Alonso-Ovalle & Menéndez-Benito (2010): Modal Indefinites #
@cite{alonso-ovalle-menendez-benito-2010}
Formalization of the core analysis: algún imposes an anti-singleton constraint on its domain of quantification, and the Modal Variation effect (speaker ignorance) is derived as a conversational implicature via scalar competition with singleton-domain alternatives.
Subset Selection Functions (§4.1) #
Indefinite determiners take a subset selection function f as argument
(@cite{alonso-ovalle-menendez-benito-2010}, building on von Fintel 1999a).
The function f maps a predicate P to a contextually relevant subset
f(P):
- (50):
⟦un⟧ = λf.λP.λQ. ∃x[f(P)(x) ∧ Q(x)] - (54):
⟦algún⟧ = λf.λP.λQ : anti-singleton(f). ∃x[f(P)(x) ∧ Q(x)]
The sole lexical difference: algún presupposes that f is
anti-singleton (|f(P)| > 1). Un allows singleton f.
Two Derivation Paths (§§4.2, 4.3) #
The Modal Variation effect is derived by scalar competition. The paper presents two parallel derivations:
- §4.2 (Necessity/ASSERT): Singleton competitors □(bedroom), □(living room), □(bathroom) are too strong — the speaker would have used one if she could. Negating them yields: the speaker doesn't know which room.
- §4.3 (Possibility/◇): Anti-exhaustivity implicatures — if ◇(bedroom), then also ◇(other rooms). Rules out singleton epistemic states.
Both paths derive the same Modal Variation effect.
Modal Variation vs Free Choice (§§2, 4.4) #
The anti-singleton constraint derives a WEAKER modal effect than the domain widening of irgendein (@cite{kratzer-shimoyama-2002}):
- Modal Variation (algún): at least two domain members are epistemic possibilities — competitors are singleton subdomains only.
- Free Choice (irgendein): EVERY domain member is a possibility — competitors are ALL subdomains.
Typology (Table 1) #
Two parameters (uniqueness × domain constraint) yield a 2×2 typology:
- Cell 1: irgendein, uno qualsiasi (widening + uniqueness → FC)
- Cell 4: algún (anti-singleton + non-uniqueness → MV + number ignorance)
- Cells 2, 3: predicted but unattested at time of publication
A subset selection function maps predicates to predicates ((50)).
In @cite{alonso-ovalle-menendez-benito-2010}'s analysis, f models
contextual domain restriction: f(P) selects the subset of P that
the determiner quantifies over. Following von Fintel (1999a), different
indefinite determiners impose different constraints on f.
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- Phenomena.ModalIndefinites.Studies.AlonsoOvalleMenendezBenito2010.SubsetSelFn Entity = ((Entity → Bool) → Entity → Bool)
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The cardinality of the selected subdomain: |f(P)| in a finite domain.
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- Phenomena.ModalIndefinites.Studies.AlonsoOvalleMenendezBenito2010.selectedSize f domain P = (List.filter (f P) domain).length
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Singleton SSF ((52)): f is singleton iff |f(P)| = 1.
Singleton selection functions yield specific indefinites — the speaker has a particular witness in mind. Un allows these; algún does not.
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Anti-singleton SSF ((53)): f is anti-singleton iff |f(P)| > 1.
Algún presupposes that its selection function is anti-singleton: the domain of quantification must contain more than one individual. This is the paper's core lexical-semantic claim.
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⟦un⟧ = λf.λP.λQ. ∃x[f(P)(x) ∧ Q(x)] ((50)).
The plain indefinite un: existential quantification over f(P).
No constraint on f — the domain CAN be a singleton.
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- Phenomena.ModalIndefinites.Studies.AlonsoOvalleMenendezBenito2010.un_sat f domain P Q = domain.any fun (x : Entity) => f P x && Q x
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⟦algún⟧ = λf.λP.λQ : anti-singleton(f). ∃x[f(P)(x) ∧ Q(x)] ((54)).
Same assertive content as un but with an anti-singleton
presupposition on f. Returns (presupposition, assertion).
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When the presupposition is satisfied, algún and un have the
same assertive content. The ONLY lexical difference is the
presupposition on f.
The three rooms of the house ((57)).
Scenario (15): María, Juan, and Pedro are playing hide-and-seek. Pedro believes Juan is inside but not in the bathroom or kitchen.
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Un is felicitous with a singleton domain ((46)): "Juan compró un libro que resultó ser el más caro."
Algún rejects singleton domains ((47)): "# Juan compró algún libro que resultó ser el más caro." The anti-singleton presupposition fails.
Algún accepts non-singleton domains.
The covert assertoric operator ((20)): ⟦ASSERT⟧ᶜ = λp.λw. ∀w' ∈ Epistemicₛₚₑₐₖₑᵣ(w)[p(w')]
Ranges over the speaker's epistemic alternatives. Following @cite{kratzer-shimoyama-2002}, unembedded algún sentences are in the scope of this operator, unifying the modal and non-modal cases.
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Full sentence under ASSERT: □[∃r ∈ f(room). Juan is in r] = ∀w' ∈ Epistemic. ∃r ∈ f(room). juanIn(r)(w')
Adapted from (55b) to the room scenario.
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With the full domain: Pedro believes Juan is in SOME room. Trivially true.
With a singleton domain {bedroom}: ASSERT requires Pedro to believe Juan is in the bedroom in ALL epistemic alternatives. He doesn't — he also considers the living room. So this is FALSE.
With anti-singleton {bedroom, living room}: Pedro believes Juan is in one of these two rooms. Holds in all his alternatives.
The Modal Variation effect is NOT stipulated — it is DERIVED:
- Speaker used algún (anti-singleton f) rather than singleton competitors □(Juan is in the bedroom), □(… living room), etc.
- By the quantity maxim, the speaker cannot assert any singleton.
- Therefore: the speaker doesn't know which room → ignorance.
The competitors ((58)) are SINGLETON subdomain alternatives. The implicature ((59b)) negates each: ¬□(bedroom) ∧ ¬□(living room) ∧ ¬□(bathroom)
The strengthened meaning ((59)): assertion + negated singleton competitors.
assertion: □(∃r ∈ f(room). Juan is in r) implicature: ¬□(bedroom) ∧ ¬□(living room) ∧ ¬□(bathroom)
The conjunction derives the Modal Variation effect.
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The strengthened meaning holds for Pedro: he believes Juan is in some room (bedroom or living room) but cannot assert which.
Each singleton competitor individually fails — Pedro doesn't know which room Juan is in. This IS the Modal Variation effect.
The strengthened meaning correctly RULES OUT scenarios where the speaker knows the answer. If Pedro's epistemic state were a singleton (only the bedroom world), the strengthened meaning would fail.
The paper gives a SECOND derivation path when algún is under a possibility modal (§4.3, (60)–(68)). The reasoning:
- Speaker uses ◇(algún room) rather than a singleton ◇(bedroom).
- The hearer infers anti-exhaustivity: if one room is possible, some other room must also be possible.
- This rules out singleton epistemic states → Modal Variation.
The anti-exhaustivity implicatures ((68b)): ◇(bedroom) → ◇(living room ∨ bathroom) ◇(living room) → ◇(bedroom ∨ bathroom) ◇(bathroom) → ◇(bedroom ∨ living room)
Possibility modal ◇ ((60b)): ◇(p) = ∃w' ∈ Epistemic. p(w')
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Sentence under ◇ ((60b)): ◇[∃r ∈ f(room). Juan is in r]
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Anti-exhaustivity implicature for room r ((68b)):
◇(Juan is in r) → ◇(Juan is in some OTHER room).
As a Boolean: ¬◇(r) ∨ ◇(other rooms).
Blocks exhaustive readings where only one room is possible.
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Strengthened meaning under ◇ ((68)): assertion + anti-exhaustivity implicatures for all singleton competitors.
This is the §4.3 analog of strengthenedMeaning (§4.2).
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Under ◇, the strengthened meaning holds for Pedro.
Under ◇, singleton epistemic states are ruled out: if Pedro only considers the bedroom, the anti-exhaustivity implicature for bedroom fails (there's no other room that's possible).
Both derivation paths (§4.2 under □, §4.3 under ◇) agree: they accept the same epistemic states for Pedro.
Algún's Modal Variation is WEAKER than irgendein's Free Choice:
- Modal Variation ((18)): the witnesses vary across epistemic alternatives — at least two domain members are possibilities.
- Free Choice ((13c)): ∀x[P(w)(x) → ∃w' ∈ 𝒜ᵥ. Q(w')(x)] — EVERY domain member is a possibility.
Scenario (27)–(30): Pedro knows Juan is NOT in the bathroom. Cualquiera (FC) is ruled out: not ALL rooms are possibilities. Algún (MV) is felicitous: at least two rooms are.
Free Choice ((13c)): every room is an epistemic possibility for Juan.
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Modal Variation (counting version): at least two rooms are epistemic possibilities.
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Modal Variation ((18), formal definition): ∃w', w'' ∈ 𝒟_w [{x : P(w')(x) & Q(w')(x)} ≠ {x : P(w'')(x) & Q(w'')(x)}]
There exist two epistemic alternatives where the sets of individuals satisfying both the restrictor and the scope differ.
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The formal definition (18) and the counting definition agree in the hide-and-seek model.
Pedro's situation ((27)–(30)): MV holds but FC doesn't. Algún is appropriate; cualquiera is not.
With full uncertainty (all three rooms possible), BOTH hold.
FC entails MV (for non-trivial domains): if ALL are possibilities then certainly more than one is. But not vice versa.
The two domain constraint strategies for modal indefinites.
@cite{alonso-ovalle-menendez-benito-2010} §4.4 argues that the contrast between irgendein and algún reduces to different constraints on the selection function:
- Widening: f(P) = P (maximal domain; competitors = all subdomains)
- Anti-singleton: |f(P)| > 1 (competitors = singleton subdomains only)
Different competitor sets → different exhaustification outcomes → different modal effects (Free Choice vs Modal Variation).
- widening : DomainConstraint
- antiSingleton : DomainConstraint
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The predicted modal effect from each domain constraint type.
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Different constraints yield different modal effects.
The critical mechanism: different constraints generate different competitor sets for exhaustification.
For irgendein (domain widener, (70)): competitors = ALL proper subdomains: ◇(bedroom), ◇(living room), ◇(bathroom), ◇(bedroom ∨ living room), ◇(bedroom ∨ bathroom), ◇(living room ∨ bathroom)
The full domain ◇(bedroom ∨ living room ∨ bathroom) is the assertion (69) itself, NOT a competitor.
For algún (anti-singleton, (58)): competitors = SINGLETON subdomains: □(bedroom), □(living room), □(bathroom)
Exhaustifying over all subdomains → Free Choice (every room possible). Exhaustifying over singletons only → Modal Variation (≥2 rooms possible).
Singleton competitor set (for anti-singleton items like algún).
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Proper subdomain competitors ((70a–f), for domain wideners like irgendein). Excludes the full domain, which is the assertion itself ((69)), not a competitor.
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Anti-singleton items have strictly fewer competitors. This is WHY their modal effect is weaker: fewer negated competitors = weaker implicature (MV instead of FC).
Whether uniqueness of the existential witness is assumed.
Uniqueness: at most one individual per world satisfies the claim (e.g., María can only marry one person). Ignorance is about IDENTITY ("the speaker doesn't know who").
Non-uniqueness: multiple witnesses possible (e.g., multiple flies in the soup, (73)–(78)). Ignorance is about NUMBER ("the speaker doesn't know how many").
- uniqueness : UniquenessParam
- nonUniqueness : UniquenessParam
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A cell in the 2010 typology (Table 1).
- uniqueness : UniquenessParam
- constraint : DomainConstraint
- modalEffect : String
- numberIgnorance : Bool
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Cell 1: Widening + Uniqueness → FC, no number ignorance. Irgendein, uno qualsiasi: "any doctor" — every doctor is a permitted option, but no ignorance about how many.
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Cell 4: Anti-singleton + Non-uniqueness → MV + number ignorance. Algún: "alguna mosca" — the speaker doesn't know how many flies are in the soup ((73)–(78)).
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Cell 2: Anti-singleton + Uniqueness (predicted, unattested in 2010). Would show MV without number ignorance.
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Cell 3: Widening + Non-uniqueness (predicted, unattested in 2010). Would show FC with number ignorance.
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The two filled cells differ on BOTH dimensions.
Two cells are attested, two are predicted gaps.
The 2010 paper's anti-singleton constraint maps to the upperBounded
field in the fragment entry. upperBounded = true records that
algún's domain cannot be a singleton.
The Modal Variation component is classified as notAtIssue in the
fragment entry, consistent with the paper's §3.3 argument that it is
a conversational implicature: cancellable ((42)), disappears under
DE operators ((43)–(44)), reinforceable without redundancy ((45d)).
The 2010 paper analyzes algún as having epistemic content only: the Modal Variation inference concerns the SPEAKER's beliefs, mediated by ASSERT ((20)). The fragment entry records epistemic-only flavor.