RSA Question Embedding

@cite{geurts-2010} @cite{hamblin-1973b} @cite{groenendijk-stokhof-1984}

Models scalar implicatures embedded in questions.

The Phenomenon

"Did some students pass?"

Questions have a unique status regarding scalar implicatures:

• Not clearly upward-entailing (UE) like assertions
• Not clearly downward-entailing (DE) like negation
• Implicatures may arise, but weaker than in assertions

Theoretical Background

Questions are often analyzed as sets of propositions or partitions of logical space. The key insight:

A question "Did some students pass?" asks the hearer to choose between:

• "Yes, some passed" (weak: >=1)
• "No, no one passed"

If the implicature "not all" is computed:

• "Yes, some-but-not-all passed"
• "No, either none or all passed" (??)

The second option doesn't make sense, suggesting local SI in questions is problematic.

Predictions

Unlike assertions (where local SI strengthens) or DE contexts (where local SI weakens), questions create an asymmetry:

• The "yes" answer benefits from SI (more precise)
• The "no" answer becomes weird with SI (disjunctive)

RSA should predict: Local SI is dispreferred in questions because it makes one answer pragmatically odd.

inductive RSA.QuestionEmbedding.StudentResult : Type

Student outcome for question scenario.

• noneP : StudentResult
• someP : StudentResult
• allP : StudentResult

instance RSA.QuestionEmbedding.instDecidableEqStudentResult : DecidableEq StudentResult

Equations

• RSA.QuestionEmbedding.instDecidableEqStudentResult x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance RSA.QuestionEmbedding.instReprStudentResult : Repr StudentResult

Equations

• RSA.QuestionEmbedding.instReprStudentResult = { reprPrec := RSA.QuestionEmbedding.instReprStudentResult.repr }

def RSA.QuestionEmbedding.instReprStudentResult.repr : StudentResult → ℕ → Std.Format

Equations

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

instance RSA.QuestionEmbedding.instBEqStudentResult : BEq StudentResult

Equations

• RSA.QuestionEmbedding.instBEqStudentResult = { beq := RSA.QuestionEmbedding.instBEqStudentResult.beq }

def RSA.QuestionEmbedding.instBEqStudentResult.beq : StudentResult → StudentResult → Bool

Equations

• RSA.QuestionEmbedding.instBEqStudentResult.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance RSA.QuestionEmbedding.instInhabitedStudentResult : Inhabited StudentResult

Equations

• RSA.QuestionEmbedding.instInhabitedStudentResult = { default := RSA.QuestionEmbedding.instInhabitedStudentResult.default }

def RSA.QuestionEmbedding.instInhabitedStudentResult.default : StudentResult

Equations

• RSA.QuestionEmbedding.instInhabitedStudentResult.default = RSA.QuestionEmbedding.StudentResult.noneP

instance RSA.QuestionEmbedding.instFintypeStudentResult : Fintype StudentResult

Equations

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

def RSA.QuestionEmbedding.questionWorlds : List StudentResult

Worlds represent the actual state of affairs.

Equations

• RSA.QuestionEmbedding.questionWorlds = [RSA.QuestionEmbedding.StudentResult.noneP, RSA.QuestionEmbedding.StudentResult.someP, RSA.QuestionEmbedding.StudentResult.allP]

structure RSA.QuestionEmbedding.YesNoPartition : Type

A yes/no question partitions the world into "yes" and "no" answers.

For "Did some students pass?":

• Yes-worlds: where some (>=1) passed
• No-worlds: where none passed

• yesWorlds : List StudentResult
• noWorlds : List StudentResult

inductive RSA.QuestionEmbedding.QuestionInterpretation : Type

Interpretations of "Did some students pass?":

1. Global: "some" = at least one

• Yes: someP, allP (some passed)
• No: noneP (none passed)

2. Local: "some" = some-but-not-all

• Yes: someP (some-but-not-all passed)
• No: noneP, allP (?? disjunctive/weird)

• global : QuestionInterpretation
• local_ : QuestionInterpretation

instance RSA.QuestionEmbedding.instDecidableEqQuestionInterpretation : DecidableEq QuestionInterpretation

Equations

• RSA.QuestionEmbedding.instDecidableEqQuestionInterpretation x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def RSA.QuestionEmbedding.instReprQuestionInterpretation.repr : QuestionInterpretation → ℕ → Std.Format

Equations

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

instance RSA.QuestionEmbedding.instReprQuestionInterpretation : Repr QuestionInterpretation

Equations

• RSA.QuestionEmbedding.instReprQuestionInterpretation = { reprPrec := RSA.QuestionEmbedding.instReprQuestionInterpretation.repr }

instance RSA.QuestionEmbedding.instBEqQuestionInterpretation : BEq QuestionInterpretation

Equations

• RSA.QuestionEmbedding.instBEqQuestionInterpretation = { beq := RSA.QuestionEmbedding.instBEqQuestionInterpretation.beq }

def RSA.QuestionEmbedding.instBEqQuestionInterpretation.beq : QuestionInterpretation → QuestionInterpretation → Bool

Equations

• RSA.QuestionEmbedding.instBEqQuestionInterpretation.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def RSA.QuestionEmbedding.instInhabitedQuestionInterpretation.default : QuestionInterpretation

Equations

• RSA.QuestionEmbedding.instInhabitedQuestionInterpretation.default = RSA.QuestionEmbedding.QuestionInterpretation.global

instance RSA.QuestionEmbedding.instInhabitedQuestionInterpretation : Inhabited QuestionInterpretation

Equations

• RSA.QuestionEmbedding.instInhabitedQuestionInterpretation = { default := RSA.QuestionEmbedding.instInhabitedQuestionInterpretation.default }

instance RSA.QuestionEmbedding.instFintypeQuestionInterpretation : Fintype QuestionInterpretation

Equations

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

def RSA.QuestionEmbedding.questionPartition (interp : QuestionInterpretation) : YesNoPartition

The partition induced by each interpretation.

Equations

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theorem RSA.QuestionEmbedding.global_partition_natural : (questionPartition QuestionInterpretation.global).yesWorlds = [StudentResult.someP, StudentResult.allP] ∧ (questionPartition QuestionInterpretation.global).noWorlds = [StudentResult.noneP]

Global interpretation gives a natural partition:

• Yes = some passed (continuous region on scale)
• No = none passed (complementary region)

theorem RSA.QuestionEmbedding.local_partition_disjunctive : (questionPartition QuestionInterpretation.local_).noWorlds = [StudentResult.noneP, StudentResult.allP]

Local interpretation gives a strange partition:

• Yes = some-but-not-all (specific point)
• No = none OR all (disjunctive!)

This disjunctive "no" answer is pragmatically odd.

def RSA.QuestionEmbedding.isContiguous (worlds : List StudentResult) : Bool

The "no" answer under local interpretation is disjunctive. This is marked by the worlds being non-contiguous on the scale.

Equations

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

theorem RSA.QuestionEmbedding.global_no_contiguous : isContiguous (questionPartition QuestionInterpretation.global).noWorlds = true

theorem RSA.QuestionEmbedding.local_no_not_contiguous : isContiguous (questionPartition QuestionInterpretation.local_).noWorlds = false

def RSA.QuestionEmbedding.exhaustiveAnswer (interp : QuestionInterpretation) (actual : StudentResult) : String

Exhaustive interpretation: the "yes" answer conveys the MAXIMAL true proposition consistent with "yes".

For global "Did some pass?":

• If someP: answer conveys "some but not all"
• If allP: answer conveys "all"

This makes local SI redundant - exhaustivity handles it.

Equations

• RSA.QuestionEmbedding.exhaustiveAnswer RSA.QuestionEmbedding.QuestionInterpretation.global RSA.QuestionEmbedding.StudentResult.noneP = "No, none passed"
• RSA.QuestionEmbedding.exhaustiveAnswer RSA.QuestionEmbedding.QuestionInterpretation.global RSA.QuestionEmbedding.StudentResult.someP = "Yes, some (but not all) passed"
• RSA.QuestionEmbedding.exhaustiveAnswer RSA.QuestionEmbedding.QuestionInterpretation.global RSA.QuestionEmbedding.StudentResult.allP = "Yes, all passed"
• RSA.QuestionEmbedding.exhaustiveAnswer RSA.QuestionEmbedding.QuestionInterpretation.local_ RSA.QuestionEmbedding.StudentResult.noneP = "No (none passed)"
• RSA.QuestionEmbedding.exhaustiveAnswer RSA.QuestionEmbedding.QuestionInterpretation.local_ RSA.QuestionEmbedding.StudentResult.someP = "Yes, some-but-not-all passed"
• RSA.QuestionEmbedding.exhaustiveAnswer RSA.QuestionEmbedding.QuestionInterpretation.local_ RSA.QuestionEmbedding.StudentResult.allP = "No (all passed)"

theorem RSA.QuestionEmbedding.local_all_gets_no : exhaustiveAnswer QuestionInterpretation.local_ StudentResult.allP = "No (all passed)"

Under local interpretation, the "all" world gets a "no" answer, which is pragmatically bizarre.

theorem RSA.QuestionEmbedding.global_answers_natural : exhaustiveAnswer QuestionInterpretation.global StudentResult.noneP = "No, none passed" ∧ exhaustiveAnswer QuestionInterpretation.global StudentResult.someP = "Yes, some (but not all) passed" ∧ exhaustiveAnswer QuestionInterpretation.global StudentResult.allP = "Yes, all passed"

Under global interpretation with exhaustivity, all answers are natural.

theorem RSA.QuestionEmbedding.question_prefers_global : isContiguous (questionPartition QuestionInterpretation.local_).noWorlds = false ∧ isContiguous (questionPartition QuestionInterpretation.global).noWorlds = true

RSA predicts: global interpretation preferred for questions.

Reason: Local creates pragmatically odd partition where "all passed" is a "no" answer.

structure RSA.QuestionEmbedding.QuestionUniqueness : Prop

Questions differ from all other embedding contexts:

• Not about entailment direction (like DE vs UE)
• About pragmatic naturalness of the question-answer structure

• not_de_like : ∃ (w : StudentResult), List.elem w (questionPartition QuestionInterpretation.global).yesWorlds = true ∧ List.elem w (questionPartition QuestionInterpretation.local_).yesWorlds = false

  Global doesn't entail local (unlike DE): allP is in global-yes but not local-yes

• not_attitude_like : ∃ (w : StudentResult), List.elem w (questionPartition QuestionInterpretation.local_).yesWorlds = true ∧ List.elem w (questionPartition QuestionInterpretation.global).yesWorlds = true

  Local doesn't entail global (unlike attitude verbs): someP is in both

def RSA.QuestionEmbedding.questionIsUnique : QuestionUniqueness

Questions are indeed unique - neither DE-like nor attitude-verb-like in their entailment pattern, yet still prefer global.

Equations

• RSA.QuestionEmbedding.questionIsUnique = ⋯