[Isr01]: Minimizers, Maximizers, and the Rhetoric of Scalar Reasoning

Israel's Scalar Model classifies polarity items by scalar value × rhetorical force (Figure 1) and explains the inverted items — maximizer NPIs (wild horses) and minimizer PPIs (for peanuts) — by propositional role (§4): facilitating roles produce inverted items, impeding roles canonical ones. The pecuniary paradox dissolves: a red cent (resource, impeding) and for peanuts (reward, facilitating) share a low monetary value but occupy different roles, hence opposite canonicity. The paper's items are ScalarItems (Semantics/Polarity/ScalarModel.lean); the classifications of the Fragments/English/PolarityItems.lean entries live here in classifiedLexicon, with the theory that consumes them.

Main results

• pecuniary_paradox — same value and direction, different roles, opposite canonicity.
• paperItems/classifiedLexicon consistency — every classification agrees with predictCanonicity.
• suggestedLikelihoodEffect — the §4 fn. 6 bridge from [Dow91] proto-role entailments to likelihood effect.

Canonical items (Figure 1)

The basic Scalar Model predicts four cells:

	Emphatic	Attenuating
NPI	low: a wink, inch	high: much, long
PPI	high: tons, utterly	low: sorta, rather

Emphatic items license maximally informative interpretations, attenuating items minimally informative ones; NPI contexts are scale-reversing (DE), PPI contexts scale-preserving (UE).

def Israel2001.aWink : Polarity.ScalarItem

a wink — canonical emphatic NPI (low, impeding): I didn't sleep a wink.

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def Israel2001.insanely : Polarity.ScalarItem

insanely — canonical emphatic PPI (high): She is insanely good-looking.

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def Israel2001.sorta : Polarity.ScalarItem

sorta — canonical attenuating PPI (low): She's sorta clever.

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def Israel2001.allThat : Polarity.ScalarItem

all that — canonical attenuating NPI (high): He's not all that clever.

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Inverted items (Figure 3)

Inverted items break the simple correlation between scalar value and polarity type; propositional role (§4) explains them.

def Israel2001.ownShadow : Polarity.ScalarItem

his own shadow — inverted emphatic PPI (low, facilitating): Godfrey is scared of his own shadow.

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def Israel2001.withAFeather : Polarity.ScalarItem

with a feather — inverted emphatic PPI (low, facilitating): You could have knocked me over with a feather.

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The pecuniary paradox

Both a red cent and for peanuts denote small monetary values (§3, examples 15–16), but the first is an NPI and the second a PPI: they occupy different propositional roles — resource (what you spend, impeding) vs reward (what you gain, facilitating).

def Israel2001.redCent : Polarity.ScalarItem

a red cent — canonical NPI, resource role: He won't spend a red cent on your wedding.

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def Israel2001.forPeanuts : Polarity.ScalarItem

for peanuts — inverted PPI, reward role: He got Madonna to play for peanuts.

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theorem Israel2001.pecuniary_paradox : redCent.scalarValue = forPeanuts.scalarValue ∧ redCent.scalarDirection = forPeanuts.scalarDirection ∧ redCent.likelihoodEffect ≠ forPeanuts.likelihoodEffect ∧ redCent.canonicity ≠ forPeanuts.canonicity

The paradox dissolved: the same low value and emphatic direction, but different propositional roles and hence opposite canonicity.

The classified lexicon

def Israel2001.paperItems : List Polarity.ScalarItem

The paper's own example items.

Equations

• Israel2001.paperItems = [Israel2001.aWink, Israel2001.insanely, Israel2001.sorta, Israel2001.allThat, Israel2001.ownShadow, Israel2001.withAFeather, Israel2001.redCent, Israel2001.forPeanuts]

def Israel2001.classifiedLexicon : List Polarity.ScalarItem

The paper's classifications of the Fragments/English/PolarityItems.lean entries (Figure 1 cells; §3 inverted items; §4 roles where the paper gives them).

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The proto-role bridge (§4, fn. 6)

def Israel2001.suggestedLikelihoodEffect (p : ArgumentStructure.EntailmentProfile) : Option Polarity.LikelihoodEffect

The suggestion of [Dow91] proto-role entailments for likelihood effect: Proto-Agent dominance suggests a facilitating role, Proto-Patient dominance an impeding one, and a tie suggests nothing — a heuristic, not a theorem: the pecuniary paradox shows propositional role can diverge from proto-role counts, which is why LikelihoodEffect is an independent concept rather than a function of theta labels.

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Ambiguous superlatives (§6)

Perception verbs allow dual scalar readings (Fauconnier 1975b): Eve didn't hear even the faintest noise ranks stimuli by likely existence, … even the loudest noise ranks experiencers by acuity. Perception is bicausal — it depends on the stimulus's salience and the perceiver's acuity — and the scale type fixes the role.

inductive Israel2001.PerceptionScaleType : Type

The two scales a negated perception superlative can invoke.

• existential : PerceptionScaleType

  Stimuli ranked by likely existence (faintest).

• perceptualAbility : PerceptionScaleType

  Experiencers ranked by perceptual acuity (loudest).

@[instance_reducible]

instance Israel2001.instDecidableEqPerceptionScaleType : DecidableEq PerceptionScaleType

Equations

• Israel2001.instDecidableEqPerceptionScaleType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Israel2001.instReprPerceptionScaleType.repr : PerceptionScaleType → ℕ → Std.Format

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@[instance_reducible]

instance Israel2001.instReprPerceptionScaleType : Repr PerceptionScaleType

Equations

• Israel2001.instReprPerceptionScaleType = { reprPrec := Israel2001.instReprPerceptionScaleType.repr }

def Israel2001.PerceptionScaleType.role : PerceptionScaleType → Polarity.LikelihoodEffect

The scale type fixes the propositional role: existential scales impede (if larger things exist, smaller ones do too), ability scales facilitate (missing the most perceptible means missing everything).

Equations

• Israel2001.PerceptionScaleType.existential.role = Polarity.LikelihoodEffect.impeding
• Israel2001.PerceptionScaleType.perceptualAbility.role = Polarity.LikelihoodEffect.facilitating

Scale-reversing = DE, scale-preserving = UE

§2 connects the Scalar Model to the Fauconnier–Ladusaw tradition: scale-reversing contexts are the downward-entailing ones, and scale-preserving contexts the upward-entailing ones, except that the relevant inferences may be pragmatic entailments within a scalar model rather than strictly logical — which is why the Scalar Model handles cases pure monotonicity misses.

inductive Israel2001.ScaleDirection : Type

Israel's scale directions: reversing (= DE, NPI-licensing) and preserving (= UE, PPI-licensing).

• reversing : ScaleDirection
• preserving : ScaleDirection

@[instance_reducible]

instance Israel2001.instDecidableEqScaleDirection : DecidableEq ScaleDirection

Equations

• Israel2001.instDecidableEqScaleDirection x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Israel2001.instReprScaleDirection : Repr ScaleDirection

Equations

• Israel2001.instReprScaleDirection = { reprPrec := Israel2001.instReprScaleDirection.repr }

def Israel2001.instReprScaleDirection.repr : ScaleDirection → ℕ → Std.Format

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def Israel2001.expectedScaleDirection (e : Polarity.Item) : Option ScaleDirection

Expected scale direction in licensing contexts, from the item's parameters: strength-licensed items need scale reversal (= DE), PPIs scale preservation (= UE), pure FCIs neither (non-veridicality).

Equations

• Israel2001.expectedScaleDirection e = if e.ppi = true then some Israel2001.ScaleDirection.preserving else if e.licensor.isSome = true then some Israel2001.ScaleDirection.reversing else none