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Donkey disjunctions and overlapping updates

Elliott, P. D. (2023). In Proceedings of SALT 33 (pp. 666–685). Linguistic Society of America

Source Bibliography Paper

This file formalizes Elliott's analysis of donkey disjunctions such as either there's no bathroom in this house, or it's in a funny place in Bilateral Update Semantics. The denial of the negated existential in the first disjunct introduces the pronoun's referent, so the positive update keeps the points with no bathroom and extends the others to a bathroom in a funny place, a heterogeneous existential reading, and the disjunction and its negation hold together where there are bathrooms of both kinds. Asserted exhaustively relative to its negation, the disjunction keeps the worlds where every bathroom is in a funny place, unless the question under discussion distinguishes no worlds. Since the negative update of an existential is defined by subsistence, the anaphoric presupposition of a free variable in its scope projects.

Implementation notes

  • Assertion and exhaustive assertion are a definedness condition, BilateralUpdate.Assertable, and an output, the positive update or exhAssert.
  • A question under discussion is a partition of all worlds, so the cell-mate of exhaustive assertion is required to be a world of the context, which the paper's partition of the context set guarantees.
  • The projection of the presupposition of y holds at states whose points all define the same referents; where two points of one world differ in whether they define y, the one that does not subsists in the updates of the one that does.

Errata

  • Assertion (26a) and exhaustive assertion (45a), (46a) are defined when the unknown update is nonempty; the surrounding prose requires it to be empty, and that is how they are read here.
  • Descent (21) requires only a larger domain, though the prose glosses it as extension; the substrate takes extension.
  • Heimian familiarity is said to follow from (26), but an open sentence can be assertable where its variable is not familiar.

References

Donkey disjunctions (§2.4.3)

L77–79
L82–92
L95–105

The positive update of the donkey disjunction keeps the points without a bathroom and extends the others to a bathroom in a funny place, (33).

theorem positive_donkeyL107–113
{W : Type u_1} {V : Type u_2} {E : Type u_3} [DecidableEq V] {s : Set (DynamicSemantics.Possibility W V (Flat E))} {x : V} (B U : E → W → Prop) [Nonempty E] (hx : DynamicSemantics.State.Novel s x) :
(donkey B U x).positive s = {p : DynamicSemantics.Possibility W V (Flat E) | p ∈ s ∧ ∀ (e : E), ¬B e p.world} ∪ (DynamicSemantics.BilateralUpdate.pred1 U x).positive ((DynamicSemantics.BilateralUpdate.exists_ x (DynamicSemantics.BilateralUpdate.pred1 B x)).positive s)

Built from

The law of non-contradiction (§2.4.3)

The donkey disjunction and its negation hold together exactly where there is a bathroom in a funny place and a bathroom elsewhere, (37).

theorem nonempty_lnc_iffL159–173
{W : Type u_1} {V : Type u_2} {E : Type u_3} [DecidableEq V] {s : Set (DynamicSemantics.Possibility W V (Flat E))} {x : V} (B U : E → W → Prop) [Nonempty E] (hx : DynamicSemantics.State.Novel s x) :
(((donkey B U x).conj (donkey B U x).neg).positive s).Nonempty ↔ ∃ p ∈ s, (∃ (e : E), B e p.world ∧ U e p.world) ∧ ∃ (e : E), B e p.world ∧ ¬U e p.world

Built from

Exhaustive assertion (§3)

L177–184

Asserted exhaustively at the fact-finding question, the donkey disjunction keeps exactly the worlds where every bathroom is in a funny place.

theorem world_exhAssert_botL192–214
{W : Type u_1} {V : Type u_2} {E : Type u_3} [DecidableEq V] {s : Set (DynamicSemantics.Possibility W V (Flat E))} {x : V} (B U : E → W → Prop) [Nonempty E] (hx : DynamicSemantics.State.Novel s x) :
DynamicSemantics.Possibility.world '' exhAssert ⊥ (donkey B U x) s = DynamicSemantics.Possibility.world '' s ∩ {w : W | ∀ (e : E), B e w → U e w}

Built from ,

Existential statements (§2.3)

At a state whose points all define the same referents, the denial of ∃x R(x, y) at a novel x keeps the points that define y and whose value for y nothing bears R to: the anaphoric presupposition of y projects, (23).

theorem negative_exists_pred2L314–334
{W : Type u_1} {V : Type u_2} {E : Type u_3} [DecidableEq V] {s : Set (DynamicSemantics.Possibility W V (Flat E))} {x : V} [Nonempty E] (R : E → E → W → Prop) {X : Set V} {y : V} (hs : DynamicSemantics.State.UniformAt X s) (hx : x ∉ X) (hyx : y ≠ x) :
(DynamicSemantics.BilateralUpdate.exists_ x (DynamicSemantics.BilateralUpdate.pred2 R x y)).negative s = {p : DynamicSemantics.Possibility W V (Flat E) | p ∈ s ∧ ∃ (a : E), p.assignment y = ↑a ∧ ∀ (e : E), ¬R e a p.world}

Assertion (§2.4.1)