Gradient Word-Order Measures

@cite{levshina-stoynova-2023}

Levshina, Namboodiripad, Allassonnière-Tang, Kramer, Talamo, Verkerk, Wilmoth, Garrido Rodriguez, Gupton, Kidd, Liu, Naccarato, Nordlinger, @cite{levshina-stoynova-2023} "Why we need a gradient approach to word order" (Linguistics 61(4):825–883) argues that word-order typology should use continuous measures (proportions, Shannon entropy, mutual information) rather than categorical labels (SVO, SOV, "rigid", "flexible").

Key Claims

1. SO proportion is continuous across languages (Figure 1)
2. Head-finality is continuous (Figure 2: 123 SUD corpora)
3. Case marking MI correlates with SO entropy (Figure 3: ~30 languages)
4. Register affects word-order proportions (Figure 7: Russian VO varies by register)
5. Flexibility scores form a continuum (Figure 8: Avar–English)

Data Sources

• OSF Dataset1.txt: per-language SO proportion (31 languages)
• OSF Dataset3.txt: per-language SO entropy, case MI (30 languages)
• OSF Dataset6.txt: Russian register variation (conversation/fiction/news)
• https://osf.io/w9u6v/

def Phenomena.WordOrder.Gradience.harmonicProportion1000 (t : Typology.CrossTab) : ℕ

Proportion of harmonic languages × 1000 (integer permille).

Equations

• Phenomena.WordOrder.Gradience.harmonicProportion1000 t = t.harmonicCount * 1000 / t.totalCount

def Phenomena.WordOrder.Gradience.disharmonicProportion1000 (t : Typology.CrossTab) : ℕ

Proportion of disharmonic languages × 1000.

Equations

• Phenomena.WordOrder.Gradience.disharmonicProportion1000 t = t.disharmonicCount * 1000 / t.totalCount

def Phenomena.WordOrder.Gradience.hiProportion1000 (t : Typology.CrossTab) : ℕ

Proportion of head-initial languages × 1000 (hihi + hihf cells).

Equations

• Phenomena.WordOrder.Gradience.hiProportion1000 t = (t.hihi.count + t.hihf.count) * 1000 / t.totalCount

theorem Phenomena.WordOrder.Gradience.voAdposition_harmonic_proportion : harmonicProportion1000 Typology.voAdposition = 943

Table 1: 94.3% harmonic (VO × Adposition).

theorem Phenomena.WordOrder.Gradience.voSubordinator_harmonic_proportion : harmonicProportion1000 Typology.voSubordinator = 861

Table 2: 86.1% harmonic (VO × Subordinator).

theorem Phenomena.WordOrder.Gradience.voRelativeClause_harmonic_proportion : harmonicProportion1000 Typology.voRelativeClause = 822

Table 3: 82.2% harmonic (VO × Relative clause).

theorem Phenomena.WordOrder.Gradience.harmony_decreases_with_complexity : harmonicProportion1000 Typology.voAdposition > harmonicProportion1000 Typology.voSubordinator ∧ harmonicProportion1000 Typology.voSubordinator > harmonicProportion1000 Typology.voRelativeClause

Harmonic proportion decreases with construction complexity: adposition > subordinator > relative clause.

Levshina et al.'s point: not all constructions are equally categorical. Even the "best" universal (VO ↔ preposition) is only 94.3% harmonic.

theorem Phenomena.WordOrder.Gradience.categorical_consistent_with_gradient : (Typology.allTables.all fun (t : Typology.CrossTab) => decide (harmonicProportion1000 t > 500) == t.harmonicDominant) = true

Gradient and categorical measures agree: harmonicProportion1000 > 500 ↔ harmonicDominant. The gradient measure refines, not contradicts, the binary one.

structure Phenomena.WordOrder.Gradience.GradientWOProfile : Type

Per-language gradient word-order data from Levshina, @cite{levshina-stoynova-2023} OSF datasets. SO proportion from Dataset1.txt, entropy and case MI from Dataset3.txt. All values × 1000, rounded to nearest integer.

• name : String
• isoCode : String
• soProportion1000 : ℕ

  Proportion of SO (subject before object) orders × 1000 (from Dataset1.txt)

• soEntropy1000 : ℕ

  Shannon entropy of S-O order × 1000 (0 = deterministic, 1000 = maximal; Dataset3.txt)

• caseMI1000 : ℕ

  Mutual information between case markers and grammatical role × 1000 (Dataset3.txt)

instance Phenomena.WordOrder.Gradience.instReprGradientWOProfile : Repr GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.instReprGradientWOProfile = { reprPrec := Phenomena.WordOrder.Gradience.instReprGradientWOProfile.repr }

def Phenomena.WordOrder.Gradience.instReprGradientWOProfile.repr : GradientWOProfile → ℕ → Std.Format

Equations

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def Phenomena.WordOrder.Gradience.instDecidableEqGradientWOProfile.decEq (x✝ x✝¹ : GradientWOProfile) : Decidable (x✝ = x✝¹)

Equations

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instance Phenomena.WordOrder.Gradience.instDecidableEqGradientWOProfile : DecidableEq GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.instDecidableEqGradientWOProfile = Phenomena.WordOrder.Gradience.instDecidableEqGradientWOProfile.decEq

instance Phenomena.WordOrder.Gradience.instBEqGradientWOProfile : BEq GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.instBEqGradientWOProfile = { beq := Phenomena.WordOrder.Gradience.instBEqGradientWOProfile.beq }

def Phenomena.WordOrder.Gradience.instBEqGradientWOProfile.beq : GradientWOProfile → GradientWOProfile → Bool

Equations

• One or more equations did not get rendered due to their size.
• Phenomena.WordOrder.Gradience.instBEqGradientWOProfile.beq x✝¹ x✝ = false

def Phenomena.WordOrder.Gradience.arabic : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.arabic = { name := "Arabic", isoCode := "ar", soProportion1000 := 933, soEntropy1000 := 345, caseMI1000 := 36 }

def Phenomena.WordOrder.Gradience.bulgarian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.bulgarian = { name := "Bulgarian", isoCode := "bg", soProportion1000 := 965, soEntropy1000 := 218, caseMI1000 := 28 }

def Phenomena.WordOrder.Gradience.croatian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.croatian = { name := "Croatian", isoCode := "hr", soProportion1000 := 856, soEntropy1000 := 586, caseMI1000 := 415 }

def Phenomena.WordOrder.Gradience.czech : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.czech = { name := "Czech", isoCode := "cs", soProportion1000 := 781, soEntropy1000 := 760, caseMI1000 := 525 }

def Phenomena.WordOrder.Gradience.danish : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.danish = { name := "Danish", isoCode := "da", soProportion1000 := 989, soEntropy1000 := 74, caseMI1000 := 0 }

def Phenomena.WordOrder.Gradience.dutch : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.dutch = { name := "Dutch", isoCode := "nl", soProportion1000 := 970, soEntropy1000 := 183, caseMI1000 := 0 }

def Phenomena.WordOrder.Gradience.english : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.english = { name := "English", isoCode := "en", soProportion1000 := 994, soEntropy1000 := 47, caseMI1000 := 0 }

def Phenomena.WordOrder.Gradience.estonian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.estonian = { name := "Estonian", isoCode := "et", soProportion1000 := 842, soEntropy1000 := 634, caseMI1000 := 692 }

def Phenomena.WordOrder.Gradience.finnish : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.finnish = { name := "Finnish", isoCode := "fi", soProportion1000 := 912, soEntropy1000 := 426, caseMI1000 := 314 }

def Phenomena.WordOrder.Gradience.french : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.french = { name := "French", isoCode := "fr", soProportion1000 := 995, soEntropy1000 := 42, caseMI1000 := 5 }

def Phenomena.WordOrder.Gradience.german : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.german = { name := "German", isoCode := "de", soProportion1000 := 916, soEntropy1000 := 386, caseMI1000 := 288 }

def Phenomena.WordOrder.Gradience.greek : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.greek = { name := "Greek", isoCode := "el", soProportion1000 := 896, soEntropy1000 := 490, caseMI1000 := 70 }

def Phenomena.WordOrder.Gradience.hindi : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.hindi = { name := "Hindi", isoCode := "hi", soProportion1000 := 874, soEntropy1000 := 509, caseMI1000 := 334 }

def Phenomena.WordOrder.Gradience.hungarian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.hungarian = { name := "Hungarian", isoCode := "hu", soProportion1000 := 727, soEntropy1000 := 858, caseMI1000 := 738 }

def Phenomena.WordOrder.Gradience.indonesian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.indonesian = { name := "Indonesian", isoCode := "id", soProportion1000 := 999, soEntropy1000 := 12, caseMI1000 := 0 }

def Phenomena.WordOrder.Gradience.italian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.italian = { name := "Italian", isoCode := "it", soProportion1000 := 969, soEntropy1000 := 192, caseMI1000 := 6 }

def Phenomena.WordOrder.Gradience.japanese : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.japanese = { name := "Japanese", isoCode := "ja", soProportion1000 := 953, soEntropy1000 := 246, caseMI1000 := 582 }

def Phenomena.WordOrder.Gradience.korean : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.korean = { name := "Korean", isoCode := "ko", soProportion1000 := 978, soEntropy1000 := 146, caseMI1000 := 357 }

def Phenomena.WordOrder.Gradience.latvian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.latvian = { name := "Latvian", isoCode := "lv", soProportion1000 := 767, soEntropy1000 := 784, caseMI1000 := 726 }

def Phenomena.WordOrder.Gradience.lithuanian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.lithuanian = { name := "Lithuanian", isoCode := "lt", soProportion1000 := 608, soEntropy1000 := 968, caseMI1000 := 788 }

def Phenomena.WordOrder.Gradience.persian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.persian = { name := "Persian", isoCode := "fa", soProportion1000 := 924, soEntropy1000 := 315, caseMI1000 := 219 }

def Phenomena.WordOrder.Gradience.portuguese : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.portuguese = { name := "Portuguese", isoCode := "pt", soProportion1000 := 986, soEntropy1000 := 102, caseMI1000 := 14 }

def Phenomena.WordOrder.Gradience.romanian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.romanian = { name := "Romanian", isoCode := "ro", soProportion1000 := 966, soEntropy1000 := 216, caseMI1000 := 7 }

def Phenomena.WordOrder.Gradience.russian : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.russian = { name := "Russian", isoCode := "ru", soProportion1000 := 861, soEntropy1000 := 580, caseMI1000 := 335 }

def Phenomena.WordOrder.Gradience.slovene : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.slovene = { name := "Slovene", isoCode := "sl", soProportion1000 := 873, soEntropy1000 := 536, caseMI1000 := 478 }

def Phenomena.WordOrder.Gradience.spanish : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.spanish = { name := "Spanish", isoCode := "es", soProportion1000 := 978, soEntropy1000 := 143, caseMI1000 := 21 }

def Phenomena.WordOrder.Gradience.swedish : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.swedish = { name := "Swedish", isoCode := "sv", soProportion1000 := 988, soEntropy1000 := 86, caseMI1000 := 0 }

def Phenomena.WordOrder.Gradience.tamil : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.tamil = { name := "Tamil", isoCode := "ta", soProportion1000 := 715, soEntropy1000 := 824, caseMI1000 := 59 }

def Phenomena.WordOrder.Gradience.turkish : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.turkish = { name := "Turkish", isoCode := "tr", soProportion1000 := 922, soEntropy1000 := 353, caseMI1000 := 167 }

def Phenomena.WordOrder.Gradience.vietnamese : GradientWOProfile

Equations

• Phenomena.WordOrder.Gradience.vietnamese = { name := "Vietnamese", isoCode := "vi", soProportion1000 := 981, soEntropy1000 := 105, caseMI1000 := 0 }

def Phenomena.WordOrder.Gradience.allProfiles : List GradientWOProfile

All 30 gradient word-order profiles from OSF Dataset1.txt + Dataset3.txt.

Equations

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theorem Phenomena.WordOrder.Gradience.allProfiles_count : allProfiles.length = 30

theorem Phenomena.WordOrder.Gradience.rigid_languages_low_entropy : ((List.filter (fun (x : GradientWOProfile) => decide (x.soProportion1000 > 960)) allProfiles).all fun (x : GradientWOProfile) => decide (x.soEntropy1000 < 300)) = true

Languages with near-deterministic SO order (proportion > 960) have low SO entropy (< 300). 13 languages: Bulgarian, Danish, Dutch, English, French, Indonesian, Italian, Korean, Portuguese, Romanian, Spanish, Swedish, Vietnamese.

theorem Phenomena.WordOrder.Gradience.case_rich_flexible_languages_high_mi : ((List.filter (fun (p : GradientWOProfile) => decide (p.soEntropy1000 > 700 ∧ p.caseMI1000 > 0)) allProfiles).all fun (p : GradientWOProfile) => decide (p.caseMI1000 > 400 ∨ (p.name == "Tamil") = true)) = true

Among Indo-European languages with case morphology, high SO entropy (> 700) implies high case MI (> 400). Czech, Hungarian, Latvian, Lithuanian all use case marking to compensate for word-order freedom.

theorem Phenomena.WordOrder.Gradience.tamil_counterexample : tamil.soEntropy1000 > 800 ∧ tamil.caseMI1000 < 100

Tamil is a counterexample to the simple "flexibility requires case marking" story: high SO entropy (824) but low case MI (59). Tamil uses verb agreement and animacy rather than case morphology for role disambiguation.

This makes the gradient approach especially valuable — it reveals that the case–flexibility correlation is a tendency with principled exceptions, not a law.

def Phenomena.WordOrder.Gradience.meanSOEntropy (ps : List GradientWOProfile) : ℕ

Case MI correlates with SO entropy: languages with caseMI > 300 have higher mean SO entropy than languages with caseMI ≤ 300.

Equations

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

theorem Phenomena.WordOrder.Gradience.case_mi_correlates_with_so_entropy : meanSOEntropy (List.filter (fun (x : GradientWOProfile) => decide (x.caseMI1000 > 300)) allProfiles) > meanSOEntropy (List.filter (fun (x : GradientWOProfile) => decide (x.caseMI1000 ≤ 300)) allProfiles)

theorem Phenomena.WordOrder.Gradience.so_proportion_is_continuous : indonesian.soProportion1000 - lithuanian.soProportion1000 > 350

SO proportion spans a wide range: from Lithuanian (608) to Indonesian (999). This 391-point spread refutes any simple binary classification.

theorem Phenomena.WordOrder.Gradience.gradient_implies_categorical : (Typology.allTables.all fun (t : Typology.CrossTab) => if harmonicProportion1000 t > 500 then t.harmonicDominant else true) = true

For all three WALS tables, harmonicProportion1000 > 500 → harmonicDominant = true.

theorem Phenomena.WordOrder.Gradience.harmony_is_gradient_not_binary : harmonicProportion1000 Typology.voAdposition ≠ harmonicProportion1000 Typology.voSubordinator ∧ harmonicProportion1000 Typology.voSubordinator ≠ harmonicProportion1000 Typology.voRelativeClause

The three tables have different harmonic proportions (943 vs 861 vs 822), showing harmony is a matter of degree, not a binary universal.

def Phenomena.WordOrder.Gradience.shared_gradient_hahn_isoCodes : List String

Shared languages between our gradient profiles and the 54-language Hahn et al. dataset.

Equations

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

theorem Phenomena.WordOrder.Gradience.shared_gradient_hahn_count : shared_gradient_hahn_isoCodes.length = 28

theorem Phenomena.WordOrder.Gradience.high_so_entropy_implies_high_branch_entropy : have highSOEntropy := List.filter (fun (p : GradientWOProfile) => decide (p.soEntropy1000 > 600)) allProfiles; have sharedHighSO := List.filter (fun (p : GradientWOProfile) => Studies.HahnDegenFutrell2021.allLanguages.any fun (x : Studies.HahnDegenFutrell2021.LanguageEfficiency) => x.isoCode == p.isoCode) highSOEntropy; (sharedHighSO.all fun (p : GradientWOProfile) => match List.find? (fun (x : Studies.HahnDegenFutrell2021.LanguageEfficiency) => x.isoCode == p.isoCode) Studies.HahnDegenFutrell2021.allLanguages with | some h => match h.branchDirEntropy1000 with | some e => decide (e > 250) | none => false | none => false) = true

Languages with high SO entropy (> 600) in Levshina that also appear in Hahn et al. all have high branching direction entropy (> 250) in Hahn et al.

Two independent measures of word-order freedom (SO entropy from corpus counts, branching direction entropy from dependency trees) converge.

def Phenomena.WordOrder.Gradience.shared_gradient_futrell_isoCodes : List String

Shared languages between gradient profiles and Futrell et al.'s DLM dataset.

Equations

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

theorem Phenomena.WordOrder.Gradience.shared_gradient_futrell_count : shared_gradient_futrell_isoCodes.length = 26

theorem Phenomena.WordOrder.Gradience.head_final_correlates_with_so : have shared := List.filter (fun (p : GradientWOProfile) => Phenomena.WordOrder.Gradience.futrellLanguages✝.any fun (x : DependencyLength.FutrellEtAl2020.LanguageDLM) => x.isoCode == p.isoCode) allProfiles; have highHF := List.filter (fun (p : GradientWOProfile) => match List.find? (fun (x : DependencyLength.FutrellEtAl2020.LanguageDLM) => x.isoCode == p.isoCode) Phenomena.WordOrder.Gradience.futrellLanguages✝¹ with | some f => decide (f.propHeadFinal > 700) | none => false) shared; (highHF.all fun (x : GradientWOProfile) => decide (x.soProportion1000 > 700)) = true

Languages with high propHeadFinal (> 700) in Futrell have high soProportion (> 700) in Levshina: head-final ≈ SOV ≈ high SO proportion.

theorem Phenomena.WordOrder.Gradience.high_entropy_languages_include_exceptions : have highEntropy := List.filter (fun (x : GradientWOProfile) => decide (x.soEntropy1000 > 600)) allProfiles; have hahnExceptions := Studies.HahnDegenFutrell2021.exceptionLanguages; (highEntropy.any fun (p : GradientWOProfile) => hahnExceptions.any fun (x : Studies.HahnDegenFutrell2021.LanguageEfficiency) => x.isoCode == p.isoCode) = true

Languages with soEntropy > 600 in Levshina include at least one of the Hahn et al. exception languages. Latvian appears in both: high SO entropy in Levshina, and a memory-surprisal optimization exception in Hahn et al. Flexible word order weakens memory-surprisal optimization.

structure Phenomena.WordOrder.Gradience.RegisterProfile : Type

Russian VO probability by register, from OSF Dataset6.txt (100 clauses per register). Demonstrates within-language variation that a categorical label obscures.

• register : String
• voProbability1000 : ℕ

instance Phenomena.WordOrder.Gradience.instReprRegisterProfile : Repr RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.instReprRegisterProfile = { reprPrec := Phenomena.WordOrder.Gradience.instReprRegisterProfile.repr }

def Phenomena.WordOrder.Gradience.instReprRegisterProfile.repr : RegisterProfile → ℕ → Std.Format

Equations

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

def Phenomena.WordOrder.Gradience.instDecidableEqRegisterProfile.decEq (x✝ x✝¹ : RegisterProfile) : Decidable (x✝ = x✝¹)

Equations

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

instance Phenomena.WordOrder.Gradience.instDecidableEqRegisterProfile : DecidableEq RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.instDecidableEqRegisterProfile = Phenomena.WordOrder.Gradience.instDecidableEqRegisterProfile.decEq

instance Phenomena.WordOrder.Gradience.instBEqRegisterProfile : BEq RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.instBEqRegisterProfile = { beq := Phenomena.WordOrder.Gradience.instBEqRegisterProfile.beq }

def Phenomena.WordOrder.Gradience.instBEqRegisterProfile.beq : RegisterProfile → RegisterProfile → Bool

Equations

• Phenomena.WordOrder.Gradience.instBEqRegisterProfile.beq { register := a, voProbability1000 := a_1 } { register := b, voProbability1000 := b_1 } = (a == b && a_1 == b_1)
• Phenomena.WordOrder.Gradience.instBEqRegisterProfile.beq x✝¹ x✝ = false

def Phenomena.WordOrder.Gradience.russianConversation : RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.russianConversation = { register := "conversation", voProbability1000 := 390 }

def Phenomena.WordOrder.Gradience.russianFiction : RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.russianFiction = { register := "fiction", voProbability1000 := 830 }

def Phenomena.WordOrder.Gradience.russianNews : RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.russianNews = { register := "news", voProbability1000 := 830 }

def Phenomena.WordOrder.Gradience.russianRegisters : List RegisterProfile

Equations

• Phenomena.WordOrder.Gradience.russianRegisters = [Phenomena.WordOrder.Gradience.russianConversation, Phenomena.WordOrder.Gradience.russianFiction, Phenomena.WordOrder.Gradience.russianNews]

theorem Phenomena.WordOrder.Gradience.russian_vo_varies_by_register : russianConversation.voProbability1000 < russianFiction.voProbability1000

Russian conversation has lower VO probability than fiction: spoken language permits more OV orders.

theorem Phenomena.WordOrder.Gradience.register_variation_is_large : russianFiction.voProbability1000 - russianConversation.voProbability1000 > 400

The register variation is large: fiction - conversation > 400 (a 44 percentage-point spread). A single categorical label cannot capture this within-language variation.