Beltrama 2025: Evaluation, thresholds, and practical commitments

@cite{beltrama-2025}

Beltrama, A. (2025). Evaluation, thresholds, and practical commitments: the grammar of adjectival mildness. Natural Language Semantics 33, 169--205.

Core claims

1. MPAs (decent, acceptable, adequate) are a novel class of gradable predicates with a hybrid profile overlapping both relative and absolute adjectives.

2. MPAs encode a necessity standard: the minimum value required for an object's pursuit to be possible given the norms and circumstances in the current world. This is a functional standard (cf. @cite{kagan-alexeyenko-2011}), not a distributional (contextual) or endpoint standard.

3. The middling inference ("not great") is a scalar implicature, not hardwired semantics: it is cancelable, reinforceable, and suspended in DE contexts.

4. The standard function s must be generalized beyond @cite{kennedy-2007}'s Principle of Interpretive Economy to handle functional standards.

Key formal definitions

• ⟦MPA⟧ = λx. μ_value(x) — basic entry (measure function returning value)
• s(MPA) = Max({d : ∀w' ∈ Acc(w)[PURSUED(x)(w') → μ_value(x)(w') ≥ d]}) — necessity standard (functional)
• ⟦POS MPA⟧ = λx. μ_value(x)(w) ≥ s(MPA) — positive form

structure Phenomena.Gradability.Studies.Beltrama2025.EmpiricalProfile : Type

Properties distinguishing MPAs from good (Table 1, p. 179).

Each field is true when the property holds for the adjective class.

• contextSensitive : Bool
• gradable : Bool
• zoneOfIndifference : Bool
• borderlineCases : Bool
• combinesBarely : Bool
• emphasisInDE : Bool
• ableDerivation : Bool

instance Phenomena.Gradability.Studies.Beltrama2025.instReprEmpiricalProfile : Repr EmpiricalProfile

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instReprEmpiricalProfile = { reprPrec := Phenomena.Gradability.Studies.Beltrama2025.instReprEmpiricalProfile.repr }

def Phenomena.Gradability.Studies.Beltrama2025.instReprEmpiricalProfile.repr : EmpiricalProfile → ℕ → Std.Format

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def Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqEmpiricalProfile.decEq (x✝ x✝¹ : EmpiricalProfile) : Decidable (x✝ = x✝¹)

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instance Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqEmpiricalProfile : DecidableEq EmpiricalProfile

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• Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqEmpiricalProfile = Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqEmpiricalProfile.decEq

def Phenomena.Gradability.Studies.Beltrama2025.instBEqEmpiricalProfile.beq : EmpiricalProfile → EmpiricalProfile → Bool

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• Phenomena.Gradability.Studies.Beltrama2025.instBEqEmpiricalProfile.beq x✝¹ x✝ = false

instance Phenomena.Gradability.Studies.Beltrama2025.instBEqEmpiricalProfile : BEq EmpiricalProfile

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instBEqEmpiricalProfile = { beq := Phenomena.Gradability.Studies.Beltrama2025.instBEqEmpiricalProfile.beq }

def Phenomena.Gradability.Studies.Beltrama2025.mpaProfile : EmpiricalProfile

MPAs: context-sensitive, gradable, no zone of indifference, no borderline cases, combines with barely, emphatic in DE contexts, productively derived via -able.

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def Phenomena.Gradability.Studies.Beltrama2025.goodProfile : EmpiricalProfile

good: context-sensitive, gradable, zone of indifference present, borderline cases present, resists barely (unless special context), mostly no emphasis in DE, no -able derivation.

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theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_good_share_context_gradability : mpaProfile.contextSensitive = goodProfile.contextSensitive ∧ mpaProfile.gradable = goodProfile.gradable

MPAs and good agree on context-sensitivity and gradability.

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_good_diverge : mpaProfile.zoneOfIndifference ≠ goodProfile.zoneOfIndifference ∧ mpaProfile.borderlineCases ≠ goodProfile.borderlineCases ∧ mpaProfile.combinesBarely ≠ goodProfile.combinesBarely ∧ mpaProfile.emphasisInDE ≠ goodProfile.emphasisInDE

MPAs and good diverge on zone of indifference, borderline cases, barely compatibility, and emphasis.

def Phenomena.Gradability.Studies.Beltrama2025.valueScaleBoundedness : Core.Scale.Boundedness

The value scale is lower-bounded at 0 (purpose-thwarting → purpose-serving), open above. Following @cite{wolfsdorf-2019} and @cite{qing-2021}.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.valueScaleBoundedness = Core.Scale.Boundedness.lowerBounded

theorem Phenomena.Gradability.Studies.Beltrama2025.ie_underpredicts_for_value_scale : Semantics.Degree.interpretiveEconomy valueScaleBoundedness = Semantics.Degree.PositiveStandard.minEndpoint

Interpretive Economy would predict a minEndpoint standard for a lower-bounded scale — but this is wrong for both good (which gets a contextual standard) and MPAs (which get a functional standard). This shows IE must be generalized for evaluative predicates.

def Phenomena.Gradability.Studies.Beltrama2025.goodStandard : Semantics.Degree.PositiveStandard

good receives a contextual standard despite being on a lower-bounded scale — an exception to Interpretive Economy. The standard is determined by the distribution of objects in the comparison class, as with other relative (Class A) adjectives like tall.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.goodStandard = Semantics.Degree.PositiveStandard.contextual

def Phenomena.Gradability.Studies.Beltrama2025.mpaStandard : Semantics.Degree.PositiveStandard

MPAs receive a functional (necessity) standard — the minimum value required for the object's pursuit to be circumstantially possible.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.mpaStandard = Semantics.Degree.PositiveStandard.functional

def Phenomena.Gradability.Studies.Beltrama2025.minsaaStandard : Semantics.Degree.PositiveStandard

MinSAAs (wet, profitable) receive a minEndpoint standard — any nonzero degree suffices.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.minsaaStandard = Semantics.Degree.PositiveStandard.minEndpoint

theorem Phenomena.Gradability.Studies.Beltrama2025.three_standards_distinct : goodStandard ≠ mpaStandard ∧ mpaStandard ≠ minsaaStandard ∧ goodStandard ≠ minsaaStandard

All three standard types for evaluative predicates on the value scale are distinct.

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_good_both_cc_sensitive : mpaStandard.requiresComparisonClass = true ∧ goodStandard.requiresComparisonClass = true ∧ minsaaStandard.requiresComparisonClass = false

MPAs and good both require comparison classes (context-sensitive), unlike MinSAAs.

structure Phenomena.Gradability.Studies.Beltrama2025.MinSAAHypothesis : Type

The MinSAA hypothesis: MPAs are minimum-standard absolute adjectives. This is initially plausible — lower-bounded scale, s(MPA) = 0 — but §4.2 shows it makes incorrect predictions.

• slightlyOk : Bool

  slightly compatible (MinSAAs: yes; MPAs: no)

• contextSensitive : Bool

  Context-sensitive (MinSAAs: no; MPAs: yes)

• compEntailsPos : Bool

  Comparative entails positive form (MinSAAs: yes; MPAs: no)

• denialCancels : Bool

  Denial cancels all degree (MinSAAs: yes; MPAs: no)

instance Phenomena.Gradability.Studies.Beltrama2025.instReprMinSAAHypothesis : Repr MinSAAHypothesis

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instReprMinSAAHypothesis = { reprPrec := Phenomena.Gradability.Studies.Beltrama2025.instReprMinSAAHypothesis.repr }

def Phenomena.Gradability.Studies.Beltrama2025.instReprMinSAAHypothesis.repr : MinSAAHypothesis → ℕ → Std.Format

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instance Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqMinSAAHypothesis : DecidableEq MinSAAHypothesis

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• Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqMinSAAHypothesis = Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqMinSAAHypothesis.decEq

def Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqMinSAAHypothesis.decEq (x✝ x✝¹ : MinSAAHypothesis) : Decidable (x✝ = x✝¹)

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def Phenomena.Gradability.Studies.Beltrama2025.instBEqMinSAAHypothesis.beq : MinSAAHypothesis → MinSAAHypothesis → Bool

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• Phenomena.Gradability.Studies.Beltrama2025.instBEqMinSAAHypothesis.beq x✝¹ x✝ = false

instance Phenomena.Gradability.Studies.Beltrama2025.instBEqMinSAAHypothesis : BEq MinSAAHypothesis

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instBEqMinSAAHypothesis = { beq := Phenomena.Gradability.Studies.Beltrama2025.instBEqMinSAAHypothesis.beq }

def Phenomena.Gradability.Studies.Beltrama2025.minsaaPredictions : MinSAAHypothesis

MinSAA predictions: slightly OK, not CC-sensitive, comparative entails positive form, denial cancels all degree.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.minsaaPredictions = { slightlyOk := true, contextSensitive := false, compEntailsPos := true, denialCancels := true }

def Phenomena.Gradability.Studies.Beltrama2025.mpaActual : MinSAAHypothesis

MPA actual behavior: slightly blocked, CC-sensitive, comparative does NOT entail positive form, denial does NOT cancel all degree.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.mpaActual = { slightlyOk := false, contextSensitive := true, compEntailsPos := false, denialCancels := false }

theorem Phenomena.Gradability.Studies.Beltrama2025.minsaa_hypothesis_rejected : minsaaPredictions ≠ mpaActual

MinSAA predictions diverge from MPA behavior on ALL four diagnostics. This is the core argument of §4.2 for rejecting the MinSAA analysis.

structure Phenomena.Gradability.Studies.Beltrama2025.NecStandardModel (W : Type) : Type

A simplified finite model for the necessity standard.

The accessibility relation is exclusively **circumstantial**
(@cite{beltrama-2025} p. 196): the accessible worlds are those
compatible with the norms and circumstances in the current world.

• W — possible worlds
• μ — measure function (value of the object in each world)
• acc — circumstantially accessible worlds from the evaluation world
• pursued — whether the object is pursued in a given world

• μ : W → ℕ
• acc : List W
• pursued : W → Bool

def Phenomena.Gradability.Studies.Beltrama2025.necessityStandard {W : Type} (m : NecStandardModel W) : ℕ

The necessity standard: the maximum degree d such that in ALL circumstantially accessible worlds where the object is pursued, its value is at least d.

s(MPA) = Max({d : ∀w' ∈ Circ(w)[PURSUED(x)(w') → μ(x)(w') ≥ d]})

This returns the minimum value of μ across pursued accessible worlds. If nothing is pursued, the universal is vacuously true for all d, so the standard is maximally high (we return max of μ over all accessible worlds, defaulting to 0 if acc is empty).

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def Phenomena.Gradability.Studies.Beltrama2025.mpaPositiveForm {W : Type} (m : NecStandardModel W) (actualValue : ℕ) : Bool

The positive form: μ_value(x)(w) ≥ s(MPA).

Equations

• Phenomena.Gradability.Studies.Beltrama2025.mpaPositiveForm m actualValue = decide (actualValue ≥ Phenomena.Gradability.Studies.Beltrama2025.necessityStandard m)

structure Phenomena.Gradability.Studies.Beltrama2025.ModifierCompatibility : Type

Degree modifier compatibility data for MPAs.

@cite{beltrama-2025} §2.2.2, §6.4--6.5:

• quite/pretty/somewhat: OK (moderate degree)
• very/extremely/super: degraded (conflict with middling flavor)
• barely: OK (necessity standard provides crisp boundary)
• slightly: blocked (value scale minimum is 0, but MPA standard ≠ 0)

• form : String
• quite : Bool
• veryExtremely : Bool
• barely : Bool
• slightly : Bool

instance Phenomena.Gradability.Studies.Beltrama2025.instReprModifierCompatibility : Repr ModifierCompatibility

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instReprModifierCompatibility = { reprPrec := Phenomena.Gradability.Studies.Beltrama2025.instReprModifierCompatibility.repr }

def Phenomena.Gradability.Studies.Beltrama2025.instReprModifierCompatibility.repr : ModifierCompatibility → ℕ → Std.Format

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def Phenomena.Gradability.Studies.Beltrama2025.decentModifiers : ModifierCompatibility

Equations

• Phenomena.Gradability.Studies.Beltrama2025.decentModifiers = { form := "decent", quite := true, veryExtremely := false, barely := true, slightly := false }

def Phenomena.Gradability.Studies.Beltrama2025.goodModifiers : ModifierCompatibility

Equations

• Phenomena.Gradability.Studies.Beltrama2025.goodModifiers = { form := "good", quite := true, veryExtremely := true, barely := false, slightly := false }

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_barely_vs_good : decentModifiers.barely = true ∧ goodModifiers.barely = false

MPAs combine with barely while good doesn't.

theorem Phenomena.Gradability.Studies.Beltrama2025.both_resist_slightly : decentModifiers.slightly = false ∧ goodModifiers.slightly = false

Both MPAs and good resist slightly.

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_resists_strong_intensifiers : decentModifiers.veryExtremely = false ∧ goodModifiers.veryExtremely = true

MPAs resist strong intensifiers while good accepts them.

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_entries_value_dimension : Fragments.English.Predicates.Adjectival.decent.dimension = Core.Dimension.value ∧ Fragments.English.Predicates.Adjectival.acceptable.dimension = Core.Dimension.value ∧ Fragments.English.Predicates.Adjectival.adequate.dimension = Core.Dimension.value

All MPA fragment entries use the value dimension.

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_entries_lower_bounded : Fragments.English.Predicates.Adjectival.decent.scaleType = Core.Scale.Boundedness.lowerBounded ∧ Fragments.English.Predicates.Adjectival.acceptable.scaleType = Core.Scale.Boundedness.lowerBounded ∧ Fragments.English.Predicates.Adjectival.adequate.scaleType = Core.Scale.Boundedness.lowerBounded

All MPA fragment entries have lower-bounded scales.

theorem Phenomena.Gradability.Studies.Beltrama2025.good_same_scale_as_mpas : Fragments.English.Predicates.Adjectival.good.scaleType = Fragments.English.Predicates.Adjectival.decent.scaleType ∧ Fragments.English.Predicates.Adjectival.good.dimension = Fragments.English.Predicates.Adjectival.decent.dimension

good shares the same scale structure as MPAs.

theorem Phenomena.Gradability.Studies.Beltrama2025.good_vs_mpa_standard_override : goodStandard ≠ mpaStandard

Despite sharing scale structure, good and MPAs receive different standards. IE predicts minEndpoint for both, but good overrides to contextual and MPAs override to functional. This division of labor is what makes the system maximally informative.

inductive Phenomena.Gradability.Studies.Beltrama2025.EvaluativeAlternative : Type

The middling inference is a scalar implicature, not lexical semantics. Evidence: cancelability, reinforceability, suspension in DE contexts.

Scale: ⟨decent, good, great, fantastic⟩ Using decent implicates ¬good (speaker would have used good if true). This is the standard Horn/Grice reasoning over evaluative scales.

• decent : EvaluativeAlternative
• good : EvaluativeAlternative
• great : EvaluativeAlternative
• fantastic : EvaluativeAlternative

instance Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqEvaluativeAlternative : DecidableEq EvaluativeAlternative

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instDecidableEqEvaluativeAlternative x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Gradability.Studies.Beltrama2025.instBEqEvaluativeAlternative.beq : EvaluativeAlternative → EvaluativeAlternative → Bool

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instBEqEvaluativeAlternative.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Gradability.Studies.Beltrama2025.instBEqEvaluativeAlternative : BEq EvaluativeAlternative

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instBEqEvaluativeAlternative = { beq := Phenomena.Gradability.Studies.Beltrama2025.instBEqEvaluativeAlternative.beq }

instance Phenomena.Gradability.Studies.Beltrama2025.instReprEvaluativeAlternative : Repr EvaluativeAlternative

Equations

• Phenomena.Gradability.Studies.Beltrama2025.instReprEvaluativeAlternative = { reprPrec := Phenomena.Gradability.Studies.Beltrama2025.instReprEvaluativeAlternative.repr }

def Phenomena.Gradability.Studies.Beltrama2025.instReprEvaluativeAlternative.repr : EvaluativeAlternative → ℕ → Std.Format

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def Phenomena.Gradability.Studies.Beltrama2025.EvaluativeAlternative.rank : EvaluativeAlternative → ℕ

Evaluative scale ordering: decent < good < great < fantastic.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.EvaluativeAlternative.decent.rank = 0
• Phenomena.Gradability.Studies.Beltrama2025.EvaluativeAlternative.good.rank = 1
• Phenomena.Gradability.Studies.Beltrama2025.EvaluativeAlternative.great.rank = 2
• Phenomena.Gradability.Studies.Beltrama2025.EvaluativeAlternative.fantastic.rank = 3

def Phenomena.Gradability.Studies.Beltrama2025.middlingInference (uttered alt : EvaluativeAlternative) : Bool

The middling inference: using a weaker alternative implicates the negation of all stronger alternatives.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.middlingInference uttered alt = decide (alt.rank > uttered.rank)

theorem Phenomena.Gradability.Studies.Beltrama2025.decent_implicates_not_good : middlingInference EvaluativeAlternative.decent EvaluativeAlternative.good = true ∧ middlingInference EvaluativeAlternative.decent EvaluativeAlternative.great = true ∧ middlingInference EvaluativeAlternative.decent EvaluativeAlternative.fantastic = true

Using decent generates implicatures against good, great, fantastic.

theorem Phenomena.Gradability.Studies.Beltrama2025.good_no_middling_against_decent : middlingInference EvaluativeAlternative.good EvaluativeAlternative.decent = false

Using good does NOT generate the middling inference (no stronger alternative is conventionally excluded).

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_no_indifference_zone : mpaProfile.zoneOfIndifference = false ∧ goodProfile.zoneOfIndifference = true

MPAs lack a zone of indifference (@cite{beltrama-2025} §6.3, p. 199--200).

The necessity standard provides a crisp boundary: an object either meets the minimum value for pursuit (MPA) or falls below it (¬MPA). The neither MPA nor not-MPA construction is defective because MPA has existential force (pursuit is possible in some accessible world) while its negation has universal force (no accessible world supports pursuit); combining these yields a contradiction.

By contrast, good/bad are lexical contraries with separate distributional thresholds, permitting a gap region where an object fails to "stand out" with respect to either standard.

def Phenomena.Gradability.Studies.Beltrama2025.decentClass : Semantics.Degree.AdjectiveClass

MPAs don't fit Kennedy's three-way classification. They share context-sensitivity with relative adjectives and crisp judgments with absolute adjectives, forming a genuinely hybrid category.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.decentClass = Semantics.Degree.AdjectiveClass.mildlyPositive

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_not_relative : decentClass.isRelative = false

MPAs are neither relative nor absolute in the Kennedy 2007 sense.

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_licensed_by_pipeline : Core.Scale.LicensingPipeline.isLicensed Fragments.English.Predicates.Adjectival.decent.scaleType = true ∧ Core.Scale.LicensingPipeline.isLicensed Fragments.English.Predicates.Adjectival.acceptable.scaleType = true ∧ Core.Scale.LicensingPipeline.isLicensed Fragments.English.Predicates.Adjectival.adequate.scaleType = true

MPAs (lower-bounded scale) are licensed for degree modification by @cite{kennedy-2007}'s scale-structure licensing pipeline. This is consistent with MPAs combining with barely and moderate modifiers, while their resistance to very/extremely is pragmatic (§6.2), not structural.

theorem Phenomena.Gradability.Studies.Beltrama2025.good_also_licensed : Core.Scale.LicensingPipeline.isLicensed Fragments.English.Predicates.Adjectival.good.scaleType = true

good is also licensed (same scale structure). The difference between MPAs and good is in standard type, not in structural licensing.

def Phenomena.Gradability.Studies.Beltrama2025.mpaValence : Semantics.Lexical.Adjective.Intensification.EvaluativeValence

MPAs have positive evaluative valence: they denote a favorable (if mild) assessment. This connects to @cite{nouwen-2024}'s evaluative measure semantics.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.mpaValence = Semantics.Lexical.Adjective.Intensification.EvaluativeValence.positive

def Phenomena.Gradability.Studies.Beltrama2025.goodValence : Semantics.Lexical.Adjective.Intensification.EvaluativeValence

good shares positive valence with MPAs.

Equations

• Phenomena.Gradability.Studies.Beltrama2025.goodValence = Semantics.Lexical.Adjective.Intensification.EvaluativeValence.positive

theorem Phenomena.Gradability.Studies.Beltrama2025.mpa_good_same_valence : mpaValence = goodValence

Both MPAs and good are positively evaluative, distinguishing them from neutral (usual) or negative (terrible) bases.

def Phenomena.Gradability.Studies.Beltrama2025.enoughParallel : Semantics.Degree.DegPType

MPAs encode the same necessity component as enough (@cite{beltrama-2025} §5.3; Nadathur 2023): the minimum degree required for the complement/pursuit to be circumstantially possible.

The parallel: "old enough to drink" ≈ "acceptable (for the purpose)". Both introduce a functional standard via a circumstantial modal base. The key difference: enough takes an overt complement clause while MPAs get their purpose from context (action-guidance).

Equations

• Phenomena.Gradability.Studies.Beltrama2025.enoughParallel = Semantics.Degree.DegPType.sufficiency

theorem Phenomena.Gradability.Studies.Beltrama2025.ie_divergence_on_value_scale : Semantics.Degree.interpretiveEconomy valueScaleBoundedness = Semantics.Degree.PositiveStandard.minEndpoint ∧ goodStandard = Semantics.Degree.PositiveStandard.contextual ∧ mpaStandard = Semantics.Degree.PositiveStandard.functional ∧ minsaaStandard = Semantics.Degree.PositiveStandard.minEndpoint ∧ goodStandard ≠ mpaStandard ∧ mpaStandard ≠ minsaaStandard

Interpretive Economy maps lower-bounded → minEndpoint, but on the value scale THREE different standards coexist:

1. good: contextual (distributional, like tall)
2. MPAs: functional (necessity, like enough)
3. MinSAAs (wet, profitable): minEndpoint (as IE predicts)

This demonstrates that IE must be generalized: the s function can assign functional standards when the adjective's lexical semantics introduces practical commitments (@cite{beltrama-2025} §5.4, p. 195).