Ahn (2015): The Semantics of Additive Either

@cite{ahn-2015}

Proceedings of Sinn und Bedeutung 19, pp. 20–35.

Key Proposal

Too and either are two-place predicates taking the host proposition p and a silent propositional anaphor q as arguments:

• ⟦too⟧(q)(p) = q ⊓ p (meet in the Boolean algebra of propositions)
• ⟦either⟧(q)(p) = q ⊔ p (join)

The propositional anaphor q must be a distinct focus alternative of p (@cite{rooth-1985}), capturing three properties shared by both particles:

1. Antecedent Requirement: q must be salient in discourse
2. Focus Sensitivity: q must be a focus alternative of p
3. Distinctness: q must be distinct from p

This replaces the traditional fully presuppositional account (@cite{heim-1992}, Rullmann 2003) with an assertive one. The key advantage is that it handles cases where too's additive meaning does not seem fully presupposed (e.g., "I don't know if Mary is in the elevator, but if John is in the elevator too, we will go over the weight limit").

NPI Distribution via Boolean Algebra

Prop' World is a BooleanAlgebra (via Mathlib's pointwise instances), and the entire NPI asymmetry falls out of the ⊓/⊔ duality:

• ⊓ entails its components (inf_le_left, inf_le_right), so exhaustification of too is always vacuous — not an NPI.
• ⊔ does not entail its components, so exhaustification of either negates them. In positive contexts this yields ⊥ (inf_compl_eq_bot via De Morgan). In negative contexts, complementation reverses the ordering (compl_le_compl), making all alternatives entailed — vacuous.

The ∨/∧ scale parallels ∃/∀: either (⊔, weak) is an NPI for exactly the same structural reason as any (∃, weak). The bridge theorem either_npi_via_chierchia instantiates @cite{chierchia-2013}'s SI–NPI generalization.

Note: the full NPI derivation uses the naive exhaustification operator O_ALT (which negates ALL non-entailed alternatives including domain alternatives), not the more conservative exh_IE (which only negates innocently excludable alternatives). Under exh_IE, the domain alternatives q and p are not both innocently excludable (negating both is inconsistent with q ∨ p), so exh_IE yields exclusive disjunction rather than contradiction. The NPI result requires Chierchia's assumption that NPIs obligatorily activate domain alternatives exhaustified by O_ALT.

@[reducible, inline]

abbrev Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.tooDenotation {World : Type u_1} (q p : Prop' World) : Prop' World

⟦too⟧(q)(p) = q ⊓ p — meet in the Boolean algebra of propositions.

Equations

• Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.tooDenotation q p = q ⊓ p

@[reducible, inline]

abbrev Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.eitherDenotation {World : Type u_1} (q p : Prop' World) : Prop' World

⟦either⟧(q)(p) = q ⊔ p — join in the Boolean algebra of propositions.

Equations

• Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.eitherDenotation q p = q ⊔ p

theorem Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.too_demorgan {World : Type u_1} (q p : Prop' World) : (tooDenotation q p)ᶜ = qᶜ ⊔ pᶜ

¬(q ⊓ p) = qᶜ ⊔ pᶜ — De Morgan for too.

theorem Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.either_demorgan {World : Type u_1} (q p : Prop' World) : (eitherDenotation q p)ᶜ = qᶜ ⊓ pᶜ

¬(q ⊔ p) = qᶜ ⊓ pᶜ — De Morgan for either.

theorem Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.too_entails_all_alts {World : Type u_1} (q p : Prop' World) : tooDenotation q p ≤ q ∧ tooDenotation q p ≤ p ∧ tooDenotation q p ≤ eitherDenotation q p

Too entails all alternatives — exhaustification always vacuous (not NPI). q ⊓ p ≤ q, q ⊓ p ≤ p, q ⊓ p ≤ q ⊔ p.

theorem Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.either_positive_contradiction {World : Type u_1} (q p : Prop' World) : eitherDenotation q p ⊓ qᶜ ⊓ pᶜ = ⊥

Either in positive context: O_ALT(q ⊔ p) = (q ⊔ p) ⊓ qᶜ ⊓ pᶜ = ⊥. By De Morgan, qᶜ ⊓ pᶜ = (q ⊔ p)ᶜ, so this is a ⊓ aᶜ = ⊥.

theorem Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.either_negative_vacuous {World : Type u_1} (q p : Prop' World) : (eitherDenotation q p)ᶜ ≤ qᶜ ⊓ pᶜ ⊓ (q ⊓ p)ᶜ

Either under negation: all alternatives entailed (vacuous). (q ⊔ p)ᶜ ≤ qᶜ ⊓ pᶜ ⊓ (q ⊓ p)ᶜ by compl_le_compl.

theorem Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.either_npi_via_chierchia {World : Type u_1} (C : Exhaustification.FreeChoice.Ctx World) (hDE : Antitone C) (q p : Prop' World) : Exhaustification.FreeChoice.siVacuous C (eitherDenotation q p) (q ⊓ p)

Either's NPI behavior as an instance of @cite{chierchia-2013}'s SI–NPI generalization: for any antitone (DE) context C, C(q ⊔ p) ∧ ¬C(q ⊓ p) is vacuous.

def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.eitherBasic : AdditiveParticleDatum

"either" with parallel negation — licensed (negative context).

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.eitherVerb : AdditiveParticleDatum

Verb focus with "either" — licensed.

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.eitherPositiveUngrammatical : AdditiveParticleDatum

"either" in positive context — ungrammatical (Ahn's (41)).

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.tooUnderNegation : AdditiveParticleDatum

"too" under negation — grammatical (Ahn's (24)–(25)).

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.almostDoesNotLicenseEither : AdditiveParticleDatum

"almost" does not license "either" (Ahn's (45)). Rullmann's licensing condition wrongly predicts almost licenses either. The exhaustification account correctly rules it out.

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.rullmannProblem : AdditiveParticleDatum

Rullmann's problem case: either blocked after positive antecedent (Ahn's (5)).

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.eitherNegAntecedent : AdditiveParticleDatum

Negative antecedent with parallel structure — basic "either" (Ahn's (32)).

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.elevatorToo : AdditiveParticleDatum

Non-presuppositional "too" — the elevator example (Ahn's intro). Under the conjunctive account, too asserts q ⊓ p, and the whole assertion is entailed, so q need not be presupposed.

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def Phenomena.Focus.AdditiveParticles.Studies.Ahn2015.examples : List AdditiveParticleDatum

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