Maximal Informativity and Temporal in-Adverbials #
@cite{beck-rullmann-1999} @cite{fox-hackl-2006} @cite{kennedy-2007} @cite{krifka-1989} @cite{krifka-1998} @cite{rouillard-2026}
@cite{rouillard-2026} "Maximal informativity accounts for the distribution of temporal in-adverbials" (Linguistics and Philosophy 49:1–56).
Core Contribution #
Temporal in-adverbials (TIAs) lead a double life:
- E-TIAs ("wrote a paper in three days"): measure event durations. Acceptable only with telic VPs.
- G-TIAs ("hasn't been sick in three days"): measure gap durations. Negative polarity items: acceptable only in negative perfects.
Both distributional restrictions derive from a single principle:
Maximal Informativity Principle (MIP): for some constituent of the LF in which a TIA appears, the numeral in the TIA's measure phrase must be capable of being maximally informative. If no number can be maximally informative (information collapse), the TIA is blocked.
Architecture #
Mereology (CUM, DIV, QUA) ──▷ scalar properties of derived TIA meanings
│ │
telicity informativity
│ │
▼ ▼
E-TIA licensing ◁───── MIP ─────▷ G-TIA polarity sensitivity
│
open PTS + closed runtime
(from Time.lean)
Sections #
- Maximal informativity (max⊨)
- Measure functions and
pts(prior time spans) - E-TIA semantics:
inas event-level map function - G-TIA semantics:
inas perfect-level identity map - The MIP as a licensing condition
- Information collapse theorems
- Bridge: Kennedy open/closed scales ↔ TIA licensing
Upward scalarity for ℕ-indexed families.
Specialization of Core.Scale.IsUpwardMonotone to ℕ.
@cite{rouillard-2026} p. 24: outputs of (77) are totally ordered by entailment,
smaller values entail larger ones.
@cite{beck-rullmann-1999}: the maximally informative number is the smallest
value returning a true proposition.
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Downward scalarity for ℕ-indexed families.
Specialization of Core.Scale.IsDownwardMonotone to ℕ.
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Upward scalar properties have maximally informative elements: the smallest true value is max⊨ (it entails all others). This is the abstract version of Rouillard's argument for why E-TIAs with telic VPs are acceptable.
Bimonotone ℕ-families (both upward and downward scalar) make every
value true at every world where any value is true.
@cite{rouillard-2026} p. 25: the property in (80) is both upward and downward
scalar — equivalently, it is a constant function. This is information
collapse. Derived from Core.Scale.bimonotone_constant.
A temporal measure function: assigns positive durations to intervals. @cite{rouillard-2026} §2.2.2, eq. (5)–(7): μ is additive over non-overlapping times and positive.
- μ : Core.Time.Interval Time → ℕ
The measure of an interval in some unit φ
- extensible (i : Core.Time.Interval Time) (m : ℕ) : self.μ i ≤ m → ∃ (j : Core.Time.Interval Time), i.subinterval j ∧ self.μ j = m
Any interval can be extended to a superinterval with a given larger measure. @cite{rouillard-2026}: temporal measure units (days, hours) are additive, so any interval can be padded to achieve any target measure ≥ current.
- subdivisible (i : Core.Time.Interval Time) (m : ℕ) : m ≤ self.μ i → ∃ (j : Core.Time.Interval Time), j.subinterval i ∧ self.μ j = m
Any interval can be subdivided to a subinterval with a given smaller measure. @cite{rouillard-2026}: temporal measure units are additive, so any interval can be trimmed to achieve any target measure ≤ current.
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Prior time span: the maximal interval right-bounded by t with measure n. @cite{rouillard-2026} eq. (50): pts(n, φ, t¹) := max⊑ᵢ(λt².t² ∈ S ∧ ∃t³[μ_φ(t³) = n ∧ rb(t¹, t²) ∧ t² ⊑ᵢ t³]) Simplified: pts(n, μ, s) is the interval consisting of every moment inclusively ordered between s and the moment n φ-units prior to s.
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The preposition in as an event-level adverbial (E-TIA reading). @cite{rouillard-2026} eq. (62): ⟦in⟧ := λM_σi λt λx_σ. M(x) ⊑ᵢ t. For E-TIAs, M = τ (runtime function): the event's runtime is included in the time denoted by the measure phrase.
Type: (e → Interval Time) → Time → (e → Prop) Instantiation for E-TIA: M = Eventuality.τ
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- Semantics.Montague.Sentence.MaximalInformativity.inETIA e bound = e.τ.subinterval bound
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E-TIA derived property: for each number n, the property that at world w there exists a P-event whose runtime is included in an n-unit time ending at g(1). @cite{rouillard-2026} eq. (77): λnλw.∃t[μ_d(t) = n ∧ ∃e[P(e)(w) ∧ τ_w(e) ⊑ᵢ g(1) ∧ τ_w(e) ⊑ᵢ t]]
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- One or more equations did not get rendered due to their size.
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G-TIA derived property: for each number n, the property that at world w an event's runtime is included in the prior time span pts(n, d, s). @cite{rouillard-2026} eq. (94), revised with open PTS (101): λnλw.∃e[P(e)(w) ∧ τ_w(e) ⊑ᵢ o(pts(n, d, s))]
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- One or more equations did not get rendered due to their size.
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Maximal Informativity Principle (MIP). @cite{rouillard-2026} eq. (92): Given a numeral N, a measure word M, an index j, and a map function F, a constituent of the form [ [ N M] j … [ in F] tⱼ] is licensed only if it is contained in a constituent γ such that, for some w¹, max⊨(w¹, λnλw². ⟦γ[N ↦ proₖ]⟧) = ⟦N⟧.
The TIA is licensed iff the numeral can be maximally informative in some constituent γ containing it.
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- Semantics.Montague.Sentence.MaximalInformativity.MIP_Licensed derivedProp n = ∃ (w : W), Core.Scale.IsMaxInf derivedProp n w
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E-TIA is MIP-licensed iff the derived E-TIA property has a maximally informative numeral at some world.
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- One or more equations did not get rendered due to their size.
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G-TIA is MIP-licensed (in positive environment) iff the derived G-TIA property has a maximally informative numeral at some world.
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- One or more equations did not get rendered due to their size.
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G-TIA under negation: the negated G-TIA property.
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G-TIA is MIP-licensed under negation.
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- One or more equations did not get rendered due to their size.
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E-TIA Information Collapse for Atelic VPs. When a VP predicate has the subinterval property (is atelic/DIV), the E-TIA property is both upward and downward monotone (constant), so no numeral is maximally informative → the E-TIA is blocked. @cite{rouillard-2026} §4.1.1: the interaction of the subinterval property with E-TIAs results in information collapse.
Proof: For any world w where eTIAProperty holds at n, we show it holds at m for arbitrary m. Given P-event e at w:
- If m ≤ μ(τ(e)): subdivide τ(e) to get j ⊆ τ(e) with μ(j) = m. By SUB, ⟨j⟩ is a P-event. Since j ⊆ τ(e) ⊆ g1, j ⊆ g1.
- If m ≥ μ(τ(e)): extend τ(e) to get j ⊇ τ(e) with μ(j) = m. Same event e works.
Quantized predicates yield upward scalar E-TIA properties. @cite{rouillard-2026} §4.1.1: when P is QUA (telic), the E-TIA property in (77) is upward scalar — propositions from smaller n entail those from larger n. The maximally informative number is the smallest n returning a true proposition (= the actual duration of the event).
This is the structural explanation for why E-TIAs are acceptable with
telic VPs: there exists a unique maximally informative number.
Proof: The same event e works. Extend the containing time t (with
μ(t) = n) to j ⊇ t with μ(j) = m ≥ n via MeasureFun.extensible.
Since τ(e) ⊆ t ⊆ j, the witness transfers.
No smallest open PTS can include a closed runtime. @cite{rouillard-2026} §4.2.2, key insight: given dense time, if event runtimes are closed and PTSs are open, there can never be a smallest open interval to include a closed time — because by density, there is always a moment between the open boundary and the closed boundary, giving a smaller PTS.
This is the structural reason why G-TIAs in positive environments suffer information collapse: the G-TIA property is downward scalar with no minimum.
Proof: By density (DenselyOrdered), find m with pts_open.left < m <
runtime.start. The open interval]m, pts_open.right[ still contains
runtime (m < runtime.start and runtime.finish ≤ pts_open.right) and
is strictly contained in pts_open (pts_open.left < m).
The analogy between @cite{kennedy-2007}'s scale typology for gradable adjectives
and @cite{rouillard-2026}'s TIA licensing is structural — both use
Core.Scale.Boundedness to classify their scales, and
Boundedness.isLicensed derives the same licensing prediction from
the classification:
| Kennedy (Adjectives) | Rouillard (TIAs) |
|------------------------------------|--------------------------------------|
| Open scale (tall, expensive) | Atelic VP / DIV predicate |
| → context-dependent threshold | → information collapse, no max⊨ |
| → RGA: "??completely tall" | → E-TIA blocked: "*was sick in 3d" |
| | |
| Closed scale (full, empty) | Telic VP / QUA predicate |
| → scale-structure standard (max/min)| → upward scalar, max⊨ exists |
| → AGA: "completely full" ✓ | → E-TIA licensed: "wrote in 3d" ✓ |
| | |
| Interpretive Economy | Maximal Informativity Principle |
| → maximize conventional meaning | → maximize numeral contribution |
Kennedy's `Boundedness.open_` (= Krifka's DIV),
and Kennedy's `Boundedness.closed` (= Krifka's QUA).
For G-TIAs, the relevant "scale" is the PTS: open PTSs behave like
open scales (no inherent bound → no maximum standard), while closed
runtimes behave like closed scales (inherent bound → standard exists).
Vendler class determines scale boundedness via the Kennedy–Rouillard
isomorphism (Core.Scale), derived through a compositional chain:
VendlerClass →.telicity Telicity →.toMereoTag MereoTag →.toBoundedness Boundedness.
Telic VPs → QUA → closed/bounded (max⊨ exists).
Atelic VPs → CUM → open/unbounded (information collapse).
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Telic VPs map to closed/bounded scales (E-TIA licensed).
Atelic VPs map to open/unbounded scales (E-TIA blocked).
Telic VPs are licensed: telicity → closed boundedness → isLicensed.
Atelic VPs are blocked: atelicity → open boundedness → ¬isLicensed.
Revised PERFECT: the perfect's domain of quantification is restricted to open intervals (S ∩ O in Rouillard's notation). @cite{rouillard-2026} eq. (107): ⟦PERF⟧ := λI_it λt¹.∃t² ∈ S ∩ O[rb(t¹, t²) ∧ I(t²)]
This revision is the key to deriving G-TIA polarity sensitivity:
- In positive environments, the smallest open PTS to include a closed runtime cannot exist (by density), so no number is maximally informative.
- In negative environments, the largest open PTS to exclude a closed runtime always exists (the one abutting the runtime's boundary), so a maximally informative number exists.
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- Semantics.Montague.Sentence.MaximalInformativity.PERF_open p s = ∃ (pts : Core.Time.Interval Time), Semantics.Tense.Aspect.Core.RB pts s.time ∧ p s.world pts
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E-TIA expressions are upward monotone in the numeral (@cite{rouillard-2026} §3): "wrote a paper in three days" entails "wrote a paper in four days" because τ(e) ⊑ t with |t| = 3 implies τ(e) ⊑ t' with |t'| = 4.
This upward monotonicity is a property of the *expression* "in n days"
(due to the containment semantics of *in*), NOT of the numeral itself.
The numeral "three" can be exact (@cite{kennedy-2015} de-Fregean type-shift:
closed scale → Class B → exact meaning), and the expression is still
upward monotone because containment is monotone in the interval size.
Indeed, exact numerals are arguably REQUIRED: with LB numerals (|t| ≥ n),
∃t. |t| ≥ n ∧ τ(e) ⊑ t is trivially true for any event (pick a large
enough interval). Only exact numerals (|t| = n) make "in n days"
informative: it then asserts |τ(e)| ≤ n.
The exact reading of the DURATION ("took exactly 3 days, not 2") arises
via scalar implicature over the upward-monotone expression, parallel to
Kennedy's analysis of "the glass is full" (endpoint standard + exactness).
E-TIA expressions are upward monotone in the numeral: if the event fits in an n-unit time, it fits in an m-unit time for m ≥ n. This follows from the containment semantics of in (τ(e) ⊑ t), not from the numeral being lower-bounded. Compatible with @cite{kennedy-2015} exact numerals on closed scales.
Rouillard's closed subinterval property (CSUB, eq. 111) is the temporal analog of Krifka/Champollion's DIV (divisive reference). DIV says every part of a P-entity is itself P; CSUB says the closed counterpart of every subinterval of a P-event's runtime is itself a P-event's runtime.
The bridge: if an event predicate is DIV (in the mereological sense from
`Mereology.lean`), then it has the closed subinterval property (in the
temporal sense needed for G-TIA polarity).
DIV event predicates have the (plain) subinterval property. If P is downward-closed under the part-of relation on events, then any subinterval of a P-event's runtime that is itself some event's runtime is a P-event's runtime.