Height as Domain Restriction in ASL

[DG22]

Davidson, K. & Gagné, D. (2022). "More is up" for domain restriction in ASL. Semantics & Pragmatics 15, Article 1: 1–52.

Core Idea

Vertical height in ASL signing space overtly marks quantifier domain restriction — the contextual variable C that spoken languages leave covert. Higher signing maps monotonically to mereologically larger domains. This height enters the grammar via pronominal elements (IX-arc), not bare nouns: pronouns, directional verbs (which incorporate pronouns), and quantifiers (which take a pronominal restrictor via a partitive-like structure).

Architecture

VerticalHeight is a 3-level linear order (low < neutral < high) that instantiates DDRP from DomainRestriction.lean. This gives all monotonicity and nesting theorems there for free.

The presuppositional denotations of IX-arc follow the phi-feature pattern from PhiFeatures.lean: height marking is a presuppositional partial identity function on plural individuals, analogous to gender on personal pronouns or proximal/distal on demonstratives.

Key Denotations

• ⟦arc⟧ = λx: ¬atomic(x). x — plural pronoun presupposes non-atomicity
• ⟦neutral⟧ = λx: ∀y(y ≤ x → y ∈ Cₑ). x — all parts in context
• ⟦domain-k⟧ = ⟦high⟧ = λx: Cₑ ⊂ {y : y ≤ x}. x — context is proper subpart

inductive ASL.Height.VerticalHeight : Type

Vertical height in ASL signing space.

[DG22] establish empirically that at least three height levels are semantically distinguished (ex. 15–16, 34): neutral signing height (default contextual domain), a mid level (superset of the default), and a high level (further superset). The contrast between (15) and (16) demonstrates that heights are relative, not absolute — a 3-level system can be compressed to 2 levels when the context provides only two relevant domains.

low < neutral < high, with higher = wider domain.

• low : VerticalHeight
• neutral : VerticalHeight
• high : VerticalHeight

@[implicit_reducible]

instance ASL.Height.instDecidableEqVerticalHeight : DecidableEq VerticalHeight

Equations

• ASL.Height.instDecidableEqVerticalHeight x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance ASL.Height.instReprVerticalHeight : Repr VerticalHeight

Equations

• ASL.Height.instReprVerticalHeight = { reprPrec := ASL.Height.instReprVerticalHeight.repr }

def ASL.Height.instReprVerticalHeight.repr : VerticalHeight → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance ASL.Height.instInhabitedVerticalHeight : Inhabited VerticalHeight

Equations

• ASL.Height.instInhabitedVerticalHeight = { default := ASL.Height.instInhabitedVerticalHeight.default }

@[implicit_reducible]

instance ASL.Height.instFintypeVerticalHeight : Fintype VerticalHeight

Equations

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

def ASL.Height.VerticalHeight.toFin : VerticalHeight → Fin 3

Rank embedding for the linear order on VerticalHeight.

Equations

• ASL.Height.VerticalHeight.low.toFin = 0
• ASL.Height.VerticalHeight.neutral.toFin = 1
• ASL.Height.VerticalHeight.high.toFin = 2

@[implicit_reducible]

instance ASL.Height.instLinearOrderVerticalHeight : LinearOrder VerticalHeight

low < neutral < high.

Equations

• ASL.Height.instLinearOrderVerticalHeight = LinearOrder.lift' ASL.Height.VerticalHeight.toFin ASL.Height.VerticalHeight.toFin_injective✝

@[implicit_reducible]

instance ASL.Height.instOrderTopVerticalHeight : OrderTop VerticalHeight

Equations

• ASL.Height.instOrderTopVerticalHeight = { top := ASL.Height.VerticalHeight.high, le_top := ASL.Height.instOrderTopVerticalHeight._proof_1 }

@[implicit_reducible]

instance ASL.Height.instOrderBotVerticalHeight : OrderBot VerticalHeight

Equations

• ASL.Height.instOrderBotVerticalHeight = { bot := ASL.Height.VerticalHeight.low, bot_le := ASL.Height.instOrderBotVerticalHeight._proof_1 }

@[reducible, inline]

abbrev ASL.Height.HeightDDRP (E : Type u_1) : Type u_1

Height-indexed domain restrictor for ASL signing space.

A HeightDDRP E is a DDRP parameterized by VerticalHeight: each height level maps to a domain predicate, with higher height giving a superset. The same nesting structure is instantiated by SpatialScale ([RS24a]) and by comparison class inference ([TG22]).

The nesting theorems DDRP.every_nesting, DDRP.some_nesting, and DDRP.no_nesting apply automatically.

Equations

• ASL.Height.HeightDDRP E = Quantification.DomainRestriction.DDRP ASL.Height.VerticalHeight E

def ASL.Height.heightToDDRP {E : Type u_1} (zone : E → VerticalHeight) : HeightDDRP E

Construct a height-indexed DDRP from a height assignment function.

Parallels sceneToDDRP in [RS24a]: given a function assigning entities to their minimum visible height level, builds the nested domain restrictor where region h contains all entities visible at height ≤ h.

Equations

• ASL.Height.heightToDDRP zone = { region := fun (h : ASL.Height.VerticalHeight) (e : E) => zone e ≤ h, monotone := ⋯, top_total := ⋯ }

def ASL.Height.arcPresup (E : Type u_1) [PartialOrder E] : Semantics.Presupposition.PartialProp E

⟦arc⟧ presupposition: the plural pronoun IX-arc presupposes that its referent is non-atomic (a plural individual).

⟦arc⟧ = λx: ¬Atom(x). x

This uses Mereology.Atom from Semantics/Mereology.lean. The at-issue content is the identity — arc contributes only a presupposition, not assertoric content.

Equations

• ASL.Height.arcPresup E = { presup := fun (x : E) => ¬Mereology.Atom x, assertion := fun (x : E) => True }

def ASL.Height.domainPresup {E : Type u_1} [PartialOrder E] (C : E → Prop) : VerticalHeight → Semantics.Presupposition.PartialProp E

Height-dependent domain presupposition.

• ⟦high⟧ = ⟦domain-k⟧: Cₑ ⊂ {y : y ≤ x} — context is a proper subpart, i.e., the domain widens beyond the contextual default. [DG22] eq. (44).

• ⟦neutral⟧: ∀y(y ≤ x → y ∈ Cₑ) — all parts are in the contextual domain. This is the unmarked default.

• ⟦low⟧: Mereology.Atom x — the referent is atomic (a singleton). [DG22] (p. 39) report that SOME-low yields specific indefinite readings ("someone — and maybe you — know who"), consistent with [Sch02]'s view of specificity as extreme domain restriction. The presupposition is maximally restrictive: the domain collapses to a single individual.

Equations

• ASL.Height.domainPresup C ASL.Height.VerticalHeight.high = { presup := fun (x : E) => (∀ (y : E), C y → y ≤ x) ∧ ∃ y ≤ x, ¬C y, assertion := fun (x : E) => True }
• ASL.Height.domainPresup C ASL.Height.VerticalHeight.neutral = { presup := fun (x : E) => ∀ y ≤ x, C y, assertion := fun (x : E) => True }
• ASL.Height.domainPresup C ASL.Height.VerticalHeight.low = { presup := fun (x : E) => Mereology.Atom x, assertion := fun (x : E) => True }

def ASL.Height.ixArcHeightPresup {E : Type u_1} [PartialOrder E] (C : E → Prop) (h : VerticalHeight) : Semantics.Presupposition.PartialProp E

Full IX-arc-height denotation: conjunction of the arc (non-atomicity) and domain (height-dependent containment) presuppositions.

⟦[IX_i]-arc-height⟧ = ιz: ¬Atom(z) ∧ domainPresup(height). (z = g(i))

Equations

• ASL.Height.ixArcHeightPresup C h = { presup := fun (x : E) => (ASL.Height.arcPresup E).presup x ∧ (ASL.Height.domainPresup C h).presup x, assertion := fun (x : E) => True }

theorem ASL.Height.high_incompatible_with_neutral {E : Type u_1} [PartialOrder E] {C : E → Prop} {x : E} (hHigh : (domainPresup C VerticalHeight.high).presup x) : ¬(domainPresup C VerticalHeight.neutral).presup x

The high presupposition entails the neutral presupposition is false: if Cₑ is a proper subpart of {y : y ≤ x}, then NOT all parts of x are in Cₑ. These are complementary conditions on the same domain.

theorem ASL.Height.arc_low_contradictory {E : Type u_1} [PartialOrder E] {C : E → Prop} {x : E} : ¬(ixArcHeightPresup C VerticalHeight.low).presup x

IX-arc-low is presuppositionally contradictory: arc requires non-atomicity (¬Atom), while low requires atomicity (Atom). This correctly predicts that the plural pronoun IX-arc cannot take low height — only singular IX can be specific/low. [DG22] (p. 39): the specific indefinite reading with low height uses SOME (an existential quantifier), not IX-arc (the plural pronoun).

[DG22] §5.3 derives quantifier truth conditions step-by-step through the partitive type function ⟦of⟧ = λxλy. y ≤ x (Ladusaw 1982). The pronoun IX-arc denotes a discourse referent z (the maximal plural individual), and ⟦of⟧ maps z to Set.Iic z — the principal down-set, i.e., the set of z's mereological parts. This set becomes the semantic restrictor of FS-ALL via the generalized quantifier structure.

The composition:
1. ⟦IX-arc-h⟧ = ιz : presup(h). z = g(i) — pronoun with presuppositions
2. ⟦of⟧ z = `Set.Iic z` = {y | y ≤ z} — parts of z
3. ⟦FS-ALL⟧(`Set.Iic z`)(Q) = ∀x ∈ Set.Iic z, Q x

The height presuppositions constrain `Set.Iic z` relative to C:
- neutral: `Set.Iic z ⊆ C` (default domain)
- high: `C ⊆ Set.Iic z` with `¬(Set.Iic z ⊆ C)` (expanded domain)

The proofs below are definitionally trivial — the presuppositions
encode exactly the `Set.Iic` containment relations. This is the
design payoff: `domainPresup` and `Set.Iic` are structurally
aligned by construction, not by accident.

`Set.Iic` is Mathlib's principal down-set — mereologically, the
parts of an entity. Key lemmas from Mathlib that apply directly:
- `Set.Iic_subset_Iic`: `Set.Iic a ⊆ Set.Iic b ↔ a ≤ b`
- `Set.Iic_top`: `Set.Iic ⊤ = Set.univ`
- `Set.mem_Iic`: `x ∈ Set.Iic b ↔ x ≤ b` 

theorem ASL.Height.partitive_neutral_subset {E : Type u_1} [PartialOrder E] {C : Set E} {z : E} (hNeut : (domainPresup C VerticalHeight.neutral).presup z) : Set.Iic z ⊆ C

Under the neutral presupposition (∀y, y ≤ z → C y), the partitive domain Set.Iic z is contained in C.

This IS the neutral presupposition — domainPresup .neutral encodes exactly Set.Iic z ⊆ C. The proof is id.

[DG22] eq. (50c): ⟦(of) IX-arc-neutral⟧ = λy. y ∈ Cₑ.

theorem ASL.Height.partitive_high_superset {E : Type u_1} [PartialOrder E] {C : Set E} {z : E} (hHigh : (domainPresup C VerticalHeight.high).presup z) : C ⊆ Set.Iic z ∧ ∃ y ∈ Set.Iic z, ¬C y

Under the high presupposition, C is a proper subset of Set.Iic z: all of C is among the parts of z, but z has parts outside C.

This IS the high presupposition — domainPresup .high encodes exactly C ⊆ Set.Iic z with a witness outside C. The proof is id.

[DG22] eq. (51c): ⟦(of) IX-arc-high⟧ = λy. y ∈ C', where C ⊂ C'.

theorem ASL.Height.partitive_ddrp_equiv {E : Type u_1} [PartialOrder E] (ddrp : HeightDDRP E) (h : VerticalHeight) (z : E) (align : ∀ (e : E), e ∈ ddrp.region h ↔ e ≤ z) (e : E) : e ∈ Set.Iic z ↔ e ∈ ddrp.region h

The partitive domain Set.Iic z aligns with a HeightDDRP region when the discourse referent z's mereological parts are exactly the entities visible at height h. This bridges the two construction paths:

• DDRP path: heightToDDRP zone → region h → quantifier restrictor
• Partitive path: IX-arc-h → Set.Iic z → ∀x ≤ z, Q x

The alignment condition align is satisfied when z is the maximal group entity at height h — the entity whose parts are exactly the individuals in the DDRP region. When this holds, the DDRP-based computation is a correct implementation of the compositional derivation.