The semantics of additive either

Ahn takes too and additive either to be two-place predicates over the host proposition and a silent propositional anaphor q, presupposed to be a distinct focus alternative of the host, and lets them assert a conjunction and a disjunction respectively (too, either, Antecedent). Negation then scopes over the assertion: Mary didn't buy them too asserts ¬(q ∧ p), which with the discourse-given q is ¬p (not_too_of_antecedent), while John didn't leave either asserts ¬(q ∨ p), so both the host and the antecedent are denied (either_compl). Since the additive component is asserted rather than presupposed, denying it with an anaphoric as well is consistent where denying an existential is not (abrusan_contrast).

The distribution of either follows from the disjunction under the exhaustification account of polarity items: the alternatives of a disjunction are the disjuncts and their conjunction, and the naive exhaustifier, Chierchia's O (Exhaustification.exh), negates every alternative the prejacent fails to entail. In a positive context that leaves the disjunction without either disjunct, a contradiction (either_positive_contradiction); under negation the negated disjunction entails every alternative, so exhaustification is vacuous (either_negative_vacuous), and likewise in any downward-entailing context (either_vacuous_in_de), which is why the account overgenerates to all such contexts and leaves the strong-NPI restriction to negation open. Too entails all of its alternatives, so it is exhaustified vacuously everywhere and is no polarity item (too_entails_alts). The paper's judgments are its rows: either is acceptable exactly in a negative host whose antecedent is not positive, and too fails only for want of an antecedent (either_iff_negative, too_any_polarity).

References

• [Ahn15]
• [Rul03]
• [Hei92]
• [Kri09c]
• [Roo92]
• [Chi13]
• [Sau04]

Too and either

def Ahn2015.Antecedent {World : Type u_1} (q p : Set World) (C : Set (Set World)) : Prop

Too and either share the presupposition that the anaphor is a focus alternative of the host distinct from it.

Equations

• Ahn2015.Antecedent q p C = (q ∈ C ∧ q ≠ p)

@[reducible, inline]

abbrev Ahn2015.too {World : Type u_1} (q p : Set World) : Set World

Too asserts its additive meaning as the conjunction q ⊓ p with the anaphor.

Equations

• Ahn2015.too q p = q ⊓ p

@[reducible, inline]

abbrev Ahn2015.either {World : Type u_1} (q p : Set World) : Set World

Either asserts the disjunctive counterpart, q ⊔ p.

Equations

• Ahn2015.either q p = q ⊔ p

theorem Ahn2015.too_compl {World : Type u_1} (q p : Set World) : (too q p)ᶜ = qᶜ ⊔ pᶜ

Negation above too denies the conjunction.

theorem Ahn2015.not_too_of_antecedent {World : Type u_1} (q p : Set World) {w : World} (hq : w ∈ q) : w ∈ (too q p)ᶜ ↔ w ∉ p

With the antecedent settled true, negated too denies the host.

theorem Ahn2015.either_compl {World : Type u_1} (q p : Set World) : (either q p)ᶜ = qᶜ ⊓ pᶜ

Negation above either denies both the host and the antecedent.

theorem Ahn2015.abrusan_contrast {World : Type u_1} (q p : Set World) {e : Set World} (he : p ⊆ e) : p ⊓ eᶜ = ⊥ ∧ p ⊓ (too q p)ᶜ = p ⊓ qᶜ

Denying an existential the host entails is contradictory, denying the anaphoric conjunction is not: it denies the antecedent.

Exhaustification

def Ahn2015.disjAlts {World : Type u_1} (q p : Set World) : Set (Set World)

The alternatives of a disjunction are the disjuncts and their conjunction.

Equations

• Ahn2015.disjAlts q p = {q ⊔ p, q, p, q ⊓ p}

theorem Ahn2015.too_entails_alts {World : Type u_1} (q p : Set World) : Exhaustification.exh {q ⊓ p, q, p, q ⊔ p} (too q p) = too q p

Too entails every alternative of the conjunction, so exhaustification is vacuous.

theorem Ahn2015.either_positive_contradiction {World : Type u_1} (q p : Set World) (hq : ¬either q p ⊆ q) (hp : ¬either q p ⊆ p) : Exhaustification.exh (disjAlts q p) (either q p) = ⊥

In a positive context either is exhaustified to a contradiction, provided neither disjunct is entailed by the disjunction.

theorem Ahn2015.either_negative_vacuous {World : Type u_1} (q p : Set World) : Exhaustification.exh (compl '' disjAlts q p) (either q p)ᶜ = (either q p)ᶜ

Under negation the negated disjunction entails every negated alternative, so exhaustification is vacuous.

theorem Ahn2015.either_vacuous_in_de {World : Type u_1} (q p : Set World) (C : Set World → Set World) (hDE : Antitone C) : C (either q p) ⊆ C (q ⊓ p)

In any downward-entailing context the scalar alternative of the disjunction is entailed, so exhaustification is vacuous there too: the account licenses either beyond negation.

The data

theorem Ahn2015.either_iff_negative (e : Datum) : e ∈ Examples.all → e.feature? "particle" = some "either" → (e.judgment = Judgment.acceptable ↔ e.feature? "polarity" = some "negative" ∧ e.feature? "antecedent" ≠ some "positive")

Either is acceptable exactly in a negative host whose antecedent is not positive.

theorem Ahn2015.too_any_polarity (e : Datum) : e ∈ Examples.all → e.feature? "particle" = some "too" → ¬e.JudgedFelicitous → e.feature? "polarity" = some "positive" ∧ e.context ≠ ""

Too occurs in both polarities; its infelicity needs a missing antecedent.