@cite{khoo-2015}: Modal Disagreements

Empirical data from Khoo's experiment on epistemic modal disagreements. The key finding: speakers reject might-claims (high rejection rating) without judging them false (low falsity rating). This dissociation between rejection and falsity judgments is predicted by @cite{rudin-2025}'s Neo-Stalnakerian Framework, which derives it from the fact that truth depends on the assertor's information while rejection depends on the rejector's information.

Experimental Design (§II)

• N = 60 participants on Amazon Mechanical Turk
• 2 × 2 mixed design: between-subjects on response type (False vs Rejection), within-subjects on sentence type (Control vs Modal)
• 7-point Likert scale (1 = completely disagree, 7 = completely agree)
• Control vignette: non-modal assertion ("Jim is at home right now")
• Modal vignette: epistemic might ("Fat Tony might be dead")
• False condition: "Do you agree that what [speaker] said is false?"
• Rejection condition: "Would you respond by saying 'No,...'?"

Key Finding: The Difference Observation (footnote 13)

When presented with Modal, ordinary speakers are strongly inclined to reject Smith's assertion (M = 5.03) but are also strongly inclined to disagree that what Smith said is false (M = 2.42). This dissociation is absent in Control, where rejection and falsity ratings are similar.

The interaction is the core result: the rejection–falsity gap reverses direction between Modal (rejection ≫ falsity) and Control (falsity ≥ rejection).

Experimental Conditions

inductive Phenomena.Modality.Studies.Khoo2015.SentenceType : Type

Sentence type: control (non-modal) vs modal (epistemic might).

• control : SentenceType
• modal : SentenceType

instance Phenomena.Modality.Studies.Khoo2015.instDecidableEqSentenceType : DecidableEq SentenceType

Equations

• Phenomena.Modality.Studies.Khoo2015.instDecidableEqSentenceType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Modality.Studies.Khoo2015.instBEqSentenceType.beq : SentenceType → SentenceType → Bool

Equations

• Phenomena.Modality.Studies.Khoo2015.instBEqSentenceType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Modality.Studies.Khoo2015.instBEqSentenceType : BEq SentenceType

Equations

• Phenomena.Modality.Studies.Khoo2015.instBEqSentenceType = { beq := Phenomena.Modality.Studies.Khoo2015.instBEqSentenceType.beq }

def Phenomena.Modality.Studies.Khoo2015.instReprSentenceType.repr : SentenceType → ℕ → Std.Format

Equations

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

instance Phenomena.Modality.Studies.Khoo2015.instReprSentenceType : Repr SentenceType

Equations

• Phenomena.Modality.Studies.Khoo2015.instReprSentenceType = { reprPrec := Phenomena.Modality.Studies.Khoo2015.instReprSentenceType.repr }

inductive Phenomena.Modality.Studies.Khoo2015.ResponseType : Type

Response type: falsity judgment vs rejection inclination.

• false_ : ResponseType
• rejection : ResponseType

instance Phenomena.Modality.Studies.Khoo2015.instDecidableEqResponseType : DecidableEq ResponseType

Equations

• Phenomena.Modality.Studies.Khoo2015.instDecidableEqResponseType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Modality.Studies.Khoo2015.instBEqResponseType.beq : ResponseType → ResponseType → Bool

Equations

• Phenomena.Modality.Studies.Khoo2015.instBEqResponseType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Modality.Studies.Khoo2015.instBEqResponseType : BEq ResponseType

Equations

• Phenomena.Modality.Studies.Khoo2015.instBEqResponseType = { beq := Phenomena.Modality.Studies.Khoo2015.instBEqResponseType.beq }

instance Phenomena.Modality.Studies.Khoo2015.instReprResponseType : Repr ResponseType

Equations

• Phenomena.Modality.Studies.Khoo2015.instReprResponseType = { reprPrec := Phenomena.Modality.Studies.Khoo2015.instReprResponseType.repr }

def Phenomena.Modality.Studies.Khoo2015.instReprResponseType.repr : ResponseType → ℕ → Std.Format

Equations

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

structure Phenomena.Modality.Studies.Khoo2015.Condition : Type

An experimental condition is a sentence × response pair.

• sentence : SentenceType
• response : ResponseType

def Phenomena.Modality.Studies.Khoo2015.instDecidableEqCondition.decEq (x✝ x✝¹ : Condition) : Decidable (x✝ = x✝¹)

Equations

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

instance Phenomena.Modality.Studies.Khoo2015.instDecidableEqCondition : DecidableEq Condition

Equations

• Phenomena.Modality.Studies.Khoo2015.instDecidableEqCondition = Phenomena.Modality.Studies.Khoo2015.instDecidableEqCondition.decEq

def Phenomena.Modality.Studies.Khoo2015.instBEqCondition.beq : Condition → Condition → Bool

Equations

• Phenomena.Modality.Studies.Khoo2015.instBEqCondition.beq { sentence := a, response := a_1 } { sentence := b, response := b_1 } = (a == b && a_1 == b_1)
• Phenomena.Modality.Studies.Khoo2015.instBEqCondition.beq x✝¹ x✝ = false

instance Phenomena.Modality.Studies.Khoo2015.instBEqCondition : BEq Condition

Equations

• Phenomena.Modality.Studies.Khoo2015.instBEqCondition = { beq := Phenomena.Modality.Studies.Khoo2015.instBEqCondition.beq }

def Phenomena.Modality.Studies.Khoo2015.instReprCondition.repr : Condition → ℕ → Std.Format

Equations

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

instance Phenomena.Modality.Studies.Khoo2015.instReprCondition : Repr Condition

Equations

• Phenomena.Modality.Studies.Khoo2015.instReprCondition = { reprPrec := Phenomena.Modality.Studies.Khoo2015.instReprCondition.repr }

def Phenomena.Modality.Studies.Khoo2015.meanRating : Condition → Float

Mean Likert rating for each condition.

Equations

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

def Phenomena.Modality.Studies.Khoo2015.sdRating : Condition → Float

Standard deviation for each condition.

Equations

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

The Difference Observation

The crucial finding: for modal sentences, rejection is high but falsity is low. For control sentences, both are high.

theorem Phenomena.Modality.Studies.Khoo2015.modal_rejection_high : meanRating { sentence := SentenceType.modal, response := ResponseType.rejection } > 4.0

Modal rejection is high (above midpoint 4).

theorem Phenomena.Modality.Studies.Khoo2015.modal_falsity_low : meanRating { sentence := SentenceType.modal, response := ResponseType.false_ } < 4.0

Modal falsity is low (below midpoint 4).

theorem Phenomena.Modality.Studies.Khoo2015.modal_rejection_exceeds_falsity : meanRating { sentence := SentenceType.modal, response := ResponseType.rejection } > meanRating { sentence := SentenceType.modal, response := ResponseType.false_ }

The dissociation: modal rejection exceeds modal falsity.

theorem Phenomena.Modality.Studies.Khoo2015.control_no_dissociation : meanRating { sentence := SentenceType.control, response := ResponseType.false_ } ≥ meanRating { sentence := SentenceType.control, response := ResponseType.rejection }

Control shows no dissociation: falsity ≥ rejection.

theorem Phenomena.Modality.Studies.Khoo2015.modal_gap_large : meanRating { sentence := SentenceType.modal, response := ResponseType.rejection } - meanRating { sentence := SentenceType.modal, response := ResponseType.false_ } > 2.0

The rejection–falsity gap is large for modal (> 2 points).

theorem Phenomena.Modality.Studies.Khoo2015.control_gap_small : meanRating { sentence := SentenceType.control, response := ResponseType.false_ } - meanRating { sentence := SentenceType.control, response := ResponseType.rejection } < 1.0

The rejection–falsity gap is small for control (< 1 point).

theorem Phenomena.Modality.Studies.Khoo2015.interaction_reversal : meanRating { sentence := SentenceType.modal, response := ResponseType.rejection } > meanRating { sentence := SentenceType.modal, response := ResponseType.false_ } ∧ meanRating { sentence := SentenceType.control, response := ResponseType.false_ } ≥ meanRating { sentence := SentenceType.control, response := ResponseType.rejection }

The interaction: the rejection–falsity gap reverses direction between Modal and Control. For Modal, rejection exceeds falsity; for Control, falsity exceeds rejection. This is Khoo's "Difference Observation."

theorem Phenomena.Modality.Studies.Khoo2015.nsf_predicts_khoo_pattern {W : Type u_1} (p : BProp W) (smith beth : List W) (h_smith : smith.any p = true) (h_beth : beth.any p = false) : Semantics.Dynamic.NeoStalnakerian.MI (Semantics.Dynamic.NeoStalnakerian.mightSimple p) smith ∧ Semantics.Dynamic.NeoStalnakerian.rejectionLicensed (Semantics.Dynamic.NeoStalnakerian.mightSimple p) beth

The Mobster scenario has the structure predicted by the NSF: Smith (assertor) has examined evidence consistent with Fat Tony being dead, so his epistemic state contains p-worlds (p = "Fat Tony is dead"). Beth (rejector) knows Fat Tony is alive, so her epistemic state has no p-worlds.

The NSF predicts:

1. Smith's assertion is true (his state is in MI(might-p))
2. Beth's rejection is licensed (her state is not might-p-compatible)

This matches Khoo's finding: speakers reject the might-claim without judging it false.