Evaluative Measure Semantics for Deadjectival Intensifiers

@cite{nouwen-2024} proposes that deadjectival intensifiers derive their degree function from the evaluative meaning of their adjectival base.

Core Idea

An evaluative adjective (e.g., "horrible") has a measure function μ that assigns high values to degrees that are evaluated negatively.

For "horrible": μ_horrible(d) peaks at extreme degrees (far from the norm). For "pleasant": μ_pleasant(d) peaks at moderate degrees (near the norm).

Intensified Meaning

Simplified from @cite{nouwen-2024} eq. 44–45; the RSA model (§4, eq. 72) uses this direct degree-level intersection:

⟦horribly warm⟧ = λd. warm(d) ∧ horrible(d)

The intensified positive form is the conjunction (intersection) of:

1. The base adjective's positive meaning: d > θ_adj
2. The evaluative measure exceeding its own threshold: μ_eval(d) > θ_eval

Note: the full compositional semantics (eq. 45) applies μ_D to a proposition about the degree, not directly to the degree. This simplification suffices for the RSA pragmatic model.

structure Semantics.Gradability.Intensification.EvaluativeMeasure (max : ℕ) : Type

An evaluative measure function assigns a rational-valued "goodness of fit" score to each degree on a scale.

• form: the adjectival base (e.g., "horrible")
• valence: evaluative valence from the Phenomena layer
• mu: the measure function μ : Nat → ℚ (takes degree's Nat value)

The measure function captures how well a degree matches the evaluative meaning of the base adjective.

• form : String
• valence : Features.EvaluativeValence
• mu : ℕ → ℚ

def Semantics.Gradability.Intensification.muHorrible (max : ℕ) : EvaluativeMeasure max

Evaluative measure for negative-evaluative bases (horrible, terrible, etc.).

μ_horrible(d) = |d - norm|

Peaks at extremes (d = 0 and d = max), lowest at the norm. Negative-evaluative adjectives evaluate extreme degrees as more salient, which is why "horribly warm" targets high degrees.

Equations

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

def Semantics.Gradability.Intensification.muPleasant (max : ℕ) : EvaluativeMeasure max

Evaluative measure for positive-evaluative bases (pleasant, nice, etc.).

μ_pleasant(d) = norm - |d - norm|

Peaks at the norm (middle degrees), lowest at extremes. Positive-evaluative adjectives evaluate moderate degrees as best, which is why "pleasantly warm" targets moderate degrees.

Equations

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

def Semantics.Gradability.Intensification.intensifiedMeaning {max : ℕ} (eval : EvaluativeMeasure max) (d : Core.Scale.Degree max) (θ_adj θ_eval : Core.Scale.Threshold max) : Prop

Intensified positive meaning (simplified from @cite{nouwen-2024} eq. 44–45).

⟦ADV-ly ADJ⟧(d, θ_adj, θ_eval) = (d > θ_adj) ∧ (μ_eval(d) > θ_eval)

The intensified form is the conjunction (intersection) of:

1. The base adjective's positive form: d > θ_adj
2. The evaluative threshold: μ_eval(d) > θ_eval

Equations

• Semantics.Gradability.Intensification.intensifiedMeaning eval d θ_adj θ_eval = (Semantics.Degree.positiveMeaning d θ_adj ∧ eval.mu d.toNat > ↑θ_eval.toNat)

@[implicit_reducible]

instance Semantics.Gradability.Intensification.instDecidableIntensifiedMeaning {max : ℕ} (eval : EvaluativeMeasure max) (d : Core.Scale.Degree max) (θ_adj θ_eval : Core.Scale.Threshold max) : Decidable (intensifiedMeaning eval d θ_adj θ_eval)

Equations

• Semantics.Gradability.Intensification.instDecidableIntensifiedMeaning eval d θ_adj θ_eval = id inferInstance

theorem Semantics.Gradability.Intensification.intensified_implies_positive {max : ℕ} (eval : EvaluativeMeasure max) (d : Core.Scale.Degree max) (θ_adj θ_eval : Core.Scale.Threshold max) (h : intensifiedMeaning eval d θ_adj θ_eval) : Degree.positiveMeaning d θ_adj

Intensified meaning entails the positive form.

If "horribly warm" is true, then "warm" is true. This is because the intensified meaning is a conjunction that includes the positive meaning as one conjunct.

theorem Semantics.Gradability.Intensification.muHorrible_peaks_at_extreme_10 : (muHorrible 10).mu 10 ≥ (muHorrible 10).mu 5

The horrible measure peaks at extremes: μ(max) ≥ μ(norm).

Negative-evaluative adjectives assign highest values to extreme degrees.

theorem Semantics.Gradability.Intensification.muPleasant_peaks_at_norm_10 : (muPleasant 10).mu 5 ≥ (muPleasant 10).mu 10

The pleasant measure peaks at norm: μ(norm) ≥ μ(max).

Positive-evaluative adjectives assign highest values to moderate degrees.

theorem Semantics.Gradability.Intensification.goldilocks_at_extreme_10 : (muHorrible 10).mu 10 ≥ (muPleasant 10).mu 10

Goldilocks structural theorem: at extreme degrees (d = max), the horrible measure exceeds the pleasant measure.

This is the semantic foundation of the Goldilocks effect: extreme degrees are more "horrible" than "pleasant".

theorem Semantics.Gradability.Intensification.goldilocks_at_norm_10 : (muPleasant 10).mu 5 ≥ (muHorrible 10).mu 5

Goldilocks structural theorem (converse): at moderate degrees (d = norm), the pleasant measure exceeds the horrible measure.

Moderate degrees are more "pleasant" than "horrible".

theorem Semantics.Gradability.Intensification.negative_valence_extreme_peak : (muHorrible 10).valence = Features.EvaluativeValence.negative ∧ (muHorrible 10).mu 10 ≥ (muHorrible 10).mu 5

Bridge between evaluative valence and evaluative measure behavior: negative-evaluative measures peak at extremes, positive at the norm.

This connects the Phenomena-layer EvaluativeValence to the Theory-layer EvaluativeMeasure structural properties.

theorem Semantics.Gradability.Intensification.positive_valence_norm_peak : (muPleasant 10).valence = Features.EvaluativeValence.positive ∧ (muPleasant 10).mu 5 ≥ (muPleasant 10).mu 10