Answering System Typology

@cite{holmberg-2016}

Cross-linguistic variation in how languages answer polar questions.

The Binary Parameter

@cite{holmberg-2016}'s central typological contribution: languages divide into two types based on what "yes" means in response to a negative question ("Doesn't John drink?"):

• Truth-based: "yes" affirms the proposition in the question. To "Doesn't he drink?", "yes" = "he doesn't drink" (Japanese, Mandarin, Thai).
• Polarity-based: "yes" assigns positive polarity. To "Doesn't he drink?", "yes" = "he does drink" (English, Swedish, German).

Answer Strategy

Orthogonally, languages vary in whether answers use dedicated particles or echo the finite verb:

• Particle: dedicated yes/no words (English yes/no, Japanese hai/iie)
• Verb echo: echoed finite verb (Finnish juo/ei juo, Welsh ydy/nac ydy)
• Mixed: both available (Swedish ja/nej + verb echo)

Connection to PolP

In @cite{holmberg-2016}'s syntax, every finite clause has a polarity head (PolP) bearing a valued or unvalued [±Pol] feature. In polar questions, [±Pol] is unvalued — the answer values it. The answering system parameter determines whether "yes" values the variable as [+Pol] (polarity-based) or affirms the question's primary proposition (truth-based).

inductive Semantics.Questions.AnsweringSystem : Type

How a language interprets "yes" in response to negative polar questions.

The diagnostic: "Doesn't John drink?" → "Yes" means...

• Truth-based: "He doesn't drink" (affirms the proposition)
• Polarity-based: "He does drink" (assigns positive polarity)

• truthBased : AnsweringSystem

  "Yes" affirms the proposition in the question (Japanese, Mandarin, Thai, Cantonese)

• polarityBased : AnsweringSystem

  "Yes" assigns positive polarity (English, Swedish, German, French, Finnish)

instance Semantics.Questions.instDecidableEqAnsweringSystem : DecidableEq AnsweringSystem

Equations

• Semantics.Questions.instDecidableEqAnsweringSystem x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Semantics.Questions.instReprAnsweringSystem.repr : AnsweringSystem → Nat → Std.Format

Equations

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

instance Semantics.Questions.instReprAnsweringSystem : Repr AnsweringSystem

Equations

• Semantics.Questions.instReprAnsweringSystem = { reprPrec := Semantics.Questions.instReprAnsweringSystem.repr }

def Semantics.Questions.instBEqAnsweringSystem.beq : AnsweringSystem → AnsweringSystem → Bool

Equations

• Semantics.Questions.instBEqAnsweringSystem.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Semantics.Questions.instBEqAnsweringSystem : BEq AnsweringSystem

Equations

• Semantics.Questions.instBEqAnsweringSystem = { beq := Semantics.Questions.instBEqAnsweringSystem.beq }

inductive Semantics.Questions.AnswerStrategy : Type

How a language forms answers to polar questions.

Orthogonal to AnsweringSystem — either system can combine with either strategy.

• particle : AnswerStrategy

  Dedicated yes/no particles (English yes/no, Japanese hai/iie)

• verbEcho : AnswerStrategy

  Echoed finite verb (Finnish juo/ei juo, Welsh ydy/nac ydy)

• mixed : AnswerStrategy

  Both particle and verb echo available (Swedish ja/nej + verb echo)

instance Semantics.Questions.instDecidableEqAnswerStrategy : DecidableEq AnswerStrategy

Equations

• Semantics.Questions.instDecidableEqAnswerStrategy x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Semantics.Questions.instReprAnswerStrategy.repr : AnswerStrategy → Nat → Std.Format

Equations

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

instance Semantics.Questions.instReprAnswerStrategy : Repr AnswerStrategy

Equations

• Semantics.Questions.instReprAnswerStrategy = { reprPrec := Semantics.Questions.instReprAnswerStrategy.repr }

instance Semantics.Questions.instBEqAnswerStrategy : BEq AnswerStrategy

Equations

• Semantics.Questions.instBEqAnswerStrategy = { beq := Semantics.Questions.instBEqAnswerStrategy.beq }

def Semantics.Questions.instBEqAnswerStrategy.beq : AnswerStrategy → AnswerStrategy → Bool

Equations

• Semantics.Questions.instBEqAnswerStrategy.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

structure Semantics.Questions.PolarAnswerProfile : Type

A language's polar answer profile: answering system + answer strategy.

• system : AnsweringSystem

  How "yes" is interpreted relative to negative questions

• strategy : AnswerStrategy

  How answers are formed (particle, verb echo, or both)

• hasPolarityReversal : Bool

  Does the language have a dedicated polarity-reversing particle (e.g., Swedish jo, German doch, French si)?

instance Semantics.Questions.instDecidableEqPolarAnswerProfile : DecidableEq PolarAnswerProfile

Equations

• Semantics.Questions.instDecidableEqPolarAnswerProfile = Semantics.Questions.instDecidableEqPolarAnswerProfile.decEq

def Semantics.Questions.instDecidableEqPolarAnswerProfile.decEq (x✝ x✝¹ : PolarAnswerProfile) : Decidable (x✝ = x✝¹)

Equations

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

def Semantics.Questions.instReprPolarAnswerProfile.repr : PolarAnswerProfile → Nat → Std.Format

Equations

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

instance Semantics.Questions.instReprPolarAnswerProfile : Repr PolarAnswerProfile

Equations

• Semantics.Questions.instReprPolarAnswerProfile = { reprPrec := Semantics.Questions.instReprPolarAnswerProfile.repr }

instance Semantics.Questions.instBEqPolarAnswerProfile : BEq PolarAnswerProfile

Equations

• Semantics.Questions.instBEqPolarAnswerProfile = { beq := Semantics.Questions.instBEqPolarAnswerProfile.beq }

def Semantics.Questions.instBEqPolarAnswerProfile.beq : PolarAnswerProfile → PolarAnswerProfile → Bool

Equations

• One or more equations did not get rendered due to their size.
• Semantics.Questions.instBEqPolarAnswerProfile.beq x✝¹ x✝ = false

def Semantics.Questions.AnsweringSystem.yesToNegativeQuestion : AnsweringSystem → Core.Polarity

The diagnostic prediction: what does "yes" mean in response to "Doesn't John drink?" under each answering system?

Returns the polarity of the proposition expressed by "yes".

Equations

• Semantics.Questions.AnsweringSystem.truthBased.yesToNegativeQuestion = Core.Polarity.negative
• Semantics.Questions.AnsweringSystem.polarityBased.yesToNegativeQuestion = Core.Polarity.positive

theorem Semantics.Questions.answering_systems_diverge_on_negative : AnsweringSystem.truthBased.yesToNegativeQuestion ≠ AnsweringSystem.polarityBased.yesToNegativeQuestion

Truth-based and polarity-based systems give opposite answers to negative questions.

Negation Height and Answering System Derivation

@cite{holmberg-2016} Ch 4.3-4.7 derives the answering system from whether the negation in the question can assign value to the polarity variable [±Pol]. The crucial factor is structural accessibility: if negation is close enough to [±Pol] to value it, the focused answer particle clashes with the inherited negative value → polarity-based. If negation is too distant or not in a c-command relation with [±Pol], the particle freely assigns value → truth-based.

• Low negation (VP-internal, below PolP scope): negation cannot reach [±Pol]. "Yes" affirms the (negative) proposition → truth-based. Examples: Japanese, Mandarin Chinese, Thai.

• Middle negation (NegP between TP and PolP, or merged with Pol): negation values [Pol] as [-Pol], creating a feature clash with an affirmative particle. "Yes" assigns [+Pol] → polarity-based. Examples: English (default), Swedish (§4.5), Finnish (§4.6, higher variety of middle), German.

• High negation (C-domain, above PolP scope): negation scopes over [±Pol] rather than valuing it. Used in positively-biased negative questions where the negation eliminates the negative alternative. Examples: English -n't in positively-biased readings (§4.8), English outer negation (§4.3).

This is the book's deepest explanatory contribution: the binary answering-system parameter is not stipulated but derived from independently motivated syntactic variation in negation height.

inductive Semantics.Questions.NegationHeight : Type

Structural height of sentential negation relative to PolP.

Note: this classifies constructions, not languages. A single language may have multiple negation heights (e.g., English has middle by default, low when scoped under an adverb, and high in positively-biased questions).

• low : NegationHeight

  Negation below PolP — VP-internal (Japanese, Mandarin, Thai)

• middle : NegationHeight

  Negation at PolP level — NegP between TP and PolP, or merged with Pol (English default, Swedish, Finnish, German)

• high : NegationHeight

  Negation above PolP — C-domain, scoping over [±Pol] (English -n't in positively-biased questions)

instance Semantics.Questions.instDecidableEqNegationHeight : DecidableEq NegationHeight

Equations

• Semantics.Questions.instDecidableEqNegationHeight x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Semantics.Questions.instReprNegationHeight : Repr NegationHeight

Equations

• Semantics.Questions.instReprNegationHeight = { reprPrec := Semantics.Questions.instReprNegationHeight.repr }

def Semantics.Questions.instReprNegationHeight.repr : NegationHeight → Nat → Std.Format

Equations

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

def Semantics.Questions.instBEqNegationHeight.beq : NegationHeight → NegationHeight → Bool

Equations

• Semantics.Questions.instBEqNegationHeight.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Semantics.Questions.instBEqNegationHeight : BEq NegationHeight

Equations

• Semantics.Questions.instBEqNegationHeight = { beq := Semantics.Questions.instBEqNegationHeight.beq }

def Semantics.Questions.NegationHeight.predictedSystem : NegationHeight → AnsweringSystem

Derive the answering system from negation height.

Low negation → truth-based: negation scopes below [±Pol], so the question's primary proposition includes negation. "Yes" affirms the (negative) proposition.

Middle/high negation → polarity-based: negation is at or above [±Pol], so "yes" values [±Pol] as [+Pol] regardless of negation.

Equations

• Semantics.Questions.NegationHeight.low.predictedSystem = Semantics.Questions.AnsweringSystem.truthBased
• Semantics.Questions.NegationHeight.middle.predictedSystem = Semantics.Questions.AnsweringSystem.polarityBased
• Semantics.Questions.NegationHeight.high.predictedSystem = Semantics.Questions.AnsweringSystem.polarityBased

theorem Semantics.Questions.middle_high_same_prediction : NegationHeight.middle.predictedSystem = NegationHeight.high.predictedSystem

Middle and high negation both predict polarity-based systems.

theorem Semantics.Questions.low_differs_from_middle : NegationHeight.low.predictedSystem ≠ NegationHeight.middle.predictedSystem

Low negation predicts a different system from middle negation.