Modal Indefinite Semantics #
@cite{alonso-ovalle-royer-2024} @cite{alonso-ovalle-menendez-benito-2010} @cite{hacquard-2006}
Formal denotation of modal indefinites: existential quantifiers carrying a modal component whose domain is projected from an event argument via an anchoring function. Extracted from EventRelativity §§3–7.
Core Denotation (A-@cite{alonso-ovalle-royer-2024}, (59)) #
⟦MI⟧^{f,e₁} = λP.λQ.λw.
∃x[P(x)(w) ∧ Q(x)(w)] ∧
∀y[P(y)(w) → ◇_{f(e₁)}(Q(y)(w'))]
The existential component is standard; the universal modal component adds modal variation: every restrictor member is a possible scope-satisfier in some accessible world. The event argument e₁ and anchoring function f determine the modal domain (epistemic, circumstantial, random choice).
Upper-Boundedness (A-@cite{alonso-ovalle-royer-2024}, §5) #
Some modal indefinites (algún) impose an anti-singleton inference: the speaker considers it possible that not all domain members satisfy the scope. Others (yalnhej) lack this condition.
Harmonic Interpretations (A-@cite{alonso-ovalle-royer-2024}, §4.3) #
Under external modals (imperatives, deontic), the MI's anchor can be co-indexed with the external modal's event, yielding "any X is fine" readings. Non-harmonic anchoring gives "a random X" readings.
The modal component of a modal indefinite (A-@cite{alonso-ovalle-royer-2024}, (59)):
∀y[P(y)(w) → ◇_{f(e₁)}(Q(y)(w'))]
For every individual y satisfying restrictor P in the actual world, there exists an accessible world w' (via anchoring function f applied to event e₁) where y satisfies scope predicate Q. This is the "modal variation" inference: every domain member is a possible witness.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Full modal indefinite denotation (A-@cite{alonso-ovalle-royer-2024}, (59)):
⟦MI⟧^{f,e₁} = λP.λQ.λw.
∃x[P(x)(w) ∧ Q(x)(w)] ∧
∀y[P(y)(w) → ◇_{f(e₁)}(Q(y)(w'))]
The existential component asserts that some individual satisfies both restrictor and scope. The universal modal component asserts that EVERY restrictor individual is a possible scope-satisfier in some accessible world — the free choice / modal variation effect.
Equations
- One or more equations did not get rendered due to their size.
Instances For
An upper-bounded modal indefinite additionally requires that NOT every P is Q in the actual world — the speaker does not know/intend for all domain members to satisfy Q.
⟦MI_UB⟧ = ⟦MI⟧ ∧ ¬∀x[P(x)(w) → Q(x)(w)]
This is the anti-singleton inference of algún. Items like yalnhej lack this condition and are compatible with all P being Q.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Upper-boundedness strengthens the modal indefinite: if the UB version holds, the plain MI version holds.
Equations
- Semantics.Modality.ModalIndefinites.instBEqBook.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- One or more equations did not get rendered due to their size.
A speech event and a described event.
- speech : SpeechOrDescribed
- described : SpeechOrDescribed
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
"Yalnhej bought a book" in world abc: ∃x[book(x) ∧ avail(x)] ∧ ∀y[book(y) → ◇_{EPI}(avail(y))] Every book is available in some accessible world.
Not upper-bounded: in world abc, all three books ARE available, yet the MI denotation holds. The anti-singleton condition fails (all books satisfy the scope), showing yalnhej is non-UB.
Yalnhej is compatible with partial-domain scenarios: the speaker can felicitously use yalnhej even when not all P are Q. This distinguishes it from maximal free relatives (whatever), which require every domain member to satisfy the scope. Unlike upper-boundedness (which blocks ∀P→Q), non-maximality is about COMPATIBILITY with ¬∀P→Q — a weaker property.
We demonstrate non-maximality using the existing 3-book model:
in world ab (books a,b available but NOT c), the MI denotation
still holds because every book is available in SOME accessible world,
even though not every book is available in the actual world.
MI holds in world ab where book c is NOT available. The existential component (∃x P∧Q) holds (book a is available). The modal component (∀y P→◇Q) holds (each book is available in some accessible world). Crucially, ¬∀y P→Q(y)(ab): book c is not available in ab. This shows yalnhej is compatible with not-all-P-being-Q — non-maximality.
Three-way contrast: maximality vs yalnhej vs algún. In world abc (all books available): MI holds + UB fails. In world ab (not all available): MI holds + UB holds. A maximal item (whatever) would require all books available (fail in ab). Algún (UB) would require not-all (fail in abc). Yalnhej (non-UB) succeeds in BOTH.
When a modal indefinite occurs under an external modal (imperative, deontic, attitude verb), the MI's anchoring event can be CO-INDEXED with the external modal's event. This "harmonic" configuration gives "any X is fine" readings — the MI's modal domain aligns with the embedding modal's domain.
Non-harmonic: the MI's anchor is independent of the external modal. "Grab yalnhej card" = grab a random card (MI anchors to described event). Harmonic: the MI's anchor is co-indexed with the imperative/deontic event. "Grab yalnhej card" = any card is fine (MI anchors to imperative event).
We model this with a card-grabbing scenario: three cards, worlds varying in which cards are grabbable, and two event types (local vs imperative).
Equations
- Semantics.Modality.ModalIndefinites.instBEqCard.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Non-harmonic MI fails: when the MI anchors to the local event,
only world only1 is accessible. In only1, only c1 is grabbable.
The modal component ∀y[card(y) → ◇_{local}(grab(y))] fails because
c2 and c3 are not grabbable in any locally accessible world.
Harmonic MI succeeds: when the MI's anchor is co-indexed with the
imperative event, all worlds are accessible. Every card is grabbable
in some world (c1 in only1, c2 in only2, c3 in all). The
modal component ∀y[card(y) → ◇_{imperative}(grab(y))] holds.
This gives the "any card is fine" reading.
Harmonic ≠ non-harmonic: the two readings are formally distinct. Same world of evaluation (.only1), same domain, same predicates — only the anchoring event differs.