[Bru01] — QR Obeys Superiority

[Bru01] [Lar88] [May85] [Pyl08]

Bruening's Linguistic Inquiry paper "QR Obeys Superiority: Frozen Scope and ACD" — both the empirical scope-freezing data set and the theoretical Minimalist analysis (QR locality + superiority + phase-theoretic barriers).

Part I: Empirical data

Theory-neutral scope-freezing examples primarily compiled from [Bru01], with contributions from [Lar88] (double-object construction examples) and [May85] (foundational scope-availability vocabulary).

Part II: Theoretical analysis (Minimalist QR)

Bruening's central thesis — QR obeys Superiority — is derived formally: double-object scope freezing follows from asymmetric c-command in [Pyl08]'s Voice + low-Appl tree, where the goal asymmetrically c-commands the theme. Other freezing contexts (possessor, passive, attitude) are analyzed via DP-phase / adjunct-island / clause-boundary barriers.

Sections

• ScopeFreezing: Empirical configurations where inverse scope is unavailable
• MinimalistAnalysis: Theoretical derivation of freezing from Minimalist QR

The verb-cluster word-order scope data formerly housed here was [Ste00a] §6.8 material and now lives in Linglib.Data.Examples.Steedman2000 / Linglib.Studies.Steedman2000.

Scope: QR and Scope Economy (relocated from Minimalist/Scope.lean)

Formalization of quantifier scope in the Minimalist tradition.

Core Mechanisms

1. Quantifier Raising (QR): Covert A'-movement of quantifiers to adjoin to TP/CP
2. Scope Economy: QR only applies if it yields a distinct interpretation
3. Locality: QR is clause-bounded and blocked by certain structural barriers

Scope Freezing in Minimalism

Inverse scope is unavailable when:

• Possessor: Quantifier inside possessor DP cannot escape (DP is a phase/barrier)
• Double Object: Indirect object c-commands direct object; QR would violate locality
• Passive: By-phrase is an adjunct; QR from adjuncts is blocked

Superiority from C-Command (not stipulation)

In earlier versions, PositionedQuantifier carried a stipulated inDoubleObject : Bool flag. This has been replaced: the theory layer provides superiorityFromTree, which derives superiority from asymmetric c-command in a SyntacticObject tree. This study uses it to connect DOC scope freezing to [Lar88]'s tree derivation.

inductive Bruening2001.Position : Type

Structural positions relevant for scope

• specTP : Position
• specVP : Position
• adjunct : Position
• embedded : Position
• complement : Position

@[implicit_reducible]

instance Bruening2001.instDecidableEqPosition : DecidableEq Position

Equations

• Bruening2001.instDecidableEqPosition x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Bruening2001.instReprPosition.repr : Position → ℕ → Std.Format

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@[implicit_reducible]

instance Bruening2001.instReprPosition : Repr Position

Equations

• Bruening2001.instReprPosition = { reprPrec := Bruening2001.instReprPosition.repr }

@[implicit_reducible]

instance Bruening2001.instInhabitedPosition : Inhabited Position

Equations

• Bruening2001.instInhabitedPosition = { default := Bruening2001.instInhabitedPosition.default }

structure Bruening2001.PositionedQuantifier : Type

A quantifier with its structural position

• quantifier : String
• position : Position
• insideDP : Bool

  Is this inside another DP?

• so : Option Minimalist.SyntacticObject

  The SyntacticObject this quantifier corresponds to in the tree. When provided, superiority can be derived from c-command.

@[implicit_reducible]

instance Bruening2001.instReprPositionedQuantifier : Repr PositionedQuantifier

Equations

• Bruening2001.instReprPositionedQuantifier = { reprPrec := Bruening2001.instReprPositionedQuantifier.repr }

def Bruening2001.instReprPositionedQuantifier.repr : PositionedQuantifier → ℕ → Std.Format

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structure Bruening2001.QROperation : Type

Quantifier Raising (QR) as covert movement.

QR adjoins a quantifier to a clausal node (TP or CP), allowing it to take scope over material it c-commands at LF.

• target : PositionedQuantifier

  The quantifier being raised

• landingSite : Position

  Landing site

• createsNewScope : Bool

  Does this create a new scope relation?

@[implicit_reducible]

instance Bruening2001.instReprQROperation : Repr QROperation

Equations

• Bruening2001.instReprQROperation = { reprPrec := Bruening2001.instReprQROperation.repr }

def Bruening2001.instReprQROperation.repr : QROperation → ℕ → Std.Format

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inductive Bruening2001.QRBarrier : Type

Barriers to QR movement.

Following phase theory and earlier barrier theory, certain nodes block extraction/QR.

• dpPhase : QRBarrier
• clauseBoundary : QRBarrier
• adjunctIsland : QRBarrier
• superiority : QRBarrier

@[implicit_reducible]

instance Bruening2001.instDecidableEqQRBarrier : DecidableEq QRBarrier

Equations

• Bruening2001.instDecidableEqQRBarrier x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Bruening2001.instReprQRBarrier.repr : QRBarrier → ℕ → Std.Format

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@[implicit_reducible]

instance Bruening2001.instReprQRBarrier : Repr QRBarrier

Equations

• Bruening2001.instReprQRBarrier = { reprPrec := Bruening2001.instReprQRBarrier.repr }

@[implicit_reducible]

instance Bruening2001.instInhabitedQRBarrier : Inhabited QRBarrier

Equations

• Bruening2001.instInhabitedQRBarrier = { default := Bruening2001.instInhabitedQRBarrier.default }

def Bruening2001.qrIsBlocked (q : PositionedQuantifier) : Option QRBarrier

Check if QR is blocked for a given quantifier

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def Bruening2001.superiorityFromTree (tree q1 q2 : Minimalist.SyntacticObject) : Bool

Superiority derived from a tree: QR of q2 over q1 is blocked when q1 asymmetrically c-commands q2 in tree.

This is the theory-layer primitive. Bridge files use it to derive DOC scope freezing from [Pyl08]'s Voice + low-Appl tree derivation.

Equations

• Bruening2001.superiorityFromTree tree q1 q2 = decide (Minimalist.SO.asymCCommandsIn tree q1 q2)

structure Bruening2001.ScopeEconomy : Type

Scope Economy: QR is only licensed if it creates a truth-conditional difference.

"Covert scope-shifting operations are blocked if they don't have a semantic effect (i.e., if they yield a logically equivalent interpretation)."

• surfaceInterpretation : String

  Surface scope interpretation

• inverseInterpretation : String

  Would-be inverse interpretation

• equivalent : Bool

  Are they truth-conditionally equivalent?

@[implicit_reducible]

instance Bruening2001.instReprScopeEconomy : Repr ScopeEconomy

Equations

• Bruening2001.instReprScopeEconomy = { reprPrec := Bruening2001.instReprScopeEconomy.repr }

def Bruening2001.instReprScopeEconomy.repr : ScopeEconomy → ℕ → Std.Format

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def Bruening2001.economyBlocksQR (e : ScopeEconomy) : Bool

QR is blocked by economy if interpretations are equivalent

Equations

• Bruening2001.economyBlocksQR e = e.equivalent

theorem Bruening2001.dp_phase_barrier_from_pic (φ : Minimalist.Phase) {goal : Minimalist.SyntacticObject} (hacc : goal ∈ Minimalist.SO.Acc φ.tree) (hcc : Minimalist.SO.cCommandsIn φ.tree (Minimalist.SO.lexLeaf φ.head) goal) : φ.Impenetrable goal

DP-as-barrier follows from PIC: if φ is the D-phase (D being a phase head under the extended inventory) and goal is an accessible term sitting in the phase head's c-command domain, then goal is frozen — φ.Impenetrable goal.

This derives the previously-stipulated QRBarrier.dpPhase from deeper principles: PIC makes DP-internal material inaccessible to operations outside DP. On the MCB-faithful SO phase API the interior Φ°_ℓ is the head's c-command domain (Def 1.14.3, "Z is the interior of the phase"), so a goal in that domain is impenetrable by construction (SO.mem_phaseInterior).

inductive Bruening2001.Availability : Type

Available scope readings for a sentence

• ambiguous : Availability
• surfaceOnly : Availability
• inverseOnly : Availability

@[implicit_reducible]

instance Bruening2001.instDecidableEqAvailability : DecidableEq Availability

Equations

• Bruening2001.instDecidableEqAvailability x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Bruening2001.instReprAvailability.repr : Availability → ℕ → Std.Format

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@[implicit_reducible]

instance Bruening2001.instReprAvailability : Repr Availability

Equations

• Bruening2001.instReprAvailability = { reprPrec := Bruening2001.instReprAvailability.repr }

@[implicit_reducible]

instance Bruening2001.instInhabitedAvailability : Inhabited Availability

Equations

• Bruening2001.instInhabitedAvailability = { default := Bruening2001.instInhabitedAvailability.default }

inductive Bruening2001.Confidence : Type

Confidence in the judgment

• clear : Confidence
• gradient : Confidence
• controversial : Confidence

@[implicit_reducible]

instance Bruening2001.instDecidableEqConfidence : DecidableEq Confidence

Equations

• Bruening2001.instDecidableEqConfidence x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Bruening2001.instReprConfidence.repr : Confidence → ℕ → Std.Format

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@[implicit_reducible]

instance Bruening2001.instReprConfidence : Repr Confidence

Equations

• Bruening2001.instReprConfidence = { reprPrec := Bruening2001.instReprConfidence.repr }

@[implicit_reducible]

instance Bruening2001.instInhabitedConfidence : Inhabited Confidence

Equations

• Bruening2001.instInhabitedConfidence = { default := Bruening2001.instInhabitedConfidence.default }

inductive Bruening2001.DataSource : Type

Source of the judgment

• introspective : DataSource
• experimental : DataSource
• corpus : DataSource

@[implicit_reducible]

instance Bruening2001.instDecidableEqDataSource : DecidableEq DataSource

Equations

• Bruening2001.instDecidableEqDataSource x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Bruening2001.instReprDataSource : Repr DataSource

Equations

• Bruening2001.instReprDataSource = { reprPrec := Bruening2001.instReprDataSource.repr }

def Bruening2001.instReprDataSource.repr : DataSource → ℕ → Std.Format

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@[implicit_reducible]

instance Bruening2001.instInhabitedDataSource : Inhabited DataSource

Equations

• Bruening2001.instInhabitedDataSource = { default := Bruening2001.instInhabitedDataSource.default }

inductive Bruening2001.FreezingContext : Type

Types of configurations that induce scope freezing

• none : FreezingContext
• possessor : FreezingContext
• doubleObject : FreezingContext
• passive : FreezingContext
• heavyNP : FreezingContext
• weakCrossover : FreezingContext
• adjunct : FreezingContext
• attitude : FreezingContext

@[implicit_reducible]

instance Bruening2001.instDecidableEqFreezingContext : DecidableEq FreezingContext

Equations

• Bruening2001.instDecidableEqFreezingContext x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Bruening2001.instReprFreezingContext : Repr FreezingContext

Equations

• Bruening2001.instReprFreezingContext = { reprPrec := Bruening2001.instReprFreezingContext.repr }

def Bruening2001.instReprFreezingContext.repr : FreezingContext → ℕ → Std.Format

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@[implicit_reducible]

instance Bruening2001.instInhabitedFreezingContext : Inhabited FreezingContext

Equations

• Bruening2001.instInhabitedFreezingContext = { default := Bruening2001.instInhabitedFreezingContext.default }

structure Bruening2001.Example : Type

A scope freezing example with empirical judgment

• id : String
• sentence : String
• quant1 : String
• quant2 : String
• context : FreezingContext
• observed : Availability
• confidence : Confidence
• source : DataSource
• surfaceGloss : String
• inverseGloss : String
• notes : String

@[implicit_reducible]

instance Bruening2001.instReprExample : Repr Example

Equations

• Bruening2001.instReprExample = { reprPrec := Bruening2001.instReprExample.repr }

def Bruening2001.instReprExample.repr : Example → ℕ → Std.Format

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def Bruening2001.possessor_baseline : Example

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def Bruening2001.possessor_frozen : Example

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def Bruening2001.possessor_variant1 : Example

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def Bruening2001.possessor_variant2 : Example

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def Bruening2001.dative_baseline : Example

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def Bruening2001.dative_frozen : Example

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def Bruening2001.dative_variant : Example

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def Bruening2001.passive_baseline : Example

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def Bruening2001.passive_frozen : Example

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def Bruening2001.passive_variant : Example

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def Bruening2001.heavy_baseline : Example

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def Bruening2001.heavy_frozen : Example

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def Bruening2001.crossover_baseline : Example

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def Bruening2001.crossover_frozen : Example

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def Bruening2001.attitude_frozen : Example

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def Bruening2001.possessorExamples : List Example

Equations

• Bruening2001.possessorExamples = [Bruening2001.possessor_baseline, Bruening2001.possessor_frozen, Bruening2001.possessor_variant1, Bruening2001.possessor_variant2]

def Bruening2001.doubleObjectExamples : List Example

Equations

• Bruening2001.doubleObjectExamples = [Bruening2001.dative_baseline, Bruening2001.dative_frozen, Bruening2001.dative_variant]

def Bruening2001.passiveExamples : List Example

Equations

• Bruening2001.passiveExamples = [Bruening2001.passive_baseline, Bruening2001.passive_frozen, Bruening2001.passive_variant]

def Bruening2001.heavyNPExamples : List Example

Equations

• Bruening2001.heavyNPExamples = [Bruening2001.heavy_baseline, Bruening2001.heavy_frozen]

def Bruening2001.crossoverExamples : List Example

Equations

• Bruening2001.crossoverExamples = [Bruening2001.crossover_baseline, Bruening2001.crossover_frozen]

def Bruening2001.attitudeExamples : List Example

Equations

• Bruening2001.attitudeExamples = [Bruening2001.attitude_frozen]

def Bruening2001.allExamples : List Example

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def Bruening2001.possessorFreezingIsClear : Bool

Possessor freezing is robust (clear judgments)

Equations

• Bruening2001.possessorFreezingIsClear = (Bruening2001.possessor_frozen.confidence == Bruening2001.Confidence.clear)

def Bruening2001.doubleObjectFreezingIsClear : Bool

Double object freezing is robust

Equations

• Bruening2001.doubleObjectFreezingIsClear = (Bruening2001.dative_frozen.confidence == Bruening2001.Confidence.clear)

def Bruening2001.passiveFreezingIsGradient : Bool

Passive freezing is more gradient

Equations

• Bruening2001.passiveFreezingIsGradient = (Bruening2001.passive_frozen.confidence == Bruening2001.Confidence.gradient)

def Bruening2001.frozenCount : ℕ

Count frozen examples

Equations

• Bruening2001.frozenCount = (List.filter (fun (x : Bruening2001.Example) => x.observed == Bruening2001.Availability.surfaceOnly) Bruening2001.allExamples).length

def Bruening2001.ambiguousCount : ℕ

Count ambiguous baselines

Equations

• Bruening2001.ambiguousCount = (List.filter (fun (x : Bruening2001.Example) => x.observed == Bruening2001.Availability.ambiguous) Bruening2001.allExamples).length

Connects Minimalist QR / Scope Economy theory to the empirical scope-freezing data above. The central claim is Bruening's "QR obeys superiority": double-object freezing falls out of asymmetric c-command in the [Pyl08] ditransitive tree.

def Bruening2001.analyzeFreezingContext : FreezingContext → Option QRBarrier

Analyze why a freezing context blocks inverse scope in Minimalism.

Equations

• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.none = none
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.possessor = some Bruening2001.QRBarrier.dpPhase
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.doubleObject = some Bruening2001.QRBarrier.superiority
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.passive = some Bruening2001.QRBarrier.adjunctIsland
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.heavyNP = none
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.weakCrossover = none
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.adjunct = some Bruening2001.QRBarrier.adjunctIsland
• Bruening2001.analyzeFreezingContext Bruening2001.FreezingContext.attitude = some Bruening2001.QRBarrier.clauseBoundary

def Bruening2001.predictsFreezing (ctx : FreezingContext) : Bool

Does Minimalism predict freezing for this context?

Equations

• Bruening2001.predictsFreezing ctx = (Bruening2001.analyzeFreezingContext ctx).isSome

structure Bruening2001.MinimalistScopeConfig : Type

Minimalist representation of a scope configuration.

• q1 : PositionedQuantifier

  Higher quantifier (typically subject)

• q2 : PositionedQuantifier

  Lower quantifier (typically object)

• freezingContext : FreezingContext

  Freezing context if any

• tree : Option Minimalist.SyntacticObject

  The tree in which q1 and q2 are positioned. When provided, superiority is derived from c-command rather than stipulated.

@[implicit_reducible]

instance Bruening2001.instReprMinimalistScopeConfig : Repr MinimalistScopeConfig

Equations

• Bruening2001.instReprMinimalistScopeConfig = { reprPrec := Bruening2001.instReprMinimalistScopeConfig.repr }

def Bruening2001.instReprMinimalistScopeConfig.repr : MinimalistScopeConfig → ℕ → Std.Format

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def Bruening2001.superiorityBlocked (config : MinimalistScopeConfig) : Bool

Check if superiority blocks QR in this configuration.

When a tree and SO positions are provided, superiority is DERIVED from asymmetric c-command. Otherwise falls back to the freezing context annotation.

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def Bruening2001.availableReadings (config : MinimalistScopeConfig) : Availability

Compute available scope readings in Minimalism.

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def Bruening2001.configFromExample (ex : Example) : MinimalistScopeConfig

Build config from a freezing example (fallback path — no tree).

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def Bruening2001.predictAvailability (ex : Example) : Availability

Minimalism's prediction for an example.

Equations

• Bruening2001.predictAvailability ex = Bruening2001.availableReadings (Bruening2001.configFromExample ex)

def Bruening2001.correctlyPredicts (ex : Example) : Bool

Check if Minimalism correctly predicts the example.

Equations

• Bruening2001.correctlyPredicts ex = (Bruening2001.predictAvailability ex == ex.observed)

theorem Bruening2001.possessor_freezes_scope : predictsFreezing FreezingContext.possessor = true

Possessor freezing follows from DP being a phase.

theorem Bruening2001.double_object_freezes_scope : predictsFreezing FreezingContext.doubleObject = true

Double object freezing follows from superiority.

theorem Bruening2001.passive_freezes_scope : predictsFreezing FreezingContext.passive = true

Passive freezing follows from adjunct island.

theorem Bruening2001.heavy_np_not_grammatically_frozen : predictsFreezing FreezingContext.heavyNP = false

Heavy NP is NOT predicted to freeze (it's processing).

theorem Bruening2001.baseline_is_ambiguous : predictsFreezing FreezingContext.none = false

Baseline (no context) is predicted ambiguous.

[Pyl08]'s low-Appl tree produces the DOC structure where V takes ApplP as complement, so the goal in Spec-ApplP asymmetrically c-commands the theme in complement of Appl. QR of the theme over the goal is blocked by superiority, derived from c-command rather than stipulated.

The Voice + low-Appl tree is rebuilt locally (planar-first, since the smart Merge SO.node is noncomputable) so this study stays self-contained and the decide proof reduces. It mirrors Pylkkanen2008.ditransitiveTree's structure ([John [Voice [sent [Mary [Appl letter]]]]]); a future dedup can re-point at that tree once the SO-carrier flip reaches Studies/Pylkkanen2008.

def Bruening2001.voice_ag_t : Minimalist.LIToken

Voice[AG] head (introduces the external argument, [Kra96]).

Equations

• Bruening2001.voice_ag_t = { item := Minimalist.LexicalItem.simple Minimalist.Cat.Voice [Minimalist.Cat.V] "Voice[AG]", id := 400 }

def Bruening2001.appl_low_t : Minimalist.LIToken

Low applicative head: takes the theme DP as complement.

Equations

• Bruening2001.appl_low_t = { item := Minimalist.LexicalItem.simple Minimalist.Cat.Appl [Minimalist.Cat.D] "Appl[LOW]", id := 402 }

def Bruening2001.V_sent_t : Minimalist.LIToken

The ditransitive verb, selecting ApplP.

Equations

• Bruening2001.V_sent_t = { item := Minimalist.LexicalItem.simple Minimalist.Cat.V [Minimalist.Cat.Appl] "sent", id := 404 }

def Bruening2001.DP_john_t : Minimalist.LIToken

The agent DP.

Equations

• Bruening2001.DP_john_t = { item := Minimalist.LexicalItem.simple Minimalist.Cat.D [] "John", id := 406 }

def Bruening2001.DP_mary_t : Minimalist.LIToken

The goal DP (Spec,ApplP).

Equations

• Bruening2001.DP_mary_t = { item := Minimalist.LexicalItem.simple Minimalist.Cat.D [] "Mary", id := 407 }

def Bruening2001.DP_letter_t : Minimalist.LIToken

The theme DP (complement of Appl).

Equations

• Bruening2001.DP_letter_t = { item := Minimalist.LexicalItem.simple Minimalist.Cat.D [] "a letter", id := 408 }

def Bruening2001.ditransitiveTree : Minimalist.SyntacticObject

The Voice + low-Appl ditransitive tree [John [Voice [sent [Mary [Appl letter]]]]]. The goal (Mary) in Spec-ApplP asymmetrically c-commands the theme (a letter) in the complement of Appl — the [BL86] asymmetry, structural.

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def Bruening2001.docScopeConfig : MinimalistScopeConfig

DOC scope freezing config with the local low-Appl tree: superiority is derived from goal asymmetrically c-commanding theme in the Voice + low-Appl structure.

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theorem Bruening2001.doc_superiority_from_tree : superiorityBlocked docScopeConfig = true

Superiority in the DOC is DERIVED from c-command in [Pyl08]'s tree: goal (Mary) asymmetrically c-commands theme (a letter) via low Appl, so QR of theme over goal is blocked.

Summary

Context	Minimalist Explanation
Possessor	DP phase blocks QR
Double object	Superiority: goal c-commands theme ([Pyl08] tree)
Passive	Adjunct island
Heavy NP	NOT grammatical (processing)
Attitude	Clause boundary