West Coast Conference on Formal Linguistics (WCCFL) (38th : 2020)

A QUD-based theory of quantifier conjunction with but Zhong, Jing Crystal; Collins, James N. 2020-03-07

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A QUD-based theory of quantifier conjunctionwith butWCCFL 38, University of British ColumbiaJing Crystal Zhong and James N. Collins7 March 2020University of Hawai‘i at MānoaIntroductionNot just any two quantifiers can be conjoined by but in the subjectposition.(1) a. No syntactician but every phonologist attended theplenary talk.b. No syntactician but *no/??few phonologists attended theplenary talk.1IntroductionBarwise and Cooper [1] (hereaǒter B&C) make the followinggeneralization:to use but in this way, it seems necessary or at least prefer-able to mix increasing and decreasing quantifiers (p. 196)Monotone increasing quantifiers: every NP, many NP, at least two NP(2) JhulaK ⊑ JdanceK(3) Jevery(man)(hula)K ⊑ Jevery(man)(dance)KMonotone decreasing quantifiers: no NP, few NP, at most two NP(4) Jno(man)(dance)K ⊑ Jno(man)(hula)KNon-monotone quantifiers: exactly n NP, an even number of NP(5) Jexactly.2(man)(dance)K ̸|= Jexactly.2(man)(hula)K2IntroductionBarwise and Cooper [1] (hereaǒter B&C) make the followinggeneralization:to use but in this way, it seems necessary or at least prefer-able to mix increasing and decreasing quantifiers (p. 196)Monotone increasing quantifiers: every NP, many NP, at least two NP(2) JhulaK ⊑ JdanceK(3) Jevery(man)(hula)K ⊑ Jevery(man)(dance)KMonotone decreasing quantifiers: no NP, few NP, at most two NP(4) Jno(man)(dance)K ⊑ Jno(man)(hula)KNon-monotone quantifiers: exactly n NP, an even number of NP(5) Jexactly.2(man)(dance)K ̸|= Jexactly.2(man)(hula)K2IntroductionBarwise and Cooper [1] (hereaǒter B&C) make the followinggeneralization:to use but in this way, it seems necessary or at least prefer-able to mix increasing and decreasing quantifiers (p. 196)Monotone increasing quantifiers: every NP, many NP, at least two NP(2) JhulaK ⊑ JdanceK(3) Jevery(man)(hula)K ⊑ Jevery(man)(dance)KMonotone decreasing quantifiers: no NP, few NP, at most two NP(4) Jno(man)(dance)K ⊑ Jno(man)(hula)KNon-monotone quantifiers: exactly n NP, an even number of NP(5) Jexactly.2(man)(dance)K ̸|= Jexactly.2(man)(hula)K2IntroductionBarwise and Cooper [1] (hereaǒter B&C) make the followinggeneralization:to use but in this way, it seems necessary or at least prefer-able to mix increasing and decreasing quantifiers (p. 196)Monotone increasing quantifiers: every NP, many NP, at least two NP(2) JhulaK ⊑ JdanceK(3) Jevery(man)(hula)K ⊑ Jevery(man)(dance)KMonotone decreasing quantifiers: no NP, few NP, at most two NP(4) Jno(man)(dance)K ⊑ Jno(man)(hula)KNon-monotone quantifiers: exactly n NP, an even number of NP(5) Jexactly.2(man)(dance)K ̸|= Jexactly.2(man)(hula)K2MonotonicityB&C’s monotonicity account explains the judgments in (6):(6) a. No syntactician (⇓) but every phonologist (⇑) attendedthe keynote.b. No syntactician (⇓) but *no/??few phonologists (⇓)attended the keynote.Where the monotonicity of the quantifier DPs differ, the conjunctionis acceptable.3Problem 1However B&C’s mismatching-monotonicity condition is not necessary(matching monotonicity is judged as OK sometimes):(7) a. Many phoneticians (⇑) but every pragmaticist (⇑)attended the keynote. ≫≫b. ??Every pragmaticist (⇑) but many phoneticians (⇑)attended the keynote.(8) a. Few phoneticians (⇓) but no pragmaticist (⇓) attended thekeynote. ≫≫b. ??No pragmaticist (⇓) but few phoneticians (⇓) attendedthe keynote.Generalization 1: Matching monotonicity is OK for scale-matequantifiers, so long as the weaker quantifier precedes the strongerone. JfewK ⊒ JnoK, JmanyK ⊒ JeveryK4Problem 1However B&C’s mismatching-monotonicity condition is not necessary(matching monotonicity is judged as OK sometimes):(7) a. Many phoneticians (⇑) but every pragmaticist (⇑)attended the keynote. ≫≫b. ??Every pragmaticist (⇑) but many phoneticians (⇑)attended the keynote.(8) a. Few phoneticians (⇓) but no pragmaticist (⇓) attended thekeynote. ≫≫b. ??No pragmaticist (⇓) but few phoneticians (⇓) attendedthe keynote.Generalization 1: Matching monotonicity is OK for scale-matequantifiers, so long as the weaker quantifier precedes the strongerone. JfewK ⊒ JnoK, JmanyK ⊒ JeveryK4Problem 1However B&C’s mismatching-monotonicity condition is not necessary(matching monotonicity is judged as OK sometimes):(7) a. Many phoneticians (⇑) but every pragmaticist (⇑)attended the keynote. ≫≫b. ??Every pragmaticist (⇑) but many phoneticians (⇑)attended the keynote.(8) a. Few phoneticians (⇓) but no pragmaticist (⇓) attended thekeynote. ≫≫b. ??No pragmaticist (⇓) but few phoneticians (⇓) attendedthe keynote.Generalization 1: Matching monotonicity is OK for scale-matequantifiers, so long as the weaker quantifier precedes the strongerone. JfewK ⊒ JnoK, JmanyK ⊒ JeveryK 4Problem 2B&C’s mismatching-condition, is not sufficient (mismatchingmonotonicity is judged as not-OK sometimes):(9) a. At least two thirds of Democrats (⇑) but fewer than half ofRepublicans (⇓) voted for the bill. ≫≫b. ??/∗At least a third of Democrats (⇑) but fewer than half ofRepublicans (⇓) voted for the bill.Generalization 2: Differing monotonicity is not OK if the quantifiersoverlap in reference.5Monotonicity vs. overlapfewer than halfat least two thirds ofat least a third of01Fig. 1. Overlapping and non-overlapping determiners6Monotonicity vs. overlap(10) a. At least 2/3 of Democrats (⇑) but fewer than half ofRepublicans (⇓) voted for the bill. ≫≫b. ??/∗At least 1/3 of Democrats (⇑) but fewer than half ofRepublicans (⇓) voted for the bill.Generalization 2: Differing monotonicity is not OK if the quantifiersoverlap in reference.—‘at least 2/3 of’ and ‘fewer than half’ don’t overlap, sobut-conjunction is licensed.—‘at least 1/3 of’ and ‘fewer than half’ overlap on a scale ofproportions, so but-conjunction is degraded.7Monotonicity vs. overlap(10) a. At least 2/3 of Democrats (⇑) but fewer than half ofRepublicans (⇓) voted for the bill. ≫≫b. ??/∗At least 1/3 of Democrats (⇑) but fewer than half ofRepublicans (⇓) voted for the bill.Generalization 2: Differing monotonicity is not OK if the quantifiersoverlap in reference.—‘at least 2/3 of’ and ‘fewer than half’ don’t overlap, sobut-conjunction is licensed.—‘at least 1/3 of’ and ‘fewer than half’ overlap on a scale ofproportions, so but-conjunction is degraded.7The empirical pictureTo summarize,(11) Generalization 1: Matching monotonicity is OK for scale-matequantifiers, so long as the weaker quantifier precedes thestronger one.(12) Generalization 2: Differing monotonicity is not OK if thequantifiers overlap in reference.Relevant factors— Ordering of determiners— Different vs. same monotonicity— Overlapping vs. non-overlapping reference8The empirical pictureTo summarize,(11) Generalization 1: Matching monotonicity is OK for scale-matequantifiers, so long as the weaker quantifier precedes thestronger one.(12) Generalization 2: Differing monotonicity is not OK if thequantifiers overlap in reference.Relevant factors— Ordering of determiners— Different vs. same monotonicity— Overlapping vs. non-overlapping reference8Experiment 1: OrderingIs there an effect from the order of determiners?— Order matters for scale-mate quantifiers w/ matchingmonotonicity. Otherwise, order doesn’t matter.(13) a. Many girls (⇑) but every boy (⇑) skipped class.b. ??Every girl but many boys skipped class.(14) a. Every girl (⇑) but no boy (⇓) skipped class.b. No girl but every boy skipped class.- 2× 2 factorial design crossing Same/DiffMono & Order- 4 conditions, 18 critical items, Latin square design- equal number of fillers- 4 point Likert scale judgment task- 24 English native speaker participants9Experiment 1: OrderingIs there an effect from the order of determiners?— Order matters for scale-mate quantifiers w/ matchingmonotonicity. Otherwise, order doesn’t matter.(13) a. Many girls (⇑) but every boy (⇑) skipped class.b. ??Every girl but many boys skipped class.(14) a. Every girl (⇑) but no boy (⇓) skipped class.b. No girl but every boy skipped class.- 2× 2 factorial design crossing Same/DiffMono & Order- 4 conditions, 18 critical items, Latin square design- equal number of fillers- 4 point Likert scale judgment task- 24 English native speaker participants9ResultsFig. 2. Results of experiment 1. Error bars represent standard error.10Experiment 2: Overlap vs. Same/diff. monotonicityTable 1. Experimental stimuliSameMono? Overlap? ExampleYes Yes exactly two X but an even number of YYes No exactly two X but an odd number of YNo Yes at least 1/3 of X but fewer than half of YNo No at least 2/3 of X but fewer than half of Y- 2× 2 factorial design crossing SameMono & Overlap- 4 conditions, 16 critical items (k = 4), Latin-square design- 16 fillers (8 grammatical, 8 ungrammatical)- 4 point Likert scale judgment task- 21 English native speaker participants11Experiment 2: Overlap vs. Same/diff. monotonicityTable 1. Experimental stimuliSameMono? Overlap? ExampleYes Yes exactly two X but an even number of YYes No exactly two X but an odd number of YNo Yes at least 1/3 of X but fewer than half of YNo No at least 2/3 of X but fewer than half of Y- 2× 2 factorial design crossing SameMono & Overlap- 4 conditions, 16 critical items (k = 4), Latin-square design- 16 fillers (8 grammatical, 8 ungrammatical)- 4 point Likert scale judgment task- 21 English native speaker participants11ResultsFig. 3. Results of experiment 2. Error bars represent standard error.12Our experimental results suggest:A. Conjoining scale-mate determiners (w/ matching monotonicity)is better when weaker Det precedes stronger Det.— many X but every Y ≫≫ every X but many YB. Conjoining non-monotone Qs is better if Dets don’t overlap.— exactly 2 X but an odd no. of Y ≫≫ exactly 2 X but an even no. of YC. Conjoining Qs w/ mis-matched monotonicity is better if Detsdon’t overlap.— at least 2/3 of X but fewer than half of Y ≫≫ at least 1/3 of X butfewer than half of Y13DisjointnessOur generalization focuses on the semantic properties of thedeterminers sans NP-description.— Determiners are analyzed as 2-place relations over properties.Revised generalization:Det1 X but Det2 Y is acceptable only if JDet1K ∩ JDet2K = ∅(15) for example, why is no X but every Y acceptable?a. JnoK = {⟨P,Q⟩ : P ̸= ∅, P ∩ Q = ∅}b. JeveryK = {⟨P,Q⟩ : P ̸= ∅, P ⊆ Q}c. therefore, JeveryK ∩ JnoK = ∅14DisjointnessOur generalization focuses on the semantic properties of thedeterminers sans NP-description.— Determiners are analyzed as 2-place relations over properties.Revised generalization:Det1 X but Det2 Y is acceptable only if JDet1K ∩ JDet2K = ∅(15) for example, why is no X but every Y acceptable?a. JnoK = {⟨P,Q⟩ : P ̸= ∅, P ∩ Q = ∅}b. JeveryK = {⟨P,Q⟩ : P ̸= ∅, P ⊆ Q}c. therefore, JeveryK ∩ JnoK = ∅14DisjointnessOur generalization focuses on the semantic properties of thedeterminers sans NP-description.— Determiners are analyzed as 2-place relations over properties.Revised generalization:Det1 X but Det2 Y is acceptable only if JDet1K ∩ JDet2K = ∅(15) for example, why is no X but every Y acceptable?a. JnoK = {⟨P,Q⟩ : P ̸= ∅, P ∩ Q = ∅}b. JeveryK = {⟨P,Q⟩ : P ̸= ∅, P ⊆ Q}c. therefore, JeveryK ∩ JnoK = ∅14DisjointnessWhy is determiner-disjointness relevant to but?(16) Toosarvandani [5] on but:Felicity condition on [SL but SR]: there is a QUD Q, such thata. For some sub-question of Q, {σ,¬σ}, JSLK |= σ.b. For some sub-question of Q, {τ,¬τ}, JSRK |= ¬τ .— The two conjuncts must resolve sub-questions (see Büring[2], Rojas-Esponda [4]) of the current QUD, but with opposite polarity.(17) What kinds of cakes do you sell?Do you sell chocolate cake?Do you sell carrot cake?(18) We sell carrot cake but we ??(don’t) sell chocolate cake.15DisjointnessWhy is determiner-disjointness relevant to but?(16) Toosarvandani [5] on but:Felicity condition on [SL but SR]: there is a QUD Q, such thata. For some sub-question of Q, {σ,¬σ}, JSLK |= σ.b. For some sub-question of Q, {τ,¬τ}, JSRK |= ¬τ .— The two conjuncts must resolve sub-questions (see Büring[2], Rojas-Esponda [4]) of the current QUD, but with opposite polarity.(17) What kinds of cakes do you sell?Do you sell chocolate cake?Do you sell carrot cake?(18) We sell carrot cake but we ??(don’t) sell chocolate cake.15Disjointnessbut’s function: to conjoin two partial resolutions of the current QUDwith opposing polarity.We assume the QUD is shaped by the intonation structure of thebut-conjunction.(19) everyF cát but noF dòg skateboarded.The contrasting determiners and contrasting descriptions ensure theQUD contains the following polar questions:(20)Did every cat skateboard?Did no cat skateboard?Did every dog skateboard?Did no dog skateboard? ⊑ current QUD16Disjointnessbut’s function: to conjoin two partial resolutions of the current QUDwith opposing polarity.We assume the QUD is shaped by the intonation structure of thebut-conjunction.(19) everyF cát but noF dòg skateboarded.The contrasting determiners and contrasting descriptions ensure theQUD contains the following polar questions:(20)Did every cat skateboard?Did no cat skateboard?Did every dog skateboard?Did no dog skateboard? ⊑ current QUD16Disjointness(21) everyF cát but noF dòg skateboarded.Example (21) signals the current QUD is structured at least partiallyas below:— The two conjuncts resolve two sub-questions with oppositepolarity answers, as required by but.(22) How many of which types skateboarded?...Every dog?No dog?Every cat?No cat?[every cat (skates)] but [no dog skates]denies affirms17DisjointnessWhy is there a disjointness condition on Dets conjoined by but?(23) Theorem: any pair of Dets with disjoint reference will satisfythe felicity condition of butProofLet Dα and Dβ be disjoint determinersa. For any X, Y, Dα(X)(Y) |= ¬Dβ(X)(Y) and Dβ(X)(Y) |= ¬Dα(X)(Y)b. ∴ for any A,B, C, Dα(A)(C) affirms Dα(A)(C)?and Dβ(B)(C) denies Dα(B)(C)?c. ∴ for any Q such that Dα(A)(C)?,Dα(B)(C)? ⪯ Q,“Dβ(A)(C) but Dα(B)(C)” is definedbut-conjoining two semantically disjoint determiners ensures thatthe current QUD is resolved according to but’s felicity condition.18DisjointnessWhat goes wrong with non-disjoint determiners(24) #exactly two cats but an even number of dogs skateboardedExample (24) doesn’t ensure that the QUD is resolved with opposingpolarity.(25) How many of which types skateboarded?...Even # of dog?2! dog?Even # of cat?2! cat?[exactly 2 cats (skate)] but [even no. of dogs skate]*doesn’t deny any Q affirmsThe felicity condition of but fails!— It is false that one conjunct affirms a sub-question, while theother denies a sub-question. 19Ordering effectsWhen conjoining scale-mate determiners, participants preferred“weak before strong”— “many X but every Y” judged better than “every X but many Y”(26) Our working hypothesis:a. uttered weak scalar items are pragmaticallystrengthened: many⇝ many-&-not-all, andb. the left conjunct must deny a sub-question,while the right must affirm a sub-question.(27) a. many-&-not-all(X)(Y) negatively resolves Q: every(X)(Y)?b. every(X)(Y) affirmatively resolves Q’: many(X)(Y)?This hypothesis assumes weak determiners are strengthened in theutterance, but not within the QUD.• See Chierchia [3] for the absence of strengthening ininterrogative contexts.20Ordering effectsWhen conjoining scale-mate determiners, participants preferred“weak before strong”— “many X but every Y” judged better than “every X but many Y”(26) Our working hypothesis:a. uttered weak scalar items are pragmaticallystrengthened: many⇝ many-&-not-all, andb. the left conjunct must deny a sub-question,while the right must affirm a sub-question.(27) a. many-&-not-all(X)(Y) negatively resolves Q: every(X)(Y)?b. every(X)(Y) affirmatively resolves Q’: many(X)(Y)?This hypothesis assumes weak determiners are strengthened in theutterance, but not within the QUD.• See Chierchia [3] for the absence of strengthening ininterrogative contexts.20Ordering effectsWhen conjoining scale-mate determiners, participants preferred“weak before strong”— “many X but every Y” judged better than “every X but many Y”(26) Our working hypothesis:a. uttered weak scalar items are pragmaticallystrengthened: many⇝ many-&-not-all, andb. the left conjunct must deny a sub-question,while the right must affirm a sub-question.(27) a. many-&-not-all(X)(Y) negatively resolves Q: every(X)(Y)?b. every(X)(Y) affirmatively resolves Q’: many(X)(Y)?This hypothesis assumes weak determiners are strengthened in theutterance, but not within the QUD.• See Chierchia [3] for the absence of strengthening ininterrogative contexts. 20ConclusionThe function of but:— signals the structure of the discourse→ “how do we (partially)resolve the current QUD?”— signals that its conjuncts (partially) resolve the current QUD withopposite polarities.(28) Current QUD?...Sub-Q4?Sub-Q3?Sub-Q2?Sub-Q1?[first conjunct] but [second conjunct]denies affirms— Determiner disjointness yields better empirical coverage thanB&C’s monotonicity-based theory of but-conjunction.21AcknowledgementsWe would like to thank Dylan Bumford, Thomas Kettig, Fred Zenker,the audience at the NINJAL-UHM Linguistics Workshop, and allexperimental participants.22References[1] Jon Barwise and Robin Cooper. Generalized quantifiers andnatural language. Linguistics and Philosophy, 4(2):159–219, 1981.[2] Daniel Büring. On D-trees, beans, and B-accents. Linguistics andPhilosophy, 26(5):511–545, 2003.[3] Gennaro Chierchia. Scalar implicatures, polarity phenomena, andthe syntax/pragmatics interface. In Adriana Belletti, editor,Structures and Beyond, Vol. 3. Oxford University Press, 2004.[4] Tania Rojas-Esponda. A discourse model for überhaupt.Semantics and Pragmatics, 7(1):1–45, 2013.[5] Maziar Toosarvandani. Contrast and the structure of discourse.Semantics and Pragmatics, 7(4), 2014.23Judgment task Likert scale24

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