@cite{ramotowska-santorio-2025} - Counterfactuals and Quantificational Force

Ramotowska, S., Marty, P., Romoli, J. & Santorio, P. (2025). Counterfactuals and quantificational force: Experimental evidence for selectional semantics. Semantics & Pragmatics 18, Article 6: 1–43.

Finding

Quantifier STRENGTH determines graded truth-value judgments for counterfactuals embedded under quantifiers, not polarity or QUD.

This supports the SELECTIONAL theory (Stalnaker + supervaluation) over:

• Homogeneity theory (von Fintel/Križ): predicts QUD × polarity interaction
• Universal theory (Lewis/Kratzer): predicts determinate true/false

Experimental Paradigm

Two experiments using graded truth-value judgments (0–99 slider from "completely false" to "completely true"). QUD manipulated between subjects: E-QuD (existential: "at least one has a chance to win") vs U-QuD (universal: "all are guaranteed to win").

• Experiment 1 (n=87 after exclusion): Lottery scenarios where "only some of the tickets that have been bought win a prize."
• Experiment 2 (n=94 after exclusion): Card game with 4 players; mixed scenario has 2/4 red cards (win) and 2/4 gray cards (lose). Also tested plural definite sentences alongside counterfactuals.

Test sentences (Experiment 2):

• "All/None/Some/Not all of the players would have won if they had played and finished this round."

Key Results

• Strong quantifiers (every, no): mean ratings < 15 (Exp 1), < 4 (Exp 2)
• Weak quantifiers (some, not every): mean ratings > 84 (Exp 1), > 82 (Exp 2)
• STRENGTH: β = −77.09, p < .001 (Exp 1); β = −88.7, p < .001 (Exp 2)
• QUD: not significant for counterfactuals (Exp 1: β = −0.09, p = .97; Exp 2: β = −0.6, p = 0.7)
• Plural definites WERE sensitive to QUD (Exp 2: β = −12.6, p = 0.01), confirming the QUD manipulation was effective

inductive Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Theory : Type

The three theories being tested.

• universal : Theory
• selectional : Theory
• homogeneity : Theory

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprTheory : Repr Theory

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprTheory = { reprPrec := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprTheory.repr }

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprTheory.repr : Theory → ℕ → Std.Format

Equations

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

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instDecidableEqTheory : DecidableEq Theory

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instDecidableEqTheory x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqTheory : BEq Theory

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqTheory = { beq := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqTheory.beq }

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqTheory.beq : Theory → Theory → Bool

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqTheory.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

inductive Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier : Type

Quantifiers tested in the experiment.

• every : Quantifier
• some : Quantifier
• no : Quantifier
• notEvery : Quantifier

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuantifier : Repr Quantifier

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuantifier = { reprPrec := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuantifier.repr }

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuantifier.repr : Quantifier → ℕ → Std.Format

Equations

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

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instDecidableEqQuantifier : DecidableEq Quantifier

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instDecidableEqQuantifier x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuantifier.beq : Quantifier → Quantifier → Bool

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuantifier.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuantifier : BEq Quantifier

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuantifier = { beq := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuantifier.beq }

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.strength : Quantifier → Fragments.English.Determiners.Strength

Map local quantifiers to canonical Strength (B&C Table II).

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.every.strength = Fragments.English.Determiners.Strength.strong
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.no.strength = Fragments.English.Determiners.Strength.strong
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.some.strength = Fragments.English.Determiners.Strength.weak
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.notEvery.strength = Fragments.English.Determiners.Strength.weak

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.isStrong (q : Quantifier) : Bool

Quantifier strength classification, derived from canonical Strength.

Equations

• q.isStrong = (q.strength == Fragments.English.Determiners.Strength.strong)

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.isPositive : Quantifier → Bool

Quantifier polarity classification.

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.every.isPositive = true
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.some.isPositive = true
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.no.isPositive = false
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.notEvery.isPositive = false

inductive Phenomena.Conditionals.Studies.RamotowskaEtAl2025.QuDType : Type

QUD type manipulated between subjects.

• existential : QuDType
• universal : QuDType

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuDType.repr : QuDType → ℕ → Std.Format

Equations

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

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuDType : Repr QuDType

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuDType = { reprPrec := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprQuDType.repr }

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instDecidableEqQuDType : DecidableEq QuDType

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instDecidableEqQuDType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuDType.beq : QuDType → QuDType → Bool

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuDType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuDType : BEq QuDType

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuDType = { beq := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instBEqQuDType.beq }

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.selectionalPredictedHigh (q : Quantifier) : Bool

Selectional theory predictions (Table 3): QUD-independent. Strong quantifiers → rejected (low ratings), weak quantifiers → accepted (high ratings).

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.selectionalPredictedHigh q = !q.isStrong

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneityPredictedHigh (q : Quantifier) (qud : QuDType) : Bool

Homogeneity theory predictions (Table 3): QUD-dependent. Positive quantifiers: high under E-QuD, low under U-QuD. Negative quantifiers: low under E-QuD, high under U-QuD. The predicted interaction is between QUD and polarity, not strength.

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneityPredictedHigh q Phenomena.Conditionals.Studies.RamotowskaEtAl2025.QuDType.existential = true
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneityPredictedHigh q Phenomena.Conditionals.Studies.RamotowskaEtAl2025.QuDType.universal = false
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneityPredictedHigh q Phenomena.Conditionals.Studies.RamotowskaEtAl2025.QuDType.existential = false
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneityPredictedHigh q Phenomena.Conditionals.Studies.RamotowskaEtAl2025.QuDType.universal = true

structure Phenomena.Conditionals.Studies.RamotowskaEtAl2025.ExperimentalDatum : Type

Experimental datum: mean slider rating (0–99 scale) for a condition. 0 = "completely false", 99 = "completely true".

• quantifier : Quantifier
• qud : QuDType
• meanRating : ℚ

instance Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprExperimentalDatum : Repr ExperimentalDatum

Equations

• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprExperimentalDatum = { reprPrec := Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprExperimentalDatum.repr }

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.instReprExperimentalDatum.repr : ExperimentalDatum → ℕ → Std.Format

Equations

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

Experiment 2 Results (card game, n=94)

Experiment 2 (§6) provides per-condition mean slider ratings for target counterfactual (TC) sentences in the mixed scenario, reported in §6.7.3 (p. 6:34). These are the verified values.

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.experiment2MixedResults : List ExperimentalDatum

Experiment 2: mean slider ratings for counterfactuals in mixed scenarios. Verified from paper §6.7.3 (p. 6:34).

Strong quantifiers (every/all, none): all means < 4. Weak quantifiers (not all, some): all means > 82.

Equations

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

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.strengthEffect : Bool

Key empirical observation: Strength, not polarity or QUD, determines truth-value judgments for counterfactuals in mixed scenarios.

Strong quantifiers (every, no) have uniformly low mean ratings (< 4/99). Weak quantifiers (some, not every) have uniformly high ratings (> 82/99). QUD has no significant effect on counterfactual ratings.

Equations

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

theorem Phenomena.Conditionals.Studies.RamotowskaEtAl2025.strength_effect_verified : (experiment2MixedResults.all fun (d : ExperimentalDatum) => if d.quantifier.isStrong = true then decide (d.meanRating < 5) else decide (d.meanRating > 80)) = true

Strength effect: all strong quantifier ratings are below 5/99 and all weak quantifier ratings are above 80/99 in the mixed scenario.

This extreme separation rules out chance variation and confirms that strength is the dominant factor.

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.qudNoEffect : Bool

QUD has no effect on counterfactuals: within each quantifier, E-QuD and U-QuD ratings are close (differ by < 5 points on 0–99 scale).

This is the key prediction of the selectional theory (QUD-independent) and against the homogeneity theory (which predicts QUD × polarity).

Equations

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

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.selectionalFits : Bool

Selectional theory succeeds: predictions match data.

The selectional theory predicts that quantifier strength determines ratings regardless of QUD. This matches the observed pattern: strong quantifiers uniformly rejected, weak uniformly accepted, with no QUD modulation.

Equations

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

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneityFails : Bool

Homogeneity theory fails: predicted QUD × polarity interaction absent.

The homogeneity theory predicts that positive quantifiers (every, some) should be rated HIGH under E-QuD but LOW under U-QuD, and vice versa for negative quantifiers. The data shows no such interaction:

• "every" is low under BOTH QUDs (~1.2 and ~1.5)
• "some" is high under BOTH QUDs (~97.2 and ~96.1)

Equations

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

theorem Phenomena.Conditionals.Studies.RamotowskaEtAl2025.homogeneity_wrong_count : (List.filter (fun (d : ExperimentalDatum) => have predictedHigh := homogeneityPredictedHigh d.quantifier d.qud; predictedHigh && decide (d.meanRating < 50) || !predictedHigh && decide (d.meanRating > 50)) experiment2MixedResults).length = 4

The homogeneity theory makes wrong predictions for 4 of 8 conditions.

Under U-QuD, homogeneity predicts:

• every → low (✓ observed: 1.5)
• some → low (✗ observed: 96.1)
• no → high (✗ observed: 3.3)
• not every → high (✓ observed: 82.1)

Under E-QuD, homogeneity predicts:

• every → high (✗ observed: 1.2)
• some → high (✓ observed: 97.2)
• no → low (✓ observed: 0.9)
• not every → low (✗ observed: 86.1)

Mixed-Effects Model Results (Table 5 of paper)

Experiment 2 target counterfactual sentences, linear mixed-effects model with POLARITY, STRENGTH, QUD and interactions as predictors:

Effect	β	p
INTERCEPT	46.1	< .001
STRENGTH	−88.7	< .001
QUD	−0.6	0.7
POLARITY	5.9	< .001
QUD:POLARITY	0.3	0.9
STRENGTH:QUD	3.9	0.2
STRENGTH:POLARITY	−13.2	< .001
STR:POL:QUD	−5.3	0.4

Key findings:

• STRENGTH is the dominant predictor (β = −88.7)
• QUD has no significant main effect or interactions
• POLARITY has a small effect (β = 5.9): "some" rated slightly higher than "not every" within weak quantifiers
• STRENGTH×POLARITY interaction (β = −13.2): the polarity effect is confined to weak quantifiers

def Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.selectionalResult (q : Quantifier) (results : List Core.Duality.Truth3) : Core.Duality.Truth3

Bridge: map study quantifiers to formal selectional predictions. Each quantifier maps to the corresponding projection operation from the theory layer (Counterfactual.lean).

Equations

• One or more equations did not get rendered due to their size.
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.no.selectionalResult results = Semantics.Conditionals.Counterfactual.noSelectional results
• Phenomena.Conditionals.Studies.RamotowskaEtAl2025.Quantifier.notEvery.selectionalResult results = Semantics.Conditionals.Counterfactual.notEverySelectional results

theorem Phenomena.Conditionals.Studies.RamotowskaEtAl2025.selectional_prediction_grounded (q : Quantifier) (bs : List Bool) (h_some_true : bs.any id = true) (h_some_false : (bs.any fun (x : Bool) => !x) = true) : (q.selectionalResult (List.map Core.Duality.Truth3.ofBool bs) == Core.Duality.Truth3.true) = selectionalPredictedHigh q

Grounding theorem: the study-level prediction (selectionalPredictedHigh) agrees with the formal selectional semantics for any mixed input.

This connects the theory layer's three-valued projection operations to the study file's simple strength-based classification. The classification is not stipulated — it is derived from the formal theory by construction.

Related Phenomena

1. Projection Duality: The strength effect reflects the adjoint duality between universal (right adjoint, fragile) and existential (left adjoint, robust) operators. See Counterfactual.lean for the formalization.

2. Plural Definites and QUD: Unlike counterfactuals, plural definite sentences ("The players won this round") ARE sensitive to QUD manipulation (Exp 2: E-QuD M=42.2 vs U-QuD M=29.6, β = −12.6, p = 0.01). This confirms the QUD manipulation worked and that counterfactuals' insensitivity to QUD is a genuine semantic property, not a failure of the manipulation.

3. Conditional Excluded Middle (CEM): Stalnaker's semantics validates CEM: (A □→ B) ∨ (A □→ ¬B). See Counterfactual.lean for the proof.