Deal (2026): Clausal complementation as relativization, revisited

Deal types notional complement clauses by their internal and external syntax. The relative embeddings of Nez Perce, whose edge obligatorily carries the relative pronoun yox̂ and the complementizer ke, are CPs containing an Ā-dependency launched above TP, not DPs or PPs, so clausal complementation is not uniformly relativization, against Kayne and Arsenijević. Table (79) places each construction by its extended spine, the heads projected from V up through any nominal or adpositional shell over C, and by whether the CP contains an Ā-dependency. Every cell is filled, so the internal syntax of a clause does not predict its external syntax, and factivity cross-cuts the Ā axis as well (table (80)).

The Nez Perce embedding strategy is read off the paper's judgments in Data/Examples/Deal2026.json. The Adyghe Ā flag is that 'think' takes the Fragment's ze-re- typer, and the Bulgarian flag is Krapova's double requirement over the Fragment's frames, checked against her sentences (56)–(59) in Data/Examples/Krapova2010.json.

Main results

• relative_factive, projection_rows: every relative embedding is factive, factivity being the Fragment entries' Karttunen class, which the projection trials (33)–(36) and (68) confirm row by row.
• table79, table80: each shell over C, none, D N or P D, occurs with and without an Ā-dependency, and factivity varies independently of the Ā-dependency.
• detoComplement_iff, deto_rows: the double requirement picks out Krapova's deto-takers.
• paradigm_case_discriminates: the case half of the diagnostic (21) that yox̂ is a D.

Implementation notes

The Ā column of (79) is a datum for English, Ndebele and Washo, whose Fragments carry no relativizer observable, and the Washo V D CP structure of footnote 33, the complementation spine Bochnak and Hanink propose, is a row beside the table's six. Table (81), nominalization against factivity, needs Turkish düşün- 'think', which has no Fragment.

TODO

• ke-agreement (§2, after Deal's earlier analysis): the φ-probe on C interacts with all φ-features, probing from the subject downward until [addr] satisfies it; needs value-sensitive satisfaction and ordered-goal probing.
• §6: relative embeddings block indexical shift and take matrix-matching tense as temporal de re, while simplex embeddings allow shift and relative tense; rests on Deal's clause-type semantics.

References

• [Dea26]
• [Kay08]
• [Kay14]
• [Ars09]
• [CP11a]
• [Kra10]
• [Pie19]
• [BH21]
• [HB17]
• [HM21]
• [Cho70]
• [Dea25b]
• [Dea15]
• [Kar71b]
• [TBRS13]

The embedding strategy

inductive Deal2026.EmbeddingStrategy : Type

The two embedding strategies are relative embedding, whose complement obligatorily carries the yox̂ ke edge and contains an Ā-dependency above TP, and simplex embedding of a bare CP.

• relative : EmbeddingStrategy
• simplex : EmbeddingStrategy

@[instance_reducible]

instance Deal2026.instDecidableEqEmbeddingStrategy : DecidableEq EmbeddingStrategy

Equations

• Deal2026.instDecidableEqEmbeddingStrategy x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Deal2026.instReprEmbeddingStrategy : Repr EmbeddingStrategy

Equations

• Deal2026.instReprEmbeddingStrategy = { reprPrec := Deal2026.instReprEmbeddingStrategy.repr }

def Deal2026.instReprEmbeddingStrategy.repr : EmbeddingStrategy → ℕ → Std.Format

Equations

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

def Deal2026.RelativeEmbedding (v : NezPerce.Verb) : Prop

A predicate embeds relatively when some grammatical notional complement of it carries the yox̂ ke edge ((27)–(28)); cuukwe's (66b) is only marginal.

Equations

• Deal2026.RelativeEmbedding v = ∃ row ∈ Deal2026.Examples.all, row.feature? "verb" = some v.form ∧ row.feature? "edge" = some "yoxKe" ∧ row.judgment = Judgment.acceptable

@[instance_reducible]

instance Deal2026.instDecidableRelativeEmbedding (v : NezPerce.Verb) : Decidable (RelativeEmbedding v)

Equations

• Deal2026.instDecidableRelativeEmbedding v = Deal2026.instDecidableRelativeEmbedding._aux_1 v

def Deal2026.strategy (v : NezPerce.Verb) : EmbeddingStrategy

A predicate's strategy is relative when it embeds relatively and simplex otherwise. Both strategies select a CP; the contrast is internal to the clause selected.

Equations

• Deal2026.strategy v = if Deal2026.RelativeEmbedding v then Deal2026.EmbeddingStrategy.relative else Deal2026.EmbeddingStrategy.simplex

theorem Deal2026.nominal_rows (row : Datum) : row ∈ Examples.all → row.feature? "diagnostic" = some "nominalComplement" → ∀ v ∈ NezPerce.verbs, row.feature? "verb" = some v.form → (row.judgment = Judgment.acceptable ↔ ∃ fr ∈ v.valency.frames, fr.HasNominal)

The relative-embedding predicates take no noun-phrase object, (41).

theorem Deal2026.relative_factive (v : NezPerce.Verb) : v ∈ NezPerce.verbs → strategy v = EmbeddingStrategy.relative → v.IsFactive

Every relative embedding is factive (§7).

theorem Deal2026.projection_rows (row : Datum) : row ∈ Examples.all → ∀ x ∈ row.environment?, ∀ v ∈ NezPerce.verbs, row.feature? "verb" = some v.form → (row.projective? = some true ↔ v.IsFactive)

Consultants endorse the complement under negation, in a question or in a conditional antecedent exactly for the factive predicates, the projection trials (33)–(36) and (68).

theorem Deal2026.cuukwe_conditional_projects : ¬Karttunen1971b.Projects Presupposition.Factivity.semi Presupposition.FamilyVariant.conditionalAntecedent ∧ (∃ row ∈ Examples.all, row.environment? = some Presupposition.FamilyVariant.conditionalAntecedent ∧ row.person? = some UD.Person.third ∧ row.feature? "verb" = some NezPerce.cuukwe.form ∧ row.projective? = some true) ∧ ∀ row ∈ Karttunen1971b.Examples.all, row.environment? = some Presupposition.FamilyVariant.conditionalAntecedent → row.person? = some UD.Person.first

The semi-factive cuukwe 'know' projects from a third-person conditional antecedent, (68), where the complement does not follow from [Kar71b]'s postulates for a semi-factive. His own conditionals, (25), are in the first person.

Table (79): internal against external syntax

structure Deal2026.Cell : Type

A cell of table (79) pairs the extended spine of a notional complement, the heads projected from V up through any shell over C, with whether the CP contains an Ā-dependency.

• spine : Minimalist.ClauseSpine
• internalAbar : Bool

def Deal2026.nezPerce (v : NezPerce.Verb) : Cell

A Nez Perce cell is a bare CP whose Ā-dependency is the relative strategy.

Equations

• Deal2026.nezPerce v = { spine := Minimalist.ClauseSpine.cP, internalAbar := Deal2026.strategy v == Deal2026.EmbeddingStrategy.relative }

def Deal2026.englishThink : Cell

English simplex V complementation, think, is a bare CP without an Ā-dependency.

Equations

• Deal2026.englishThink = { spine := Minimalist.ClauseSpine.cP, internalAbar := false }

def Deal2026.adygheRelative : Cell

The Adyghe relative embedding of (43) is V D N CP with an Ā-dependency. Its flag comes from the Fragment, where 'think' takes the relativizer ze- with the applicative re- on its tensed complement ([CP11a]).

Equations

• Deal2026.adygheRelative = { spine := Minimalist.ClauseSpine.cP.append [Minimalist.Cat.N, Minimalist.Cat.D], internalAbar := decide (Adyghe.gwepshesa.Takes Adyghe.zeRe) }

def Deal2026.DetoComplement (v : Bulgarian.Verb) : Prop

By Krapova's double requirement, reported at footnote 22, deto introduces the complement of a predicate that is an emotive factive, a factive preferential attitude, and takes a za phrase.

Equations

• Deal2026.DetoComplement v = ((∃ a ∈ v.attitude, a.IsPreferential) ∧ v.IsFactive ∧ ∃ fr ∈ v.valency.frames, fr.HasAdpositional)

@[instance_reducible]

instance Deal2026.instDecidableDetoComplement (v : Bulgarian.Verb) : Decidable (DetoComplement v)

Equations

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

theorem Deal2026.detoComplement_iff : (∀ v ∈ Bulgarian.verbs, DetoComplement v ↔ v.form ∈ List.map (fun (x : Bulgarian.Verb) => x.form) Bulgarian.detoTakers) ∧ ¬DetoComplement Bulgarian.vazmushtavamSe ∧ ¬DetoComplement Bulgarian.razbiram

The double requirement picks out exactly the predicates Krapova lists as deto-takers, and neither condition suffices, since văzmuštavam se 'resent' is emotive without a za phrase and razbiram 'comprehend' is factive without being emotive.

theorem Deal2026.deto_rows (row : Datum) : row ∈ Krapova2010.Examples.all → row.feature? "diagnostic" = some "detoSelection" → ∀ v ∈ Bulgarian.verbs, row.feature? "verb" = some v.form → (row.judgment = Judgment.acceptable ↔ DetoComplement v) ∧ ∀ a ∈ row.alternatives, a.2 = Judgment.acceptable

Krapova's deto sentences (56) and (58) are grammatical exactly for the predicates that meet the double requirement, and their če variants are always grammatical (fn. 46).

theorem Deal2026.zaPhrase_rows (row : Datum) : row ∈ Krapova2010.Examples.all → row.feature? "diagnostic" = some "zaPhrase" → ∀ v ∈ Bulgarian.verbs, row.feature? "verb" = some v.form → (row.judgment = Judgment.acceptable ↔ ∃ fr ∈ v.valency.frames, fr.HasAdpositional) ∧ ∀ a ∈ row.alternatives, a.2 = Judgment.ungrammatical

The nominal paraphrases (59) take the preposition za and no other, the adpositional frame of the Fragment's emotive factives.

theorem Deal2026.krapova_projection_rows (row : Datum) : row ∈ Krapova2010.Examples.all → ∀ x ∈ row.environment?, ∀ v ∈ Bulgarian.verbs, row.feature? "verb" = some v.form → (row.projective? = some true ↔ v.IsFactive)

In Krapova's factivity tests (57a–b) the complement survives negation and a question exactly for the factive predicates.

theorem Deal2026.krapova_contradiction_rows (row : Datum) : row ∈ Krapova2010.Examples.all → row.feature? "diagnostic" = some "contradiction" → ∀ v ∈ Bulgarian.verbs, row.feature? "verb" = some v.form → ((∃ a ∈ row.alternatives, a.2 = Judgment.ungrammatical) ↔ v.IsFactive)

In Krapova's contradiction tests (57c) and footnote 46, a continuation denying the complement is starred exactly under a factive predicate, under deto and če alike.

def Deal2026.englishNComplementation : Cell

English N complementation, the fact that S, is V D N CP without an Ā-dependency, the DP shell with an N co-argument of [HM21].

Equations

• Deal2026.englishNComplementation = { spine := Minimalist.ClauseSpine.cP.append [Minimalist.Cat.N, Minimalist.Cat.D], internalAbar := false }

def Deal2026.bulgarianRelative : Cell

The Bulgarian relative embedding of (49) is V P D CP with an Ā-dependency. Its flag comes from the Fragment, where săžaljavam 'regret' meets the double requirement and so takes the deto complement Krapova analyzes as a hidden relative.

Equations

• Deal2026.bulgarianRelative = { spine := Minimalist.ClauseSpine.cP.append [Minimalist.Cat.D, Minimalist.Cat.P], internalAbar := decide (Deal2026.DetoComplement Bulgarian.sazhaljavam) }

def Deal2026.ndebeleEmbedding : Cell

The Ndebele embedding of (78) is V P D CP without an Ā-dependency. The preposition nga 'about' sits over the class-15 augment, which [Pie19] takes as a D over kuthi.

Equations

• Deal2026.ndebeleEmbedding = { spine := Minimalist.ClauseSpine.cP.append [Minimalist.Cat.D, Minimalist.Cat.P], internalAbar := false }

def Deal2026.washoFactive : Cell

The Washo factive of footnote 33 is V D CP without an Ā-dependency or an N, a silent D over the nominalized clause ([HB17], [BH21]), whose index binds without movement ([Han21]).

Equations

• Deal2026.washoFactive = { spine := BochnakHanink2021.complementSpine, internalAbar := false }

def Deal2026.rows : List Cell

The rows are the six cells of table (79), with the V CP cell without an Ā-dependency witnessed twice, and the Washo structure.

Equations

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

theorem Deal2026.table79 (shell : List Minimalist.Cat) : shell ∈ [[], [Minimalist.Cat.N, Minimalist.Cat.D], [Minimalist.Cat.D, Minimalist.Cat.P]] → ∀ abar ∈ [true, false], ∃ c ∈ rows, c.spine.above Minimalist.Cat.C = shell ∧ c.internalAbar = abar

Every cell of table (79) is filled, each of the three shells over C, none, D N and P D, occurring with and without an Ā-dependency in the CP. The internal syntax of a clause does not predict its external syntax, the category selected ranging over C, D and P for one CP-internal syntax, and not all complementation is relativization.

Table (80): factivity against the Ā axis

theorem Deal2026.table80 : (∃ v ∈ NezPerce.verbs, v.IsFactive ∧ strategy v = EmbeddingStrategy.relative) ∧ (∃ v ∈ NezPerce.verbs, v.IsFactive ∧ strategy v = EmbeddingStrategy.simplex) ∧ (∃ v ∈ NezPerce.verbs, ¬v.IsFactive ∧ strategy v = EmbeddingStrategy.simplex) ∧ ∃ v ∈ Adyghe.verbs, v.Takes Adyghe.zeRe ∧ ¬v.IsFactive

Every cell of table (80) is filled, by a factive relative embedding, lilooy 'be happy', a factive simplex embedding, cuukwe 'know', a non-factive simplex embedding, neki 'think', and a non-factive relative embedding, Adyghe 'think', which requires the relative strategy for its tensed complement. Factivity and relative-embedding syntax vary independently.

The D-inflection diagnostic (21)

theorem Deal2026.paradigm_case_discriminates (p : NezPerce.RelativePronoun) : p ∈ NezPerce.relativePronounParadigm → ∀ q ∈ NezPerce.relativePronounParadigm, p.case ≠ q.case → ∀ f ∈ p.forms, f ∉ q.forms

The relative pronoun yox̂ ko inflects for case, in that cells of the paradigm with distinct cases share no forms. This is the case half of the diagnostic that yox̂ ko is a D while the invariant ke is a C.