Büring 2007: Cross-Polar Nomalies

@cite{buring-2007}

Daniel Büring. Cross-Polar Nomalies. SALT 17 (2007).

Core Puzzle

Cross-polar anomalies — comparisons pairing A⁺ with its direct antonym A⁻ — are ungrammatical: *"John is shorter than Mary is tall." But cross-polar nomalies — comparisons pairing A⁻ with a non-antonymous A⁺ from a different spatial dimension — are perfectly acceptable: "The ladder was shorter than the house was high."

Analysis

LITTLE is a degree negation operator (@cite{heim-2006}): short = LITTLE long, less = LITTLE -er. Formally, LITTLE complements a degree predicate: ⟦LITTLE⟧ = λi.λd. i(d) = 0, mapping positive extents to negative extents (@cite{kennedy-1999}).

Cross-polar nomalies work because MORE LITTLE-A in the main clause can be reinterpreted as LITTLE-er A. This reinterpretation is blocked for direct antonyms by comparative deletion (MaxElide) and for inverse configurations by the requirement that LITTLE license ellipsis only in its own clause.

Three-way pattern:

• Cross-polar anomaly: *A⁺-er than A⁻ / *A⁻-er than A⁺ (direct antonyms) — BAD
• Cross-polar nomaly: A⁻-er than A⁺ (different dimensions) — OK
• Inverse nomaly: *A⁺-er than A⁻ (different dimensions) — BAD

Two Competing Analyses (§3 vs §5)

• Analysis 1 (§3–4, preferred): LITTLE sits with -er in the main clause. "shorter than high" = LITTLE-er long than HOW the house is high. The degree negation scopes with the comparative morpheme.

• Analysis 2 (§5): A second LITTLE appears in the than-clause, turning high into LITTLE-high (= low). The main-clause LITTLE is then elided under identity with the than-clause LITTLE (comparative subdeletion).

Both predict the same truth conditions for basic cases. §6 uses modal scope as a diagnostic: universal/existential modals in the than-clause disambiguate the two, favoring Analysis 1.

Formal Connections

• LITTLE as extent complement: littlePred maps posExt to negExt, connecting to @cite{kennedy-1999}'s extent algebra in Core.Scale.
• Cross-polar anomaly = algebraic impossibility: same-dimension cross-polar comparison requires crossExtentInclusion, which crossExtent_always_false proves is impossible on any linear order.
• Cross-polar nomaly = subcomparative: different-dimension comparison is subcomparative from @cite{schwarzschild-wilkinson-2002}.
• Klein limitation bridge: @cite{von-stechow-1984}'s Klein limitation 3 ("Ede is more tall than broad") is exactly a subcomparative.

theorem Buring2007.little_positive_to_negative {Entity : Type u_1} {D : Type u_2} [LinearOrder D] [BoundedOrder D] (μ : Entity → D) (x : Entity) : (Semantics.Degree.Intervals.negativeInterval μ x).lower = (Semantics.Degree.Intervals.positiveInterval μ x).upper

LITTLE maps positive intervals to negative intervals (@cite{buring-2007} §4, def. 22): the positive interval [⊥, μ(x)] becomes the negative interval [μ(x), ⊤]. This is the interval-level counterpart of little_posExt_eq_negExt (which operates on extent sets).

The bridge connects the interval framework (Schwarzschild) to the extent framework (Kennedy) via LITTLE.

theorem Buring2007.crossPolar_anomaly_impossible {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ : Entity → D) (a b : Entity) : ¬Core.Scale.crossExtentInclusion μ a b

Cross-polar anomaly = attempting to compare a positive extent with a negative extent on the same dimension.

"?*John is shorter than Mary is tall" requires posExt(Mary) ⊆ negExt(John), but crossExtent_always_false from @cite{kennedy-1999}'s extent algebra proves this is impossible on any linear order: the boundary degree μ(a) belongs to posExt but not negExt, so posExt can never be a subset of negExt.

Note: @cite{buring-2007}'s explanation is syntactic (MaxElide §3.2), not algebraic. The algebraic impossibility is a stronger claim: even if the LF were syntactically available, the semantics would be vacuous. Büring's account is compatible — MaxElide blocks the LF before semantics applies.

inductive Buring2007.CrossPolarType : Type

Classification of cross-polar configurations (p. 3).

• anomaly : CrossPolarType
• nomaly : CrossPolarType
• inverseNomaly : CrossPolarType

@[implicit_reducible]

instance Buring2007.instDecidableEqCrossPolarType : DecidableEq CrossPolarType

Equations

• Buring2007.instDecidableEqCrossPolarType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Buring2007.instReprCrossPolarType.repr : CrossPolarType → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Buring2007.instReprCrossPolarType : Repr CrossPolarType

Equations

• Buring2007.instReprCrossPolarType = { reprPrec := Buring2007.instReprCrossPolarType.repr }

structure Buring2007.CrossPolarDatum : Type

• sentence : String
• classification : CrossPolarType
• grammatical : Bool

@[implicit_reducible]

instance Buring2007.instReprCrossPolarDatum : Repr CrossPolarDatum

Equations

• Buring2007.instReprCrossPolarDatum = { reprPrec := Buring2007.instReprCrossPolarDatum.repr }

def Buring2007.instReprCrossPolarDatum.repr : CrossPolarDatum → ℕ → Std.Format

Equations

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

def Buring2007.crossPolarData : List CrossPolarDatum

Equations

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

theorem Buring2007.subcomparative_same_dimension {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ : Entity → D) (a b : Entity) : Semantics.Degree.Intervals.subcomparative μ μ a b ↔ Semantics.Degree.Comparative.comparativeSem μ a b Core.Scale.ScalePolarity.positive

When both dimensions use the same measure function, the subcomparative collapses to the standard comparative: "a is shorter than b" = "b is taller than a".

@[implicit_reducible]

instance Buring2007.instDecidableEqThing : DecidableEq Buring2007.Thing✝

Equations

• Buring2007.instDecidableEqThing x✝ y✝ = if h : Buring2007.Thing.ctorIdx✝ x✝ = Buring2007.Thing.ctorIdx✝ y✝ then isTrue ⋯ else isFalse ⋯

def Buring2007.instReprThing.repr : Buring2007.Thing✝ → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Buring2007.instReprThing : Repr Buring2007.Thing✝

Equations

• Buring2007.instReprThing = { reprPrec := Buring2007.instReprThing.repr }

structure Buring2007.AnomalyBlockingDatum : Type

@cite{buring-2007}'s syntactic explanation for why direct-antonym cross-polar constructions are anomalous (§3.2): when A⁻ and A⁺ ARE direct antonyms (same dimension), comparative deletion (ellipsis of the whole A in the than-clause) produces a competing form. MaxElide (Takahashi and Fox 2005) prefers this deletion, blocking the cross-polar LF.

For nomalies, deletion is unavailable because the adjectives differ (long ≠ high), so no competition arises (§3.3).

Inverse nomalies (*A⁺-er than A⁻, different dimensions) are blocked because LITTLE in the main clause cannot license ellipsis in the than-clause (§3.4): the LF "the house is MORE high [than HOW the ladder is LITTLE-long]" cannot be reinterpreted as "LITTLE-er high" because LITTLE and MORE are in separate clauses.

• sentence : String
• whyBlocked : String
• competingForm : String

def Buring2007.instReprAnomalyBlockingDatum.repr : AnomalyBlockingDatum → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Buring2007.instReprAnomalyBlockingDatum : Repr AnomalyBlockingDatum

Equations

• Buring2007.instReprAnomalyBlockingDatum = { reprPrec := Buring2007.instReprAnomalyBlockingDatum.repr }

def Buring2007.anomalyBlocking : List AnomalyBlockingDatum

Equations

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

theorem Buring2007.klein_limitation_is_subcomparative {Entity : Type u_1} {D : Type u_2} [LinearOrder D] (μ₁ μ₂ : Entity → D) (a : Entity) : Semantics.Degree.Intervals.subcomparative μ₁ μ₂ a a ↔ μ₁ a > μ₂ a

@cite{von-stechow-1984}'s Klein limitation 3: "Ede is more tall than broad" is a cross-dimensional comparison that Klein's degree-free framework cannot express.

@cite{buring-2007}'s cross-polar nomalies are the same phenomenon: "shorter(length) than high(height)" compares different dimensions on a shared spatial extent scale. Both require degree ontology (specifically, subcomparative from @cite{schwarzschild-wilkinson-2002}).

Definitionally: comparing two dimensions of the same entity is subcomparative μ₁ μ₂ a a, which unfolds to μ₁ a > μ₂ a.

@[implicit_reducible]

instance Buring2007.instDecidableEqCastlePart : DecidableEq Buring2007.CastlePart✝

Equations

• Buring2007.instDecidableEqCastlePart x✝ y✝ = if h : Buring2007.CastlePart.ctorIdx✝ x✝ = Buring2007.CastlePart.ctorIdx✝ y✝ then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Buring2007.instReprCastlePart : Repr Buring2007.CastlePart✝

Equations

• Buring2007.instReprCastlePart = { reprPrec := Buring2007.instReprCastlePart.repr }

def Buring2007.instReprCastlePart.repr : Buring2007.CastlePart✝ → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Buring2007.instDecidableEqPermittedWorld : DecidableEq Buring2007.PermittedWorld✝

Equations

• Buring2007.instDecidableEqPermittedWorld x✝ y✝ = if h : Buring2007.PermittedWorld.ctorIdx✝ x✝ = Buring2007.PermittedWorld.ctorIdx✝ y✝ then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Buring2007.instReprPermittedWorld : Repr Buring2007.PermittedWorld✝

Equations

• Buring2007.instReprPermittedWorld = { reprPrec := Buring2007.instReprPermittedWorld.repr }

def Buring2007.instReprPermittedWorld.repr : Buring2007.PermittedWorld✝ → ℕ → Std.Format

Equations

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

def Buring2007.analysis1_minRequiredWidth : ℚ

Analysis 1 (preferred): LITTLE scopes with -er in main clause. The than-clause denotes {d | ∀w ∈ Deon(@). d ≤ WIDTH_w(moat)}, whose max is the minimum required width (= 30). The comparative asserts: min-required-width > bridge-length.

Truth conditions: the bridge is shorter than the moat's minimum required width. This is correct — the bridge (15ft) can't span a moat that must be at least 30ft wide.

Equations

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

theorem Buring2007.analysis1_correct : analysis1_minRequiredWidth > Buring2007.bridgeLength✝

def Buring2007.analysis2_maxPermittedNarrowness (maxWidth : ℚ) : ℚ

Analysis 2: LITTLE scopes in the than-clause (with the adjective). HAS-TO scopes over LITTLE-wide (= narrow). The than-clause denotes {d | ∀w ∈ Deon(@). NARROWNESS_w(moat) ≥ d}, whose max is the min narrowness across permitted worlds — i.e., the narrowness in the world where the moat is widest (= 40ft → narrowness is minimal).

On a bounded scale [0, maxWidth], narrowness = maxWidth - width. Min narrowness = maxWidth - max(width) = maxWidth - 40. For any reasonable maxWidth, this is smaller than bridge shortness.

Truth conditions: we could (but don't have to) build a moat narrow enough that the bridge would span it. This does NOT match the intuition of (29), which asserts the bridge is too short, period.

Equations

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

theorem Buring2007.analysis2_wrong_at_50 : ¬analysis2_maxPermittedNarrowness 50 > Buring2007.bridgeLength✝

Analysis 2 predicts narrowness of (maxWidth - 40). For any reasonable maxWidth (e.g. 50), this is 10 — less than the bridge length of 15. So Analysis 2 would predict the sentence is FALSE (the bridge IS long enough for the widest-possible moat-narrowness). This is the wrong prediction.

theorem Buring2007.analyses_diverge : analysis1_minRequiredWidth > Buring2007.bridgeLength✝ ∧ ¬analysis2_maxPermittedNarrowness 50 > Buring2007.bridgeLength✝

The two analyses diverge: Analysis 1 predicts TRUE (bridge too short), Analysis 2 predicts FALSE (bridge long enough). Native speakers judge the sentence true, confirming Analysis 1.

structure Buring2007.ModalNomalyDatum : Type

@cite{buring-2007} §6.2 (p. 14, ex. 38): existential modals produce the same disambiguation.

"The moat is narrower than drawbridges are allowed to be long."

Analysis 1: moat width < max permitted bridge length. Paraphrase: "we can get a bridge that spans the moat."

Analysis 2: moat narrowness < max permitted bridge shortness. Paraphrase: weaker — about permitted shortness, not length.

• sentence : String
• modalForce : String
• analysis1Reading : String
• analysis2Reading : String
• nativeSpeakerMatch : Bool

def Buring2007.instReprModalNomalyDatum.repr : ModalNomalyDatum → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Buring2007.instReprModalNomalyDatum : Repr ModalNomalyDatum

Equations

• Buring2007.instReprModalNomalyDatum = { reprPrec := Buring2007.instReprModalNomalyDatum.repr }

def Buring2007.modalDiagnosticData : List ModalNomalyDatum

Equations

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