Scope of negation with respect to numeral interpretation.
Following Jespersen (cited in @cite{horn-1972}), negation can target:
- The lower bound (internal): "not ≥n" = "<n"
- The exact reading (external): "not =n" = "≠n"
- internal : NegationScope
Internal negation: targets lower bound. "doesn't have 3" = has <3
- external : NegationScope
External negation: targets exact reading. "doesn't have THREE" = has ≠3
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Compute the negated meaning under a given scope.
- Internal: ¬(≥n) = <n (negates lower bound)
- External: ¬(=n) = ≠n (negates exact reading)
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Internal negation of lower-bound: "not ≥n" = "<n"
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- Implicature.lowerBound_internal_neg w k = decide (k < w.toNat)
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External negation targets exact reading: "not =n" = "≠n"
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- Implicature.external_neg w k = (k != w.toNat)
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Lower-bound internal negation gives <n
"John doesn't have 3 children" (unstressed) → ¬(≥3) → <3 → compatible with 0, 1, 2
Lower-bound external negation gives ≠n
"John doesn't have THREE children" (stressed) → ¬(=3) → compatible with 0, 1, 2, 4, 5,...
The asymmetry is predicted by lower-bound semantics
Internal negation (default) differs from external negation (marked):
- Internal: worlds where ¬(≥n), i.e., <n
- External: worlds where ≠n
At world 4: internal gives false (4 ≥ 3), external gives true (4 ≠ 3)
Problem for Exact semantics: No internal/external distinction
If "three" literally means =3, then negation gives ≠3. There's no "weaker" lower-bound assertion to negate.
Both scopes collapse to the same meaning.
The key divergence: world 4
Lower-bound: internal at 4 = false, external at 4 = true (DIFFERENT) Exact: internal at 4 = true, external at 4 = true (SAME)
Empirically, "John doesn't have 3 children" (unstressed) suggests <3, not ≠3. This requires the internal/external distinction that only lower-bound provides.
The default (unmarked) reading of negated numerals.
Following Jespersen/Horn, the default is internal negation, which targets the lower bound.
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The marked reading requires prosodic emphasis.
"John doesn't have THREE children" (stress on THREE) targets the pragmatically enriched exact reading.
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Default interpretation is internal
"John doesn't have 3 children" (unmarked) → Internal negation → <3 (not ≥3)
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- Implicature.interpretNegatedNumeral _T _w stressed = if stressed = true then Implicature.NegationScope.external else Implicature.NegationScope.internal
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Count worlds compatible with negated numeral under given scope.
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- Implicature.compatibleUnderNegation T w scope = List.filter (Implicature.negatedMeaning T w scope) T.worlds
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Lower-bound internal: fewer worlds than external
"doesn't have 3" (internal): {0, 1, 2} "doesn't have THREE" (external): {0, 1, 2} (within standard worlds)
Filter extended worlds under negation for a given numeral.
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- Implicature.compatibleExtended w scope = List.filter (fun (k : ℕ) => Implicature.negatedMeaning Semantics.Lexical.Numeral.LowerBound w scope k) Implicature.extendedWorlds
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With extended worlds, the asymmetry is clear
Internal "not 3": {0, 1, 2} (worlds < 3) External "not THREE": {0, 1, 2, 4, 5} (worlds ≠ 3)
Summary: Negation Scope Supports Lower-Bound Semantics
| Reading | Form | Meaning | Theory Support |
|---|---|---|---|
| Internal | "doesn't have 3" | <3 | Lower-bound |
| External | "doesn't have THREE" | ≠3 | Both |
The existence of two distinct readings (internal vs external) is predicted by lower-bound semantics but not by exact semantics.
Empirical fact: Unmarked negation gives <n, not ≠n. This requires a lower bound to negate.