Augurzky et al. (2023): Putting plural definites into context #
A plural definite is homogeneous, every boy opened his presents being understood of all his presents and no boy opened his presents of none, and non-maximal: if all that matters is whether any present was opened, the first sentence passes though some stayed closed. The implicature approach strengthens an existential definite by an implicature, absent in downward-entailing scopes, so that pruning yields non-maximality in positive sentences only; the non-implicature approach gives the sentence a truth-value gap that the question the context raises resolves, for positive and negative sentences alike. Two picture-verification experiments put the definite under every and no (Experiment 1) or not every (Experiment 2) while a family rule made it relevant whether any or whether all presents were opened. Experiment 1 found the context affecting every more than no, and Experiment 2 found it affecting every and not every alike, so each approach is challenged by one experiment.
Main results #
pictures_truth: the TRUTH VALUE of each picture of Figures 1 and 4 is its sentence's supervaluation.nonImplicature_iff: Križ's usability relative to the context's question is truth at the context's resolution;not_addressesIssue_respected: the question as §3 states it does not serve.mem_reading_iff_designated_cell: the reading Bar-Lev's pruning selects is that resolution.nonImplicature_nonmaximal_iff,implicature_nonmaximal_iff,table1: Tables 1 and 2 on every mixed display.implicature_existential_iff,krizChemla_asymmetry_of_existential: in an existential context the approaches agree, which yields Križ and Chemla's asymmetry between every and no.enriched_nonmaximal_iff: the amendment of §4.2 accepts every and not every in their lax contexts and no in neither.exactlyTwo_positive,exactlyTwo_negative: the non-implicature prediction for the test (20).
Implementation notes #
A picture is a display of Križ and Chemla, one trivalent cell per boy. The question §3 states,
whether the family rule was respected, is lifted boy by boy (Context.issue). Table 1 is stated
for one boy's nine presents, the model of Bar-Lev's pruning being identified with the cell
(barePlural_holds_eq_cell). The verdicts are bivalent, so they register whether the context
changes a judgment and not by how much; the printed tests are reported in docstrings, not
re-thresholded. The text of §3.2.1 describes the true control of not every as its mixed picture.
TODO #
- Bar-Lev's cover-based route to non-maximality under negation (footnote 6) is not formalized.
- The test sentences (18) and (19) of §4.2 need disjunctive alternatives and are not modelled.
References #
Quantifiers and pictures #
The number tree each quantifier denotes.
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Instances For
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- AugurzkyEtAl2023.instDecidableRelNatTree AugurzkyEtAl2023.Quantifier.every = AugurzkyEtAl2023.instDecidableRelNatTree._aux_1
- AugurzkyEtAl2023.instDecidableRelNatTree AugurzkyEtAl2023.Quantifier.no = AugurzkyEtAl2023.instDecidableRelNatTree._aux_3
- AugurzkyEtAl2023.instDecidableRelNatTree AugurzkyEtAl2023.Quantifier.notEvery = AugurzkyEtAl2023.instDecidableRelNatTree._aux_5
no and not every are downward entailing; every is not.
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- AugurzkyEtAl2023.instDecidableScopeAntitoneTree AugurzkyEtAl2023.Quantifier.every = { decide := false, reflects_decide := AugurzkyEtAl2023.instDecidableScopeAntitoneTree._proof_1 }
- AugurzkyEtAl2023.instDecidableScopeAntitoneTree AugurzkyEtAl2023.Quantifier.no = { decide := true, reflects_decide := AugurzkyEtAl2023.instDecidableScopeAntitoneTree._proof_2 }
- AugurzkyEtAl2023.instDecidableScopeAntitoneTree AugurzkyEtAl2023.Quantifier.notEvery = { decide := true, reflects_decide := AugurzkyEtAl2023.instDecidableScopeAntitoneTree._proof_3 }
A picture is the display giving each boy he opened his presents over his nine.
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- p.display = List.map KrizChemla2015.cell p.opened
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The trivalent value a level of TRUTH VALUE names, a mixed picture being one on which the sentence's two resolutions disagree.
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The TRUTH VALUE of every picture of Figures 1 and 4 is the supervaluation of its sentence.
Contexts #
The resolution of a boy's partially opened presents that each context makes.
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The question the context raises about the boys, which of them opened any of their presents or which of them opened all.
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- ctx.issue = Setoid.ker (KrizChemla2015.resolve ctx.designation)
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Over one boy the issue is the partition of (12) or (13).
The non-implicature approach #
Under any question at least as fine as the polar question on the reading at δ and at most
as fine as the resolution at δ, the trivalent sentence is usable exactly where that reading
holds.
The non-implicature verdict is the usability of the trivalent sentence at the display relative to the question the context raises.
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- AugurzkyEtAl2023.nonImplicature q ctx d = Homogeneity.usable ctx.issue (KrizChemla2015.supervaluation q) d
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The non-implicature verdict is truth at the resolution the context makes.
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- AugurzkyEtAl2023.instDecidableNonImplicature q ctx d = decidable_of_iff (KrizChemla2015.reading q d ctx.designation) ⋯
Under the outer negation of a quantifier the verdict is negated in every context.
A mixed picture for a scope-monotone quantifier is accepted exactly in the existential context.
A mixed picture for a scope-antitone quantifier is accepted exactly in the universal context.
The family rule is respected where no present was opened, or where all were; the question whether it was is the question §3 states.
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- AugurzkyEtAl2023.Context.existential.respected = {d : KrizChemla2015.Display | ∀ v ∈ d, v = -1}
- AugurzkyEtAl2023.Context.universal.respected = {d : KrizChemla2015.Display | ∀ v ∈ d, v = 1}
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Under the question whether the rule was respected, every boy opened his presents does not address the existential issue, nor no boy opened his presents the universal one, so neither is usable anywhere and the question that yields Table 2 is the boy-by-boy one.
The implicature approach #
The implicature verdict. In a downward-entailing scope exhaustification is vacuous and the
definite keeps its existential literal meaning; elsewhere each boy's definite is exhaustified
over the alternatives the context leaves, to the threshold reading the context's question
selects, which is its resolution at the context's standard (mem_reading_iff_designated_cell).
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- AugurzkyEtAl2023.implicature q ctx d = if q.ScopeAntitone then KrizChemla2015.someReading q d else KrizChemla2015.reading q d ctx.designation
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- AugurzkyEtAl2023.instDecidableImplicature q ctx d = id inferInstance
Wherever implicatures arise the two approaches agree.
Under a downward-entailing quantifier the implicature verdict ignores the context.
In the existential context the two verdicts agree on every quantifier, both being the existential resolution, so an experiment that does not control the context cannot separate the approaches (§2).
The implicature approach rejects every mixed picture of a downward-entailing quantifier.
The implicature approach accepts a mixed picture of a scope-monotone quantifier exactly in the existential context.
Table 2 and the recoding of Figures 3 and 5 #
The context favouring a non-maximal reading, as Figures 3 and 5 recode CONTEXT, is the existential one for every and the universal one for the negative quantifiers.
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On the non-implicature approach every quantifier's mixed picture is accepted exactly in its lax context (Table 2), so the context affects every quantifier alike and no interaction of context and polarity is predicted. Experiment 2 found none (χ²(1) = 2.1, p = .15, §3.2.2), but Experiment 1 found one (χ²(1) = 11, p < .001, §3.1.2).
On the implicature approach only every's mixed picture is accepted, in its lax context (Table 2), so an interaction of context and polarity is predicted in both experiments. Experiment 1 found it (χ²(1) = 11, p < .001, §3.1.2), though with a context effect on no that the approach does not predict, and Experiment 2 did not (χ²(1) = 2.1, p = .15, §3.2.2).
Table 1 is Table 2 over one boy. In a mixed scenario the positive sentence is accepted exactly in the existential context on both approaches, and the negative one is rejected in both contexts on the implicature approach and accepted exactly in the universal one on the other.
Križ and Chemla's asymmetry under an accommodated existential context (§2) #
With an existential context accommodated throughout, the non-implicature approach accepts every GAP item of every and rejects every GAP item of no in Križ and Chemla's experiments, the asymmetry they found.
The boy-wise strengthening is the reading pruning selects #
The number of presents the context's question asks about is one, or all of them.
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The questions (12) and (13) in the model whose worlds are the sets of presents opened.
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The context's question selects, in the sense of pruning, the threshold reading at the number of presents it asks about.
A boy's nine presents, the objects of a cell.
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- AugurzkyEtAl2023.nine = Finset.range 9
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The resolution a context makes of a boy's cell is the threshold its question selects for his nine presents.
The reading pruning selects for a boy's presents under the context's question holds of the presents he opened exactly when the context's standard designates his cell.
A boy's trivalent value in the model whose worlds are the sets of presents opened is his cell at the number he opened.
The implicature approach amended (§4.2) #
The quantifier together with its own scalar implicature (16) strengthens not every by the denial of its stronger alternative no. The other quantifiers have no stronger alternative to deny.
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Instances For
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- AugurzkyEtAl2023.instDecidableRelNatEnriched AugurzkyEtAl2023.Quantifier.every = AugurzkyEtAl2023.instDecidableRelNatEnriched._aux_1
- AugurzkyEtAl2023.instDecidableRelNatEnriched AugurzkyEtAl2023.Quantifier.no = AugurzkyEtAl2023.instDecidableRelNatEnriched._aux_3
- AugurzkyEtAl2023.instDecidableRelNatEnriched AugurzkyEtAl2023.Quantifier.notEvery = fun (a b : ℕ) => { decide := decide (a ≠ 0) && decide (b ≠ 0), reflects_decide := ⋯ }
Together with its implicature not every is non-monotonic, the observation of §4.2.
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- AugurzkyEtAl2023.instDecidableScopeAntitoneEnriched AugurzkyEtAl2023.Quantifier.every = isFalse AugurzkyEtAl2023.instDecidableScopeAntitoneEnriched._proof_1
- AugurzkyEtAl2023.instDecidableScopeAntitoneEnriched AugurzkyEtAl2023.Quantifier.no = isTrue Quantifier.NumberTree.scopeAntitone_no
- AugurzkyEtAl2023.instDecidableScopeAntitoneEnriched AugurzkyEtAl2023.Quantifier.notEvery = isFalse AugurzkyEtAl2023.instDecidableScopeAntitoneEnriched._proof_2
On the mixed pictures of the experiments, where some boy opened all of his presents, the amended implicature approach accepts every and not every exactly in their lax contexts and no in neither. It thus predicts an interaction of context and polarity in Experiment 1 and none in Experiment 2, the outcome of both (§4.2).
The proposed test (§4.3) #
(20) in a mixed scenario for its positive part, where two boys opened some but not all of their presents and the others none, is accepted exactly in the existential context.
(20) in a mixed scenario for its negative part, where two boys opened all of their presents and some opened some but not all, is accepted exactly in the universal context.