Holmberg (2016): The Syntax of Yes and No

@cite{holmberg-2016}

Core Contribution

A cross-linguistic typology of polar question answering. The central parameter is the answering system: truth-based vs polarity-based.

Key Claims Formalized

1. Hamblin ↔ [±Pol]: Hamblin's polar p yields exactly two answer cells, corresponding to [+Pol] and [-Pol] valuations.

2. Answering system divergence: Truth-based and polarity-based systems give opposite answers to negative questions.

3. Polarity reversal: Languages like Swedish (jo), German (doch), and French (si) have a dedicated particle that assigns [+Pol] while contradicting a negative context.

Connection to Existing Infrastructure

• Hamblin.polar (semantic question denotation)
• Minimalism.Polarity.PolFeature (syntactic [±Pol] feature)
• AnsweringSystem (typological parameter)
• NegationHeight → predictedSystem (negation height derives answering system)
• PolarAnswerProfile (per-language classification)
• VerumFocus.lean (@cite{romero-han-2004}): complementary analysis — VERUM explains structural source of bias, Holmberg explains cross-linguistic answer variation. Both derive unbalanced partitions for negative questions.

A Hamblin polar question {p, ¬p} corresponds to an unvalued [±Pol] feature. Each answer cell values the feature: - p → [+Pol] (affirmative) - ¬p → [-Pol] (negative)

The two answer propositions are the "positive cell" and "negative cell"
of the partition induced by the question. 

def Phenomena.Questions.Studies.Holmberg2016.positiveAnswer {W : Type u_1} (p : W → Bool) : W → Bool

The positive answer to a polar question: the proposition p itself.

Equations

• Phenomena.Questions.Studies.Holmberg2016.positiveAnswer p = p

def Phenomena.Questions.Studies.Holmberg2016.negativeAnswer {W : Type u_1} (p : W → Bool) : W → Bool

The negative answer: ¬p.

Equations

• Phenomena.Questions.Studies.Holmberg2016.negativeAnswer p w = !p w

theorem Phenomena.Questions.Studies.Holmberg2016.both_answers_in_polar {W : Type u_1} [BEq W] (p : W → Bool) (worlds : List W) (_hw : worlds ≠ []) : Semantics.Questions.Hamblin.polar p worlds (positiveAnswer p) = true ∧ Semantics.Questions.Hamblin.polar p worlds (negativeAnswer p) = true

Both answers are in the Hamblin denotation of polar p.

def Phenomena.Questions.Studies.Holmberg2016.positiveToPolFeature : Minimalism.Polarity.PolFeature

The positive answer maps to [+Pol] (valued positive).

Equations

• Phenomena.Questions.Studies.Holmberg2016.positiveToPolFeature = Minimalism.Polarity.PolFeature.valued Core.Polarity.positive

def Phenomena.Questions.Studies.Holmberg2016.negativeToPolFeature : Minimalism.Polarity.PolFeature

The negative answer maps to [-Pol] (valued negative).

Equations

• Phenomena.Questions.Studies.Holmberg2016.negativeToPolFeature = Minimalism.Polarity.PolFeature.valued Core.Polarity.negative

theorem Phenomena.Questions.Studies.Holmberg2016.positive_valuation : Minimalism.Polarity.PolFeature.unvalued.value Core.Polarity.positive = some positiveToPolFeature

Valuing [uPol] as positive gives [+Pol].

theorem Phenomena.Questions.Studies.Holmberg2016.negative_valuation : Minimalism.Polarity.PolFeature.unvalued.value Core.Polarity.negative = some negativeToPolFeature

Valuing [uPol] as negative gives [-Pol].

theorem Phenomena.Questions.Studies.Holmberg2016.diagnostic_prediction : Semantics.Questions.AnsweringSystem.truthBased.yesToNegativeQuestion = Core.Polarity.negative ∧ Semantics.Questions.AnsweringSystem.polarityBased.yesToNegativeQuestion = Core.Polarity.positive

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

• Truth-based: "He doesn't drink" (negative polarity)
• Polarity-based: "He does drink" (positive polarity)

theorem Phenomena.Questions.Studies.Holmberg2016.english_swedish_same_system : PolarAnswerStructure.englishProfile.system = PolarAnswerStructure.swedishProfile.system

English and Swedish are both polarity-based.

theorem Phenomena.Questions.Studies.Holmberg2016.japanese_mandarin_same_system : PolarAnswerStructure.japaneseProfile.system = PolarAnswerStructure.mandarinProfile.system

Japanese and Mandarin are both truth-based.

theorem Phenomena.Questions.Studies.Holmberg2016.english_japanese_differ : PolarAnswerStructure.englishProfile.system ≠ PolarAnswerStructure.japaneseProfile.system

English and Japanese differ in answering system.

theorem Phenomena.Questions.Studies.Holmberg2016.swedish_reversal_english_not : PolarAnswerStructure.swedishProfile.hasPolarityReversal = true ∧ PolarAnswerStructure.englishProfile.hasPolarityReversal = false

Swedish has polarity reversal; English does not.

theorem Phenomena.Questions.Studies.Holmberg2016.system_strategy_orthogonal : PolarAnswerStructure.englishProfile.strategy = PolarAnswerStructure.japaneseProfile.strategy ∧ PolarAnswerStructure.englishProfile.system ≠ PolarAnswerStructure.japaneseProfile.system

The answering system and answer strategy are orthogonal: both truth-based and polarity-based systems can use particles.

theorem Phenomena.Questions.Studies.Holmberg2016.japanese_negation_height_predicts : Semantics.Questions.NegationHeight.low.predictedSystem = PolarAnswerStructure.japaneseProfile.system

Japanese has low negation → truth-based predicted, matches actual profile.

theorem Phenomena.Questions.Studies.Holmberg2016.mandarin_negation_height_predicts : Semantics.Questions.NegationHeight.low.predictedSystem = PolarAnswerStructure.mandarinProfile.system

Mandarin has low negation → truth-based predicted, matches actual profile.

theorem Phenomena.Questions.Studies.Holmberg2016.english_negation_height_predicts : Semantics.Questions.NegationHeight.middle.predictedSystem = PolarAnswerStructure.englishProfile.system

English has middle negation → polarity-based predicted, matches actual profile.

theorem Phenomena.Questions.Studies.Holmberg2016.swedish_negation_height_predicts : Semantics.Questions.NegationHeight.middle.predictedSystem = PolarAnswerStructure.swedishProfile.system

Swedish has middle negation (exclusively, no low negation; §4.5) → polarity-based predicted, matches actual profile.

theorem Phenomena.Questions.Studies.Holmberg2016.finnish_negation_height_predicts : Semantics.Questions.NegationHeight.middle.predictedSystem = PolarAnswerStructure.finnishProfile.system

Finnish has middle negation (higher variety of middle; §4.6, p178: "still technically a middle negation position") → polarity-based predicted, matches actual profile.

theorem Phenomena.Questions.Studies.Holmberg2016.japanese_endtoend : Semantics.Questions.NegationHeight.low.predictedSystem.yesToNegativeQuestion = PolarAnswerStructure.japanese_hai_to_neg.answerPolarity

End-to-end: Japanese low negation → truth-based → "yes" to negative question has negative polarity → matches the Japanese hai datum.

theorem Phenomena.Questions.Studies.Holmberg2016.english_endtoend : Semantics.Questions.NegationHeight.middle.predictedSystem.yesToNegativeQuestion = PolarAnswerStructure.english_yes_to_neg.answerPolarity

End-to-end: English middle negation → polarity-based → "yes" to negative question has positive polarity → matches the English "yes" datum.

theorem Phenomena.Questions.Studies.Holmberg2016.endtoend_diverge : Semantics.Questions.NegationHeight.low.predictedSystem.yesToNegativeQuestion ≠ Semantics.Questions.NegationHeight.middle.predictedSystem.yesToNegativeQuestion

The end-to-end chains for Japanese and English yield opposite polarities, as predicted by their different negation heights.

@cite{holmberg-2016} §4.13: languages with a polarity-reversing particle (Swedish jo, German doch, French si) are correlated with the polarity-based system. Truth-based languages do not need a reversing particle because they can always use "no" to disconfirm the negative alternative of a negative question.

theorem Phenomena.Questions.Studies.Holmberg2016.truthBased_no_reversal : PolarAnswerStructure.japaneseProfile.hasPolarityReversal = false ∧ PolarAnswerStructure.mandarinProfile.hasPolarityReversal = false

Truth-based languages do not have polarity reversal in our profiles. (Japanese and Mandarin both lack a reversing particle.)

theorem Phenomena.Questions.Studies.Holmberg2016.polarityBased_reversal_variation : PolarAnswerStructure.swedishProfile.hasPolarityReversal = true ∧ PolarAnswerStructure.englishProfile.hasPolarityReversal = false

Among polarity-based languages, reversal is attested but not universal: Swedish has it, English does not.