Bhatt & Dayal (2020): the polar question particle kya:

[BD20]

Hindi-Urdu polar kya: is a polar question particle (PQP) — a class distinct from clause-typing Q-morphemes like Japanese ka: it occurs only in polar questions (not in wh-questions), sits in a projection above CP the paper labels ForceP, and embeds only in the quasi-subordinated configuration of [DG09] (complements of rogatives), not in ordinary subordination. Semantically kya: presupposes that its sister question has a singleton alternative set (eq. 23) and is otherwise the identity; a polar question denotes the singleton {p} (eq. 22b). The same singleton restriction models the (bias-introducing) Mandarin particle nandao in [Xu12a] (cross-linguistic picture in [Xu17]), which the paper draws on.

The singleton-alternative presupposition (eq. 23)

⟦kya:⟧ = λQ⟨st⟩t : ∃p ∈ Q[∀q[q ∈ Q → q = p]].Q — defined only when the sister question Q has a singleton alternative set, and then the identity on Q. The presupposition is Question.IsSingleton; the felicitous sister is the subtype SingletonQuestion W. In the paper a polar question denotes a singleton (eq. 22b: ⟦did John leave⟧ = {John left}) — the substrate's one-cell ofSet p, not its two-cell Hamblin polar.

theorem BhattDayal2020.kya_felicitous_singleton_polar {W : Type u_1} (p : Set W) : (Question.ofSet p).IsSingleton

Felicitous case: a polar question in the paper's sense is a singleton.

theorem BhattDayal2020.kya_interp {W : Type u_1} (p : Set W) : (Question.SingletonQuestion.ofSet p).issue = Question.ofSet p

Defined case: on a felicitous sister, kya: is the identity.

theorem BhattDayal2020.kya_infelicitous_wh {W : Type u_1} {E : Type u_2} {D : Set E} {P : E → Set W} (hD : D.Nonempty) (hne : ∀ e ∈ D, (P e).Nonempty) (hA : IsAntichain (fun (x1 x2 : Set W) => x1 ⊆ x2) (P '' D)) {e₁ e₂ : E} (h₁ : e₁ ∈ D) (h₂ : e₂ ∈ D) (hPne : P e₁ ≠ P e₂) : ¬(Question.which D P).IsSingleton

The headline restriction (ex. 4): a wh-question with two distinct answer cells is non-singleton, so kya: rejects it.

theorem BhattDayal2020.kya_infelicitous_two_cell_polar {W : Type u_1} {p : Set W} (hne : p ≠ ∅) (hnu : p ≠ Set.univ) : ¬(Question.polar p).IsSingleton

A two-cell Hamblin polar likewise fails the presupposition.

Embedding: quasi-subordination only

kya: appears in matrix polar questions and in the quasi-subordinated configuration of [DG09] (complements of rogatives, ex. 9) but not in ordinary subordination (complements of responsives, ex. 8) — read off the fragment's embedding facet.

theorem BhattDayal2020.kya_embedding_profile : HindiUrdu.Particles.kya.LicensedInEmbed Clause.EmbeddingContext.matrix ∧ HindiUrdu.Particles.kya.LicensedInEmbed Clause.EmbeddingContext.quasiSubordinated ∧ ¬HindiUrdu.Particles.kya.LicensedInEmbed Clause.EmbeddingContext.subordinated