "whale" about a person, the listener infers the referent is a person, not literally a whale.
feature_large : Finding
Metaphor elevates the "large" feature above its prior.
feature_graceful : Finding
Metaphor elevates the "graceful" feature above its prior.
feature_majestic : Finding
Metaphor elevates the "majestic" feature above its prior.
context_sensitivity : Finding
A specific QUD ("Is he large?") raises P(large=T) higher than a vague QUD ("What is he like?").
literal_correct : Finding
Hearing "person", the listener correctly infers the referent is a person.

Instances For

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instDecidableEqFinding :

DecidableEq Finding

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

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqFinding :

BEq Finding

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

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

Finding → Finding → Bool

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

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

Finding → ℕ → Std.Format

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instReprFinding :

Repr Finding

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

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def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.findings :

List Finding

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inductive Phenomena.Nonliteral.Metaphor.KaoEtAl2014.Cat :

Type

Categories: whale (metaphor vehicle) and person (literal referent). Categories double as utterance types.

whale : Cat
person : Cat

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instDecidableEqCat :

DecidableEq Cat

Equations

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

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqCat :

BEq Cat

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Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqCat = { beq := Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqCat.beq }

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def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqCat.beq :

Cat → Cat → Bool

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Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqCat.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instReprCat.repr :

Cat → ℕ → Std.Format

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instReprCat :

Repr Cat

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Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instReprCat = { reprPrec := Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instReprCat.repr }

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inductive Phenomena.Nonliteral.Metaphor.KaoEtAl2014.Goal :

Type

QUDs: which feature is the speaker trying to communicate?

large : Goal
graceful : Goal
majestic : Goal

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instDecidableEqGoal :

DecidableEq Goal

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

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instBEqGoal :

BEq Goal

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

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

Goal → Goal → Bool

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

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

Goal → ℕ → Std.Format

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instReprGoal :

Repr Goal

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

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@[reducible, inline]

abbrev Phenomena.Nonliteral.Metaphor.KaoEtAl2014.World :

Type

World = category × large × graceful × majestic.

Equations

Phenomena.Nonliteral.Metaphor.KaoEtAl2014.World = (Phenomena.Nonliteral.Metaphor.KaoEtAl2014.Cat × Bool × Bool × Bool)

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instFintypeCat :

Fintype Cat

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instance Phenomena.Nonliteral.Metaphor.KaoEtAl2014.instFintypeGoal :

Fintype Goal

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def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.featurePrior (c : Cat) (large graceful majestic : Bool) :

ℝ

Feature prior P(large, graceful, majestic | category). Unnormalized counts (×10000) from Experiment 1b / memo code.

Equations

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def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.catPrior :

Cat → ℝ

Category prior. Unnormalized: whale = 1, person = 99.

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def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.meaning (_q : Goal) (u : Cat) (w : World) :

ℝ

L0 meaning: P(features|category) when category matches utterance, 0 otherwise. The feature prior is baked into L0 following RSAConfig convention.

Equations

Phenomena.Nonliteral.Metaphor.KaoEtAl2014.meaning _q u w = if (u == w.1) = true then Phenomena.Nonliteral.Metaphor.KaoEtAl2014.featurePrior w.1 w.2.1 w.2.2.1 w.2.2.2 else 0

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

Bool

Project a world onto the QUD-relevant feature.

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

ℝ

Sum L0 over the QUD equivalence class of w under goal q.

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noncomputable def Phenomena.Nonliteral.Metaphor.KaoEtAl2014.cfg (goalPrior : Goal → ℝ) (hp : ∀ (g : Goal), 0 ≤ goalPrior g) :

RSA.RSAConfig Cat World

@cite{kao-etal-2014-hyperbole} metaphor model, parametric in goal prior.

S1 score is rpow(projected_L0, α) — the paper's Eq. 5 without utterance cost. This directly encodes the paper's equations and lets rsa_predict handle the interval arithmetic.

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