Questions/ScopeReadings.lean

@cite{dayal-2016} @cite{groenendijk-stokhof-1984} @cite{szabolcsi-1997}

Pair-List, Choice, and Mention-Some Readings (@cite{groenendijk-stokhof-1984}, Chapter VI, Sections 2-5).

The Phenomenon

Embedded questions exhibit scope interactions with quantifiers:

Pair-List Readings (Section 2.1): "Which student was recommended by each professor?" → For each professor, which student they recommended (list of pairs)

Here the universal quantifier takes WIDE scope over the wh-phrase. The answer is a functional relation: professor → student.

Choice Readings (Section 2.2): "John knows whom Mary or Sue invited" → John knows the answer, where the answer depends on whether Mary or Sue is the one asking

The disjunction/existential takes WIDE scope over the wh-phrase. Different individuals can give different complete answers.

Semantic Analysis

Pair-List: ∀y. ?x. R(y,x)

• The question is parameterized by the universal
• Answer must resolve the question for each y

Choice: ∃y. ?x. R(y,x) (or ∨)

• The question is parameterized by which y is relevant
• Answer resolves the question for SOME y (the one that matters)

Pair-List Readings

"Which student was recommended by each professor?"

Standard Reading: ?x. ∀y. Prof(y) → Recommended(y, x)

• One student recommended by ALL professors
• Very restrictive; often no such student exists

Pair-List Reading: ∀y. Prof(y) → ?x. Recommended(y, x)

• For each professor, which student they recommended
• Answer is a FUNCTION from professors to students

The pair-list reading arises when:

1. The universal takes wide scope over the wh-phrase
2. The answer is a function/relation

structure Semantics.Questions.ScopeReadings.PairListQuestion (W : Type u_1) (E : Type u_2) : Type (max u_1 u_2)

A pair-list question: for each element of a domain, ask a wh-question.

"Which X did each Y verb?" becomes: ∀y∈Y. ?x. R(y, x)

• universalDomain : List E

  The domain of the universal ("each professor")

• whDomain : List E

  The wh-domain ("which student")

• relation : W → E → E → Bool

  The relation being asked about

• universalNP : String

  Name of the universal NP

• whPhrase : String

  Name of the wh-phrase

structure Semantics.Questions.ScopeReadings.PairListAnswer (E : Type u_1) : Type u_1

A pair-list answer: a function from the universal domain to wh-answers.

• pairs : E → Option E

  The function mapping each universal element to its answer

def Semantics.Questions.ScopeReadings.PairListQuestion.toGSQuestion {W : Type u_1} {E : Type u_2} [DecidableEq E] (plq : PairListQuestion W E) : GSQuestion W

Convert a pair-list question to a GSQuestion.

The partition groups worlds by the ENTIRE function from Y to X: Two worlds are equivalent iff they have the same pair-list.

Equations

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

def Semantics.Questions.ScopeReadings.PairListQuestion.individualQuestion {W E : Type} [DecidableEq E] (plq : PairListQuestion W E) (y : E) : GSQuestion W

Build individual question for a specific universal element.

Equations

• plq.individualQuestion y = { sameAnswer := fun (w v : W) => plq.whDomain.all fun (x : E) => plq.relation w y x == plq.relation v y x, refl := ⋯, symm := ⋯, trans := ⋯ }

theorem Semantics.Questions.ScopeReadings.pairList_as_conjunction {W E : Type} [DecidableEq E] [BEq W] (plq : PairListQuestion W E) (w v : W) : plq.toGSQuestion.sameAnswer w v = (Coordination.pairListAsConjunction plq.universalDomain plq.individualQuestion).sameAnswer w v

A pair-list question is equivalent to a conjunction of individual questions.

∀y. ?x. R(y,x) = ∧_{y∈Y} ?x. R(y,x)

The partition induced by the pair-list question agrees pointwise with the conjunction of per-element individual questions.

def Semantics.Questions.ScopeReadings.PairListQuestion.maxCells {W : Type u_1} {E : Type u_2} (plq : PairListQuestion W E) : ℕ

Number of cells in pair-list: can be up to |X|^|Y| (exponential).

If each professor can recommend any student, and there are n professors and m students, the pair-list question has up to m^n cells.

Equations

• plq.maxCells = plq.whDomain.length ^ plq.universalDomain.length

Choice Readings

"John knows whom Mary or Sue invited"

Non-Choice Reading: John knows: ?x. (Mary-invited(x) ∨ Sue-invited(x))

• John knows who was invited by at least one of them

Choice Reading: John knows: Mary-invited(?x) ∨ Sue-invited(?x)

• Depending on whether Mary or Sue is relevant, John knows that answer
• This is a de re reading of the disjunction

The choice reading arises when:

1. A disjunctive/existential NP takes wide scope over the wh-phrase
2. The embedded question is "parameterized" by which disjunct is true

"Whom does John or Mary love?" can be answered:

• "If John: Bill. If Mary: Sue." This gives DIFFERENT answers depending on who is the lover.

structure Semantics.Questions.ScopeReadings.ChoiceQuestion (W : Type u_1) (E : Type u_2) : Type (max u_1 u_2)

A choice question: question parameterized by a disjunction.

"Whom does Y₁ or Y₂ verb?"

• disjuncts : List E

  The disjuncts

• whDomain : List E

  The wh-domain

• relation : W → E → E → Bool

  The relation

• disjunctionNP : String

  Description of the disjunction

• whPhrase : String

  The wh-phrase

structure Semantics.Questions.ScopeReadings.ChoiceAnswer (E : Type u_1) : Type u_1

A choice answer: potentially different answer for each disjunct.

• activeDisjunct : E

  Which disjunct is "active"

• answer : E

  The answer for that disjunct

def Semantics.Questions.ScopeReadings.instReprChoiceAnswer.repr {E✝ : Type u_1} [Repr E✝] : ChoiceAnswer E✝ → ℕ → Std.Format

Equations

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

instance Semantics.Questions.ScopeReadings.instReprChoiceAnswer {E✝ : Type u_1} [Repr E✝] : Repr (ChoiceAnswer E✝)

Equations

• Semantics.Questions.ScopeReadings.instReprChoiceAnswer = { reprPrec := Semantics.Questions.ScopeReadings.instReprChoiceAnswer.repr }

def Semantics.Questions.ScopeReadings.ChoiceQuestion.conditionalAnswer {W : Type u_1} {E : Type u_2} [DecidableEq E] (cq : ChoiceQuestion W E) (w : W) : List (E × List E)

Under choice reading, different disjuncts can give different answers.

"Whom does John or Mary love?"

• Answer: "John loves Bill; Mary loves Sue"
• This is the CONDITIONAL ANSWER: for each disjunct, the answer.

Equations

• cq.conditionalAnswer w = List.map (fun (d : E) => (d, List.filter (cq.relation w d) cq.whDomain)) cq.disjuncts

def Semantics.Questions.ScopeReadings.ChoiceQuestion.toLiftedQuestion_choice {W : Type u_1} {E : Type u_2} [BEq W] [DecidableEq E] (cq : ChoiceQuestion W E) : LiftedTypes.LiftedQuestion W

Convert choice question to LiftedQuestion under choice reading.

The choice reading involves existential quantification over disjuncts, which is like disjunction. Since disjunction of equivalence relations is NOT an equivalence relation, we must use the lifted type.

Two worlds are equivalent iff for SOME disjunct, they give the same answer.

Equations

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

def Semantics.Questions.ScopeReadings.ChoiceQuestion.toGSQuestion_nonchoice {W : Type u_1} {E : Type u_2} [DecidableEq E] (cq : ChoiceQuestion W E) : GSQuestion W

Under NON-choice reading, worlds match iff ALL disjunct-answers match.

Equations

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

Relationship between choice and non-choice readings

The choice reading (existential/disjunctive) is semantically COARSER than the non-choice reading (universal/conjunctive). However, since they now have different types (LiftedQuestion vs GSQuestion), direct comparison via refinement (⊑) is not available.

To compare them, lift the non-choice reading and show that the lifted non-choice "entails" the choice reading in the sense that any property holding of all disjunct-questions holds of at least one.

Existential Wide Scope

"Which student does some professor recommend?"

Similar to choice readings, but with existential quantifier:

Narrow Scope: ?x. ∃y. Prof(y) ∧ Recommends(y,x)

• Which student is recommended by at least one professor?

Wide Scope: ∃y. Prof(y) ∧ ?x. Recommends(y,x)

• There is some professor; which student do they recommend?

The wide scope reading is a "functional" or "choice" reading where the existential picks out a specific professor.

structure Semantics.Questions.ScopeReadings.ExistentialQuestion (W : Type u_1) (E : Type u_2) : Type (max u_1 u_2)

An existential-over-wh question.

• existentialDomain : List E

  The existential domain

• whDomain : List E

  The wh-domain

• relation : W → E → E → Bool

  The relation

def Semantics.Questions.ScopeReadings.ExistentialQuestion.narrowScope {W E : Type} [DecidableEq E] [DecidableEq (List E)] (eq : ExistentialQuestion W E) : GSQuestion W

Narrow scope: ∃y. ?x. R(y,x) collapses to ?x. ∃y. R(y,x)

Equations

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

def Semantics.Questions.ScopeReadings.ExistentialQuestion.wideScope {W : Type u_1} {E : Type u_2} [BEq W] [DecidableEq E] (eq : ExistentialQuestion W E) : LiftedTypes.LiftedQuestion W

Wide scope: for the "relevant" y, what's the answer?

This is similar to choice reading - the answer can vary based on which existential witness is considered.

Since this involves existential quantification (which is like disjunction), we must use the lifted type to avoid the transitivity problem.

Equations

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

Functional Readings

When the wh-answer is FUNCTIONALLY DEPENDENT on a wide-scope quantifier, we get readings where the answer is a function.

"Which of his teachers does every student admire?" → Answer: "John admires Smith, Mary admires Jones,..." → A function from students to teachers

The function IS the complete answer under the pair-list reading.

def Semantics.Questions.ScopeReadings.FunctionalAnswer (D : Type u_1) (R : Type u_2) : Type (max u_1 u_2)

A functional answer: maps each element of one domain to another.

Equations

• Semantics.Questions.ScopeReadings.FunctionalAnswer D R = (D → Option R)

def Semantics.Questions.ScopeReadings.numFunctionalAnswers (domainSize rangeSize : ℕ) : ℕ

The number of possible functional answers.

Equations

• Semantics.Questions.ScopeReadings.numFunctionalAnswers domainSize rangeSize = (rangeSize + 1) ^ domainSize

def Semantics.Questions.ScopeReadings.hasFunctionalReading {W : Type u_1} {E : Type u_2} [DecidableEq E] (plq : PairListQuestion W E) (worlds : List W) : Bool

A question has a functional reading if the answer is a function.

This happens when:

1. There's a pair-list configuration (universal over wh)
2. For each domain element, there's a unique wh-answer

Equations

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

Mention-Some and Scope

"Where can I buy an Italian newspaper?"

This has a mention-some reading: ONE place suffices.

The existential "an Italian newspaper" can take wide scope: ∃x. Paper(x) ∧ ?place. Sells(place, x)

But this doesn't fully explain mention-some, which is about the QUESTIONER'S GOAL, not scope.

However, wide scope CAN interact with mention-some:

• "Which student does some professor recommend?" (choice + mention-some)
• "Who speaks some Scandinavian language?" (existential + mention-some)

See MentionSome.lean for full G&S Section 5 treatment:

• I-MS rule (formal semantic operation)
• P-ANS analysis and its problems
• Embedded mention-some under "know"/"wonder"
• Verb licensing (why "depends" blocks mention-some)

def Semantics.Questions.ScopeReadings.ExistentialQuestion.toMentionSomeInterrogative {W : Type u_1} {E : Type u_2} (eq : ExistentialQuestion W E) : MentionSome.MentionSomeInterrogative W E

Convert an ExistentialQuestion to a MentionSomeInterrogative.

This connects the scope-based existential question to the G&S Section 5 formal treatment of mention-some.

Equations

• eq.toMentionSomeInterrogative = { whDomain := eq.whDomain, abstract := fun (w : W) (x : E) => eq.existentialDomain.any fun (y : E) => eq.relation w y x }

theorem Semantics.Questions.ScopeReadings.wideScope_existential_licenses_mentionSome {W : Type u_1} {E : Type u_2} [DecidableEq E] (eq : ExistentialQuestion W E) (w : W) (_hExists : (eq.existentialDomain.any fun (y : E) => eq.whDomain.any fun (x : E) => eq.relation w y x) = true) : (eq.toMentionSomeInterrogative.whDomain.any fun (x : E) => eq.toMentionSomeInterrogative.abstract w x) = true

Wide-scope existential implies mention-some is licensed.

@cite{groenendijk-stokhof-1984}, Section 5: When an existential takes wide scope over the wh-phrase, the question receives a mention-some interpretation because any witness for the existential provides a sufficient answer.

When Do These Readings Arise?

Pair-List Licensing:

1. Universal quantifier c-commands wh-phrase
2. The question has a "natural functional interpretation"
3. The answer can be presented as a list

"Which student did each professor recommend?" ✓ "Each professor recommended which student?" (marked)

Choice Licensing:

1. Disjunction/existential c-commands wh-phrase
2. The different disjuncts/witnesses are contextually salient
3. Conditional answers are felicitous

"Whom does John or Mary love?" ✓ (if both John and Mary are salient)

Factors:

• Attitude verbs: "know" licenses pair-list more than "wonder"
• Definiteness: definite subjects are harder for pair-list
• Information structure: contrastive focus on universal

structure Semantics.Questions.ScopeReadings.PairListFactors : Type

Factors affecting pair-list availability.

• attitudeVerb : Option String

  The attitude verb (if embedded)

• universalDefinite : Bool

  Is the universal definite?

• contrastiveFocus : Bool

  Does the universal have contrastive focus?

• isMatrix : Bool

  Is the question matrix or embedded?

def Semantics.Questions.ScopeReadings.predictPairList (factors : PairListFactors) : Bool

Predict pair-list availability from factors.

Equations

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