Polarity-Marking Levels #

Blühdorn (2012), @cite{turco-braun-dimroth-2014}, @cite{hohle-1992}

Languages mark polarity switches (negation → affirmation) by targeting one of two distinct semantic levels:

Polarity level: a particle directly sets [+Pol], "undoing" contextual negation. Dutch wel, French si, Swedish jo.
Assertion level: prosodic prominence on the finite verb highlights the assertion operator — the element that carries the assertive relation between topic and comment. German Verum focus.

Both achieve polarity contrast/correction at the pragmatic level, but they operate on different structural components of the sentence and make different predictions about co-occurrence with negation.

Key prediction: negation compatibility #

Because the assertion operator wraps the polarized proposition, Verum focus is compatible with either polarity value:

"Das Kind HAT nicht geweint" (VF + neg = emphatic denial) ✓
"Das Kind HAT geweint" (VF + pos = emphatic assertion) ✓

Because a polarity particle is the polarity operator, it clashes with the opposite polarity value:

"Het kind heeft wel gehuild" (wel + pos) ✓
*"Het kind heeft wel niet gehuild" (wel + neg = contradictory) ✗

Sentence decomposition #

Following Blühdorn (2012), a sentence's polarity-relevant structure decomposes into three layers:

  ASSERT [ POL [ RADICAL ] ]
    ↑           ↑       ↑
    VF         wel    content

Verum focus targets ASSERT; polarity particles target POL. The radical is the polarity-neutral propositional content.

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inductive Semantics.Focus.PolarityLevel.PolarityMarkingLevel :

Type

The semantic level at which a polarity-marking device operates.

Blühdorn (2012): Verum focus and affirmative particles target different structural components, yielding different co-occurrence patterns with negation.

polarity : PolarityMarkingLevel
Polarity level: targets [±Pol] directly. Affirmative particles (Dutch wel) set [+Pol].
assertion : PolarityMarkingLevel
Assertion level: highlights the assertion operator (finiteness). German Verum focus (@cite{hohle-1992}).

Instances For

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@[implicit_reducible]

instance Semantics.Focus.PolarityLevel.instDecidableEqPolarityMarkingLevel :

DecidableEq PolarityMarkingLevel

Equations

Semantics.Focus.PolarityLevel.instDecidableEqPolarityMarkingLevel x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

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@[implicit_reducible]

instance Semantics.Focus.PolarityLevel.instReprPolarityMarkingLevel :

Repr PolarityMarkingLevel

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Semantics.Focus.PolarityLevel.instReprPolarityMarkingLevel = { reprPrec := Semantics.Focus.PolarityLevel.instReprPolarityMarkingLevel.repr }

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def Semantics.Focus.PolarityLevel.instReprPolarityMarkingLevel.repr :

PolarityMarkingLevel → ℕ → Std.Format

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

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def Semantics.Focus.PolarityLevel.strategyLevel :

Typology.PolarityMarking.Strategy → Option PolarityMarkingLevel

Map polarity-marking strategies to their semantic level. Returns none for strategies without a clear level assignment.

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Instances For

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structure Semantics.Focus.PolarityLevel.SentenceStructure (W : Type) :

Type

A sentence decomposed into its polarity-relevant structural layers.

Blühdorn (2012): every sentence has a polarity-neutral radical, a polarity operator [±Pol], and an assertion operator. Different polarity-marking devices target different layers.

The marking field uses Option PolarityMarkingLevel rather than two independent Bools — assertion-level and polarity-level marking are mutually exclusive by construction.

radical : W → Bool
Polarity-neutral propositional content
pol : Features.Polarity
The polarity value [±Pol]
marking : Option PolarityMarkingLevel
Which structural level is overtly marked, if any

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def Semantics.Focus.PolarityLevel.SentenceStructure.eval {W : Type} (s : SentenceStructure W) :

W → Bool

Apply polarity to the radical to get truth conditions.

Polarity is the innermost operator: it wraps the radical before the assertion operator applies.

Equations

s.eval = match (motive := Features.Polarity → W → Bool) s.pol with | Features.Polarity.positive => s.radical | Features.Polarity.negative => fun (w : W) => !s.radical w

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def Semantics.Focus.PolarityLevel.PolarityMarkingLevel.compatibleWith :

PolarityMarkingLevel → Features.Polarity → Bool

Is a marking level compatible with a given polarity value?

Assertion-level marking (VF) is compatible with either polarity because it targets the assertion operator, which wraps the already-polarized proposition.

Polarity-level marking (particles) requires [+Pol] — the particle IS the polarity operator, so it cannot coexist with [−Pol].

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def Semantics.Focus.PolarityLevel.SentenceStructure.wellFormed {W : Type} (s : SentenceStructure W) :

Bool

Well-formedness constraint on polarity marking.

Delegates to PolarityMarkingLevel.compatibleWith: unmarked sentences are always well-formed; marked sentences must have a level compatible with their polarity value.

Equations

s.wellFormed = match s.marking with | none => true | some level => level.compatibleWith s.pol

Instances For

Prediction 1: VF is compatible with negation #

"Das Kind HAT nicht geweint" — VF on a negative sentence yields emphatic denial (counter-presuppositional reading, Gussenhoven 1983).

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theorem Semantics.Focus.PolarityLevel.vf_negative_wellformed {W : Type} (radical : W → Bool) :

{ radical := radical, pol := Features.Polarity.negative, marking := some PolarityMarkingLevel.assertion }.wellFormed = true

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theorem Semantics.Focus.PolarityLevel.vf_positive_wellformed {W : Type} (radical : W → Bool) :

{ radical := radical, pol := Features.Polarity.positive, marking := some PolarityMarkingLevel.assertion }.wellFormed = true

Prediction 2: polarity particles require [+Pol] #

*"Het kind heeft wel niet gehuild" — wel in a negative sentence is contradictory.

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theorem Semantics.Focus.PolarityLevel.particle_negative_illformed {W : Type} (radical : W → Bool) :

{ radical := radical, pol := Features.Polarity.negative, marking := some PolarityMarkingLevel.polarity }.wellFormed = false

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theorem Semantics.Focus.PolarityLevel.particle_positive_wellformed {W : Type} (radical : W → Bool) :

{ radical := radical, pol := Features.Polarity.positive, marking := some PolarityMarkingLevel.polarity }.wellFormed = true

The two levels differ on negation compatibility #

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theorem Semantics.Focus.PolarityLevel.levels_differ_on_negation :

PolarityMarkingLevel.polarity.compatibleWith Features.Polarity.negative ≠ PolarityMarkingLevel.assertion.compatibleWith Features.Polarity.negative

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theorem Semantics.Focus.PolarityLevel.functional_equivalence_positive {W : Type} (radical : W → Bool) :

have vf := { radical := radical, pol := Features.Polarity.positive, marking := some PolarityMarkingLevel.assertion }; have prt := { radical := radical, pol := Features.Polarity.positive, marking := some PolarityMarkingLevel.polarity }; vf.eval = prt.eval

Despite operating at different semantic levels, both strategies yield the same truth conditions when applied to a positive proposition.

@cite{turco-braun-dimroth-2014}: Dutch wel and German VF are "functionally equivalent" for polarity contrast/correction — the pragmatic effect is the same even though the structural target differs.

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theorem Semantics.Focus.PolarityLevel.particle_targets_polarity :

strategyLevel Typology.PolarityMarking.Strategy.particle = some PolarityMarkingLevel.polarity

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theorem Semantics.Focus.PolarityLevel.vf_targets_assertion :

strategyLevel Typology.PolarityMarking.Strategy.verumFocus = some PolarityMarkingLevel.assertion

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theorem Semantics.Focus.PolarityLevel.particle_vf_different_levels :

strategyLevel Typology.PolarityMarking.Strategy.particle ≠ strategyLevel Typology.PolarityMarking.Strategy.verumFocus

Particles and VF target different levels.

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theorem Semantics.Focus.PolarityLevel.reversal_targets_polarity :