Aloni and van Ormondt (2023): modified numerals and split disjunction #
Superlative modifiers (at least n, at most n) generate ignorance inferences
that comparative modifiers (more than n, fewer than n) do not, and the
inferences are obviated under universal quantifiers and modals, distribute
under universals, license free choice under modals, and vanish under negation
— exactly the profile of plain disjunction. Following Büring, superlative
modifiers are split disjunctions, at least n ↦ n ∨ more and
at most n ↦ n ∨ less, and the whole profile falls out of [Alo22]'s
neglect-zero enrichment [·]⁺ once BSML is raised to the first-order QBSML.
The QBSML facts of §5 are the universal theorems of
Logic/Team/QBSML/FreeChoice; here the denotations (14) and (16) are the
split of the ≥ and ≤ intervals of Degree.Comparison, the
results (56)–(61) and (63) are those facts for the paper's three/more
predicates in every model, under the paper's side conditions, and the
obviation claim (57) is the Fig. 14 countermodel. Its worlds are, as
throughout the paper, the sets of atomic facts true at them: w_{Pa} is "the
world w in which some object a has property P" (Example 3.1, p. 549).
The example rows record the inference profile the analysis answers to.
References #
Superlative modifiers as disjunctions #
At least n means exactly n or more than n, (14).
At most n means exactly n or fewer than n, (16).
Superlative rows carry the ignorance inference in unembedded position and comparative rows never do (the contrast of (2)–(7)).
The predicates #
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- AloniVanOrmondt2023.instDecidableEqPredicate x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- AloniVanOrmondt2023.instDecidableEqVar x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- AloniVanOrmondt2023.instReprVar = { reprPrec := AloniVanOrmondt2023.instReprVar.repr }
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threeOrMore c is three ∨ more (45b) said of the individual c.
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By Proposition 4.1, a state supports the NE-free ∀x(three(x) ∨ more(x)) iff its
first-order translation, computed by rfl, is true at every index.
The results (56)–(61) #
Ignorance (56), Fact 3, is [three ∨ more]⁺ ⊨ ◇three ∧ ◇more on a state-based
accessibility relation, the paper's epistemic reading.
At a state of maximal information, a single index, [∀x(three(x) ∨ more(x))]⁺
supports ∃x three(x) ∧ ∃x more(x), the distribution of (51) (Fact 5).
Distribution under partial information (58), Fact 6, yields the modalized conclusion
∃x◇three(x) ∧ ∃x◇more(x) on a state-based accessibility relation.
Free choice under necessity (59), Fact 7, is [□(three ∨ more)]⁺ ⊨ ◇three ∧ ◇more.
Free choice under possibility (60), Fact 8, is [◇(three ∨ more)]⁺ ⊨ ◇three ∧ ◇more.
Universal free choice (63), Fact 9, is [∀x◇(three(x) ∨ more(x))]⁺ ⊨ ∀x◇three(x) ∧ ∀x◇more(x), the inference [Che09b] attests.
Under negation the enrichment is inert, [¬(three ∨ more)]⁺ ⊨ ¬three ∧ ¬more ((61),
Fact 10), so the simpler fewer than three blocks (61a).
Obviation: the Fig. 14 countermodel #
A single index at the world w_{PaQb} with the empty assignment; that world alone
sees itself. The domain is the paper's two objects, and a world is the set of
atomic facts true at it.
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- AloniVanOrmondt2023.instDecidableEqEntity x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- AloniVanOrmondt2023.instReprEntity = { reprPrec := AloniVanOrmondt2023.instReprEntity.repr }
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In the Fig. 14 model only w_{PaQb} has an arrow, to itself.
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- AloniVanOrmondt2023.fig14Index = (AloniVanOrmondt2023.wPaQb, fun (x : AloniVanOrmondt2023.Var) => none)
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The accessibility is state-based on the Fig. 14 state, so obviation is not an artefact of dropping the frame condition behind ignorance.
As Fig. 15 shows, the universal extension splits into the x/a index supporting
[three(x)]⁺ and the x/b index supporting [more(x)]⁺.
As Fig. 16 shows, at the x/b index the only accessible world is w_{PaQb}, where
three holds of a alone, so ◇three(x) fails.
The universal quantifier obviates ignorance, [∀x(three(x) ∨ more(x))]⁺ ⊭ ∀x(◇three(x) ∧ ◇more(x)) ((57), Fact 4).