Spector 2013: Bare Numerals and Scalar Implicatures #
@cite{spector-2013} @cite{horn-1972} @cite{kennedy-2015} @cite{chierchia-fox-spector-2012} @cite{fox-2007} @cite{carston-1988} @cite{breheny-2008}
Bare numerals and scalar implicatures. Language and Linguistics Compass 7(5): 273–294.
Core Contribution #
@cite{spector-2013} evaluates four approaches to bare numeral interpretation:
- Neo-Gricean (@cite{horn-1972}): basic = ≥n, exact via scalar implicature
- Underspecification (@cite{carston-1988}): context selects ≥n, =n, or ≤n
- Exactly-only (@cite{breheny-2008}): basic = =n, other readings via context
- Ambiguity via EXH (@cite{chierchia-fox-spector-2012}): numerals have an "at least" base meaning; a covert exhaustivity operator EXH generates the "exactly" reading; both are grammatically available
The paper argues that approach 4 is necessary and sufficient to capture three generalizations about numeral interpretation (§5, (41a–c)):
- (a) "At least" readings available in all embedded environments
- (b) "Exactly" readings available in all syntactic environments
- (c) "At most" readings arise only in DE environments
Formalization #
- §1: The four approaches as an enum
- §2: The three generalizations as a checkable predicate
- §3: EXH bridge — proves
exhNumeralagrees with the generalExcluder.exhfrom @cite{fox-2007}'s innocent exclusion on numeral alternative sets - §4: Neo-Gricean failure in DE contexts + discourse coherence against exactly-only
- §5: Against underspecification (no genuine "at most" readings)
- §6: Ambiguity via EXH captures all three generalizations
- §7: Intermediate embedded implicatures distinguish EXH-ambiguity from lexical ambiguity
Integration #
- EXH bridge (§3) connects
Theories/Semantics/Lexical/Numeral/Semantics.lean'sexhNumeralto theExhaustification.innocentexcluder ExhaustivityLimit.leanproves RSA at α→∞ = EXH for ⟨some, all⟩- @cite{spector-2007} proves Max(P) = {Exhaust(P)} (Gricean ↔ exhaustive)
Inlined embedding combinators specialised to LowerBound (atLeastMeaning)
and Exact (bareMeaning) numeral semantics. These replace the polymorphic
versions previously housed in Theories/Semantics/Numerals/Embedding.lean,
which has been dissolved as part of the NumeralTheory cleanup.
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- Spector2013.notEx w n = ¬Semantics.Numerals.bareMeaning w.toNat n
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- Spector2013.possLB w acc = ∃ n ∈ acc, Semantics.Numerals.atLeastMeaning w.toNat n
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- Spector2013.possEx w acc = ∃ n ∈ acc, Semantics.Numerals.bareMeaning w.toNat n
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- Spector2013.necLB w acc = ∀ n ∈ acc, Semantics.Numerals.atLeastMeaning w.toNat n
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- Spector2013.necEx w acc = ∀ n ∈ acc, Semantics.Numerals.bareMeaning w.toNat n
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Lower-bound exhaustification: ≥w(n) ∧ ¬≥(succ w)(n).
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- Spector2013.exhLB w n = (Semantics.Numerals.atLeastMeaning w.toNat n ∧ match w.succ with | some w' => ¬Semantics.Numerals.atLeastMeaning w'.toNat n | none => True)
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- One or more equations did not get rendered due to their size.
Narrow-scope LB EXH under modal possibility: ◇(EXH(≥w)).
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- Spector2013.exhUnderPossLB w acc = ∃ n ∈ acc, Spector2013.exhLB w n
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- Spector2013.instDecidableExhUnderPossLB w acc = id inferInstance
Wide-scope LB EXH over modal possibility: EXH(◇(≥w)) —
◇(≥w) ∧ ¬◇(≥succ w).
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- Spector2013.exhOverPossLB w acc = (Spector2013.possLB w acc ∧ match w.succ with | some w' => ¬Spector2013.possLB w' acc | none => True)
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- One or more equations did not get rendered due to their size.
The four theoretical approaches to bare numeral interpretation evaluated in @cite{spector-2013} §1.
- neoGricean : Approach
Neo-Gricean (@cite{horn-1972}): basic = ≥n, exact via scalar implicature
- underspecification : Approach
Underspecification (@cite{carston-1988}): context selects ≥n, =n, or ≤n
- exactlyOnly : Approach
Exactly-only (@cite{breheny-2008}): basic = =n, other readings via context
- ambiguityEXH : Approach
Ambiguity via EXH (@cite{chierchia-fox-spector-2012}): base = ≥n, exact via covert exhaustivity operator; both readings available
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- Spector2013.instDecidableEqApproach x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- Spector2013.instReprApproach = { reprPrec := Spector2013.instReprApproach.repr }
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- One or more equations did not get rendered due to their size.
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Does the approach derive the exact reading via EXH / implicature?
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Does the approach claim "at most" is a genuine bare-numeral reading?
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- Spector2013.Approach.underspecification.claimsAtMost = true
- x✝.claimsAtMost = false
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@cite{spector-2013}'s three generalizations about numeral interpretation (41a–c). An adequate theory must satisfy all three.
- atLeastAvailable : Bool
(41a) "At least" readings available in all embedded environments.
- exactlyAvailable : Bool
(41b) "Exactly" readings available in all syntactic environments.
- atMostOnlyInDE : Bool
(41c) "At most" readings available only in DE environments.
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- g.allSatisfied = (g.atLeastAvailable && g.exactlyAvailable && g.atMostOnlyInDE)
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Neo-Gricean fails (41b): SIs are blocked/degraded in DE contexts, yet "exactly" readings persist there.
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- Spector2013.neoGriceanPredictions = { atLeastAvailable := true, exactlyAvailable := false, atMostOnlyInDE := true }
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Underspecification fails (41c): predicts "at most" should be freely available in all contexts, but it isn't (§3, example (30b)).
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- Spector2013.underspecPredictions = { atLeastAvailable := true, exactlyAvailable := true, atMostOnlyInDE := false }
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Exactly-only fails (41a): needs ad hoc mechanisms (implicit restriction, weakening) to derive "at least" readings (§4.2, examples (36)–(37)).
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- Spector2013.exactlyOnlyPredictions = { atLeastAvailable := false, exactlyAvailable := true, atMostOnlyInDE := true }
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Ambiguity via EXH satisfies all three: base = ≥n (always available), EXH derives =n (freely insertable), "at most" = =n + background.
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- Spector2013.ambiguityEXHPredictions = { atLeastAvailable := true, exactlyAvailable := true, atMostOnlyInDE := true }
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Only the ambiguity-via-EXH account satisfies all three generalizations.
Bridging numeral exhaustification to general innocent exclusion #
The numeral-specific exhNumeral (in Semantics.lean) hard-codes the scalar
alternatives {≥k} and checks only the immediate successor. The general
Excluder.exh from @cite{fox-2007} operates on arbitrary alternative sets
via innocent exclusion.
We prove these agree on the standard numeral domain Fin 4. This bridges
two previously disconnected parts of the library and validates that numerals
receive standard exhaustification — they are not a special case.
Standard four-world numeral domain. World i : Fin 4 represents
"exactly i objects".
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- Spector2013.NumW = Fin 4
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Prejacent for bare numeral m under LB: ≥m as a Bool predicate on NumW.
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- Spector2013.lbMeaning m w = decide (Semantics.Numerals.atLeastMeaning m ↑w)
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Numeral alternatives for bare numeral m under LB: {≥0, ≥1, ..., ≥(m+1)}.
Includes the prejacent and both weaker and stronger alternatives, as
a Finset of world supports.
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- Spector2013.lbAltsF m = Exhaustification.altsFromPreds (List.map (fun (k : ℕ) (w : Spector2013.NumW) => decide (Semantics.Numerals.atLeastMeaning k ↑w)) (List.range (m + 2)))
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Support of the prejacent ≥m.
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Innocent exclusion negates exactly the immediate successor ≥(m+1)
for each numeral. Equivalently: the exhaustified meaning is the
singleton {m}.
EXH bridge: The numeral-specific exhNumeral agrees with the
general innocent.exh on the four-world domain for all three bare numerals.
This proves numerals get standard @cite{fox-2007} exhaustification — they are not a special case requiring a bespoke operator.
The EXH bridge also holds for the local exhLB helper, which is
definitionally equivalent to the inlined LB exhaustification.
The conditional/tax problem (@cite{spector-2013} §2.2.2) #
"If you have three children, you do not qualify for tax exemptions."
Under neo-Gricean (base = ≥3), pragmatic strengthening can only narrow the literal meaning from ≥3 to =3. But the attested reading is "if 3 or fewer" (≤3), which is broader than ≥3 along a different dimension. The neo-Gricean approach has no mechanism to derive this.
EXH narrows ≥3 to =3. Neither ≥3 nor =3 entails ≤3. The "at most" reading requires background knowledge about monotonicity of the relevant scale (tax exemptions decrease with more children), not pragmatic strengthening.
The indirect scalar implicature problem (@cite{spector-2013} §2.2.2). "Peter didn't solve three problems" — the neo-Gricean approach predicts an indirect SI: "Peter solved exactly two." But this is not perceived. Demonstrated on the small domain {0,1,2,3} with numeral "three".
Discourse coherence against exactly-only (@cite{spector-2013} §4.2). "I have four chairs. In fact, I have five."
Under LB (≥4): the second sentence is consistent — 5 ≥ 4, so the speaker's first claim wasn't false. "In fact" cancels the implicature.
Under exactly-only (=4): the second sentence contradicts the first — 5 ≠ 4. The discourse should be infelicitous, but it isn't.
No genuine "at most" readings (@cite{spector-2013} §3) #
The decisive argument: if bare numerals could mean ≤n, then "One must be (at most) 40 to be eligible for the Fields medal" should be true. But it's necessarily false — there IS no maximum age for Fields eligibility; the constraint is a minimum (≤40 at time of award). The underspecification view wrongly predicts ≤40 is available.
The ≤n reading gives wrong truth conditions for minimum-threshold predicates. ≤40 makes ages 35, 30, ... eligible (wrong for Fields), while ≥40 correctly captures "at least 40" for voting thresholds.
(41a) "At least" = base meaning, always present. The base ≥n is true at n and above, and survives under all operators.
(41b) "Exactly" = EXH(base), available wherever EXH can scope. @cite{spector-2013} suggests that numerals may intrinsically activate their alternatives (§6.2), which would explain why EXH doesn't require prosodic marking for numerals (unlike "or" in DE contexts).
(41c) "At most" = =n + monotone background knowledge, only in DE. Under negation (DE): ¬(=3) is non-directional ({0,1,2,4,5,...}). Background monotonicity (e.g., tax exemptions decrease with children) restricts this to ≤3. In UE contexts, no such restriction applies, so the "at most" reading is unavailable.
EXH-ambiguity predicts more readings than lexical ambiguity #
(@cite{spector-2013} §6.2, examples (52)–(53))
Under lexical ambiguity, a numeral IS either ≥n or =n — no scope flexibility. Under EXH-ambiguity, EXH is an operator that can scope at different positions. For ◇(numeral), this yields three readings:
- ◇(≥n): use base meaning — "possible to do at-least-n"
- ◇(EXH(≥n)) = ◇(=n): EXH scopes under modal — "possible to do exactly n"
- EXH(◇(≥n)) = ◇(≥n) ∧ ¬◇(≥n+1): EXH scopes over modal — "possible ≥n but NOT possible ≥n+1"
Lexical ambiguity only produces readings 1 and 2. Reading 3 — the wide-scope EXH — is unique to the EXH-ambiguity account.
Three distinct readings for ◇(numeral) under EXH-ambiguity. With accessible worlds [2, 3]:
- ◇(≥2) = true (both worlds satisfy ≥2)
- ◇(EXH(≥2)) = ◇(=2) = true (world 2 satisfies =2)
- EXH(◇(≥2)) = ◇(≥2) ∧ ¬◇(≥3) = false (world 3 makes ◇(≥3) true)
Lexical ambiguity can only produce readings 1 and 2. Reading 3 distinguishes the two accounts. Under lexical ambiguity, ◇(=2) ≠ EXH(◇(≥2)) — the wide-scope EXH reading is not derivable from either lexical entry alone.
The intermediate reading (§6.2, example (53)): □(EXH(≥n)) — "required to do exactly n."
"Whenever the professor demanded [EXH(solve ≥3 problems)]" = "whenever demanded exactly 3 (not 4)"
At accessible worlds [3, 4]:
- □(≥3) = true — "required to solve at least 3" (too weak)
- □(=3) = false — "required exactly 3 in every world" (4 ≠ 3)
- □(EXH(≥3)) = false — "in every demand-world, exactly 3" (4 fails EXH)
At accessible worlds [3]:
- all three agree: true
Integration with the rest of linglib #
The results here connect to three independent lines of evidence in the library:
EXH bridge (§3):
exhNumeral=innocent.exhon the four-world numeral domain. This closes the gap betweenSemantics/Lexical/Numeral/Semantics.leanandSemantics/Exhaustification/Innocent.lean— numerals get standard @cite{fox-2007} exhaustification.RSA=EXH limit (
ExhaustivityLimit.lean):l1_weak_weakOnly_tendsto_oneproves RSA L1 at α→∞ recovers Fox's EXH for ⟨some, all⟩. Combined with the EXH bridge here, this means RSA at α→∞ on numerals should also recoverexhNumeral— the three formalisms (EXH,exhNumeral, RSA-limit) converge.Gricean foundation (@cite{spector-2007}):
max_eq_exhaustproves Max(P) = {Exhaust(P)} — Gricean reasoning derives exhaustive interpretation. @cite{spector-2013}'s EXH operator is the grammaticalized version of the same operation.
@cite{spector-2013}: the ambiguity-via-EXH account uniquely captures all three generalizations, and the EXH bridge validates that numeral exhaustification is an instance of general innocent exclusion.