list readings of conjoined singular...

list readings of conjoined singular ‘which’-phrases

Andreea C. Nicolae1 Patrick D. Elliott2 Yasutada Sudo2

NELS 46 at Concordia UniversityOctober 18, 20151Zentrum für Allgemeine Sprachwissenschaft2University College London

roadmap

1. The puzzle:Conjoined singular ‘which’-phrases have list readings withcollective predicates but not with distributive predicates.

2. Theoretical Background:∙ Intersective semantics for conjunction, Distributivity andCollectivity (Winter 2001, Champollion 2015)

∙ ANS-operator and uniqueness presupposition (Dayal 1996)

3. Our analysis

4. Conclusions

1

the puzzle

key observation

Conjoined singular ‘which’-phrases = ‘which NP and which NP’.(See appendix for ‘which NP and NP’)

They typically invite single-tuple answers.

(1) Which boy and which girl sneezed?a. #John and Mary sneezed, Fred and Sue sneezed, and Ed and

Laura sneezed.b. John and Mary sneezed.

Observation

Conjoined singular ‘which’-phrases can receive a list answer withcollective predicates like ‘live together’.

3

key observation (cont.)

(2) Which boy and which girl live together?a. % John and Mary live together, Fred and Sue live together, and

Ed and Laura live together.b. John and Mary live together.

Since we have found the judgements at issue to be fairly delicate, weran an experiment (see Appendix).

∙ (2a) is acceptable but its felicity seems to be subject tointer-speaker variation.

∙ (2b) is a felicitous answer to (2) for everyone.

We will develop an account of why distributive and collectivepredicates differ (and how speakers differ).

4

proposal in a nutshell

We propose that the list readings of conjoined singular‘which’-phrases with collective predicates are due to plurality,similarly to (3).

(3) Which boys left the party?— John, Bill and Fred left the party.

(See Appendix for why they are different from ‘pair-list answers’).

∙ Winter’s (2001) theory of intersective conjunction, distributivity andcollectivity (see also Champollion 2015).

∙ Dayal’s (1996) ANS-operator to explain single-tuple vs. list answers.∙ Contraints on the distributivity operator D (cf. De Vries 2015)

5

theoretical background

non-intersective readings

Conjoined nominals give rise to distributive and collective readings.

(4) John and MaryA boy and a girl

} {live together. (Collective)sneezed. (Distributive)

Winter (2001) derives both readings with a single meaning for ‘and’(also Champollion 2015).

(5) Intersective semantics for ‘and’JandKw = λPτ t.λQτ t.λxτ . x ∈ P ∧ x ∈ Q

Collective readings are derived with two operators, MIN and ER.

(We won’t distinguish sets and their characteristic functions)

7

conjoined proper names

Let’s first see how the distributive and collective readings of ‘Johnand Mary’ are derived.

Proper names denote Montagovian individuals of type ⟨et, t⟩.

(6) JJohnKw = λPet. j ∈ P JMaryKw = λPet. m ∈ P

(7) JJohn and MaryKw = JandKw (JJohnKw)(JMaryKw)= λPet. j ∈ P ∧m ∈ P= {Pet | j ∈ P ∧m ∈ P }

The distributive reading of ‘John and Mary’ is straightforwardlyderived.

(8) JJohn and Mary sneezedKw = j ∈ SNEEZEDw ∧m ∈ SNEEZEDw

8

min

In order to derive the collective interpretation an operator MIN(‘Minimization’) is necessary.

(9) JMINKw = λQ⟨et,t⟩.λPet.P ∈ Q ∧ ∀P′[P′ ⊂ P → P′ /∈ Q]

MIN applies to a quantifier (a set of sets) Q and gives back thesmallest predicates in Q (minimal subsets of Q).

(10) JMINKw (JJohn and MaryKw)= JMINKw ({Pet | j ∈ P ∧m ∈ P })= JMINKw ({ { j,m } , { j,m, a } , { j,m,b } , { j,m, a,b } . . . })= { { j,m } }

{ j,m } represents the plurality consisting of John and Mary.

9

er

Collective predicates are true of pluralities like { j,m }.

(11) Jlive togeherKw = λXet. X ∈ LIVE.TOGETHERw

Since JMINKw (JJohn and MaryKw) = { { j,m } } is a type-⟨et, t⟩quantifier, it cannot combine directly with a collective predicate.

To solve the type-mismatch, ER (‘Existential Raising’) is used.

(12) JERKw = λPτ t.λQτ t. ∃x[x ∈ P ∧ x ∈ Q]

(13) JERKw (JMINKw (JJohn and MaryKw))= JERKw ({ { j,m } })= λP⟨et,t⟩. ∃X[X ∈ { { j,m } } ∧ X ∈ P]= λP⟨et,t⟩. { j,m } ∈ P

(14) JER MIN John and Mary live togetherKw= { j,m } ∈ LIVE.TOGETHERw

10

conjoined quantifiers: distributive

The same mechanism accounts for the distributive and collectivereadings of conjoined quantifiers like ‘a boy and a girl’.

(15) a. Ja boyKw = λQet. ∃x[x ∈ BOYw ∧ x ∈ Q]b. Ja girlKw = λQet. ∃y[y ∈ GIRLw ∧ y ∈ Q]

By intersecting these with ‘and’, we get:

(16) Ja boy and a girlKw= λQet. ∃x∃y[x ∈ BOYw ∧ y ∈ GIRLw ∧ x, y ∈ Q]

This accounts for a distributive interpretation.

(17) JA boy and a girl sneezedKw= ∃x∃y[x ∈ BOYw ∧ y ∈ GIRLw ∧ x, y ∈ SNEEZEDw]

11

conjoined quantifiers: collective

Using MIN and ER, we can derive the collective interpretation.

(18) JMINKw (Ja boy and a girlKw)= MIN({Qet | ∃x∃y[x ∈ BOYw ∧ y ∈ GIRLw ∧ x, y ∈ Q] })= { { x, y } | x ∈ BOYw ∧ y ∈ GIRLw }

This is a set of sets containing a boy and a girl and nothing else.Unlike for ‘John and Mary’, there are multiple such sets.

Using ER, we derive the correct interpretation.

(19) JERKw (JMINKw (Ja boy and a girlKw))= JERKw ({ { x, y } | x ∈ BOYw ∧ y ∈ GIRLw })= λP⟨et,t⟩. ∃X[X ∈ { { x, y } | x ∈ BOYw ∧ y ∈ GIRLw } ∧ X ∈ P]

(20) JER MIN a boy and a girl live togetherKw= ∃X

[X ∈ { { x, y } | x ∈ BOYw ∧ y ∈ GIRLw }∧X ∈ LIVE.TOGETHERw

]12

examples

Distributivet

⟨et, t⟩

⟨et, t⟩

a boy∩and

⟨et, t⟩

a girl

et

sneezed

Collectivet

⟨⟨et, t⟩, t⟩

ER ⟨et, t⟩

MIN ⟨et, t⟩

⟨et, t⟩

a boy∩and

⟨et, t⟩

a girl

⟨et, t⟩

live together

13

summary: intersective semantics

Both distributive and collective readings are derived from theintersective semantics of ‘and’.

(21) JandKw = λPτ t.λQτ t.λxτ . x ∈ P ∧ x ∈ Q

The collective interpretation requires extracting a minimal set (=plural individual) { x, y } in the extension of the quantifier.

(22) JMINKw = λQ⟨et,t⟩.λPet.P ∈ Q ∧ ∀P′[P′ ⊂ P → P′ /∈ Q]

(23) JERKw = λPτ t.λQτ t. ∃x[x ∈ P ∧ x ∈ Q]

Both Winter and Champollion use choice-functions instead of ER, but thiscomplication is unnecessary here.

See Champollion (2015) for an extension of this account for ‘Q NPs and NPs’.

14

plural nouns

We follow Winter and assume that plural nouns denote sets of setsof individuals; they can be derived from the meaning of the singularnoun via the application of D:

(24) JboysKw = D(JboyKw)The semantics of D is as in (25):

(25) JDKw = λQτ t.λPτ t. P ̸= ∅ ∧ P ⊆ Q

D applies to a set and returns the power set of that set, minus ∅ (D= ℘+). Note that it raises the type from σt to ⟨σt, t⟩.

(26) JDKw ({b1,b2,b3}) =

{b1,b2,b3 } ,

{b1,b2 } , {b2,b3 } , {b1,b2 } ,{b1 } , {b2 } , {b3 }

15

plurals example

JD boyKw is of type ⟨et, t⟩ (a set of sets).

‘Some’ is a cross-categorial existential quantifier (type ⟨τ t, t⟩).

‘Some boys’ combines directly with a collective predicate (type⟨et, t⟩).

To combine with a distributive predicate of type ⟨e, t⟩, anotherinstance of D is necessary.

Singular: Some boy sneezedt

⟨et, t⟩

some et

boy

et

sneezed

Plural: Some boys sneezedt

⟨⟨et, t⟩, t⟩

some ⟨et, t⟩

D boy

⟨et, t⟩

D sneezed

∃x[x ∈ JboyKw ∧ x ∈ JsneezedKw] ∃X[X ∈ D(JboyKw)∧X ∈ D(JsneezedKw)]16

question semantics

Questions denote sets of propositions.

‘Which’-phrases are existential quantifiers.

(27) Jwhich boyKw = λQet.∃x[x ∈ BOYw ∧ x ∈ Q] (= Ja boyKw)‘Wh’-phrases obligatorily take scope over the question operator (forsyntactic reasons). p gets bound at the top-most node.

(28) J?pKw = λqst. p = q

λpst

which boy λx?p

tx sneezed

{p | ∃x[x ∈ BOYw ∧ p = λw′. x ∈ SNEEZEDw′ ] }

17

ans-operator

We use Dayal’s (1996) ANS-operator to derive the uniquenesspresupposition of singular ‘which’-phrases.

(29) JANSKw = λQ⟨st,t⟩ : ∃!p[p ∈ maxinfw(Q)]. ιp[p ∈ maxinfw(Q)]

(30) maxinfw(Q) = {p ∈ Q | w ∈ p ∧ ∀p′ ∈ Q[w ∈ p′ → p ⊆ p′] }

ANS applies to a question denotation Q and denotes the maximallyinformative true answer to Q.

(ANS is essentially a definite determiner)

The uniqueness presupposition of singular ‘which’-phrases comesfrom the presupposition that there is a unique maximallyinformative true answer.

18

examples: singular

(31) Which boy sneezed?

(32) w1: John sneezed, and Bill and Martin didn’t sneeze.

JANSKw1

λw′. j ∈ SNEEZEDw′ ,

λw′. b ∈ SNEEZEDw′ ,

λw′. m ∈ SNEEZEDw′

= λw′. j ∈ SNEEZEDw′

(33) w2: John and Bill sneezed, Martin didn’t sneeze.

JANSKw2



λw′. m ∈ SNEEZEDw′

= undefined

In (33) λw′. j ∈ SNEEZEDw′ and λw′. b ∈ SNEEZEDw′ are equallyinformative.

19

examples: plural

If the noun is plural and the predicate has D, there is always aunique maximally true answer.

(34) Which boys D sneezed?

(35) w2: John and Bill sneezed, Martin didn’t sneeze.

JANSKw2



λw′. m ∈ SNEEZEDw′ ,

λw′. j,b ∈ SNEEZEDw′ ,

λw′. j,m ∈ SNEEZEDw′ ,

. . .

= λw′. j,b ∈ SNEEZEDw′

20

our analysis

our analysis

Combining Winter’s theory of conjunction and collectivity andDayal’s ANS-operator, we are now in a position to account for ourobservation.

Observation

Conjoined singular ‘which’-phrases can receive a list answer withcollective predicates like ‘live together’, but not with distributivepredicates like ‘sneezed’.

We will proceed as follows:

1. Single-tuple reading with ‘live together’2. List reading with ‘live together’3. Single-tuple reading with ‘sneezed’4. *List reading with ‘sneezed’

22

single-tuple reading with a collective predicate

The single-tuple reading with ‘live together’ is derived by theoperators ER and MIN, just as in the case of ‘a boy and a girl’.

Assume BOYw = {b1,b2 } and GIRLw = { g1, g2 }.

(36) JER MIN which boy and which girl live together?Kw=

λw′. {b1, g1 } ∈ LIVE.TOGETHERw′ ,



λw′. {b2, g1 } ∈ LIVE.TOGETHERw′

Applying the ANS-operator to this set derives the uniquenesspresupposition and delivers the single-tuple answer.

(37) w1: b1 and g1 live together, and nobody else lives together.JANSKw1 ((36)) = λw′. {b1, g1 } ∈ LIVE.TOGETHERw′

23

single-tuple reading with a collective predicate

(38) Which boy and which girl live together?

⟨st, t⟩

λpst t

⟨⟨et, t⟩, t⟩

ER ⟨et, t⟩

MIN ⟨et, t⟩

⟨et, t⟩

which boy∩and

⟨et, t⟩

which girl

⟨et, t⟩

λX⟨e,t⟩ t

?p t

tX live together

24

list reading with a collective predicate

Proposal: The list reading with ‘live together’ can be derived via theinsertion of D, one at the level of the ‘which’-phrase and another atthe level of the collective predicate.

(39)q[ER D MIN which boy and which girl] [D live together]?

yw

This is no different from what we assumed for a plural‘which’-question like ‘which boys sneezed?’

(40)q[which D boy] [D sneezed]?

yw

25



⟨st, t⟩

λpst t

⟨⟨⟨et, t⟩, t⟩, t⟩

ER ⟨⟨et, t⟩, t⟩

D ⟨et, t⟩

MIN ⟨et, t⟩

⟨et, t⟩

which boy∩and

⟨et, t⟩

which girl

⟨⟨et, t⟩, t⟩

λX⟨et,t⟩ t

?p t

tX ⟨⟨et, t⟩, t⟩

D live together

26


Recall that JMINKw (Jwhich boy and which girlKw) delivers a set of allpossible boy-girl pairs.

(42) { {b1, g1 } , {b2, g1 } , {b1, g2 } , {b2, g2 } }

Applying D to this set delivers the set consisting of all possiblesubsets of JMINKw (Jwhich boy and which girlKw)

{{b1, g1 } , {b2, g1 } , {b1, g2 } , {b2, g2 }

},{

{b1, g1 } , {b2, g1 } , {b1, g2 }},

{{b1, g1 } , {b2, g1 } , {b2, g2 }

}, . . .{

{b1, g1 } , {b1, g2 }},

{{b1, g1 } , {b2, g1 }

}, . . .

{ {b1, g1 } } , { {b2, g1 } } , { {b1, g2 } } , { {b2, g2 } }

Similarly, applying D to the collective predicate ‘live together’delivers the power set of all possible cohabitants.

27


The question denotation contains propositions that make referenceto plural individuals.

(43) JER D MIN which boy and which girl D live together?Kw

=





λw′. {b1, g1 } , {b2, g2 } ∈ LIVE.TOGETHERw′ ,

λw′. {b1, g2 } , {b2, g1 } ∈ LIVE.TOGETHERw′

An answer that names a plurality, namely a list answer, is acceptable.

(44) w2: b1 and g1 live together and b2 and g2 live together.JANSKw2 ((43)) = λw′. {b1, g1 } , {b2, g2 } ∈ LIVE.TOGETHERw′

28

single-tuple reading with a distributive predicate

Turning now to the distributive predicate ‘sneezed’, the single-tuplereading is generated with the following LF without covert operators.

st

λpst t

⟨et, t⟩

⟨et, t⟩

which boy∩and

⟨et, t⟩

which girl

⟨e, t⟩

λxe?p t

tx sneezed

{pst | ∃x, y[x ∈ BOYw ∧ y ∈ GIRLw ∧ p = [λw′. x, y ∈ SNEEZEDw′ ]] }

29

list reading with a distributive predicate

We generated the list reading with a collective predicate with Dabove MIN and below ER and on the predicate.

But this will result in a type-mismatch with a distributive predicate.

λpst

⟨⟨⟨et, t⟩, t⟩, t⟩


D ⟨et, t⟩

MIN ⟨et, t⟩

⟨et, t⟩

which boy∩and

⟨et, t⟩

which girl

λX⟨et,t⟩?p

tX ⟨et, t⟩

D sneezed

30

*double d

The type mismatch would be resolved if another D could be used.st

λpst

⟨⟨⟨et, t⟩, t⟩, t⟩


D ⟨et, t⟩

MIN ⟨et, t⟩

⟨et, t⟩

which boy∩and

⟨et, t⟩

which girl

⟨⟨et, t⟩, t⟩

λX⟨et,t⟩?p t

tX ⟨⟨et, t⟩, t⟩

D ⟨et, t⟩

D sneezed

This would derive the list reading with distributive predicates.

We need to assume that D cannot be stacked like this.31

*d with a singular dp

More generally, the distribution of D needs to be constrained (Winter2001, De Vries 2015).

If covert D were freely available, it would make a singular NP plural!

(45) a. *Which boy live together?b. Which boy sneezed?

—# John, Bill and Fred sneezed.c. *A boy live(s) together.

32

distribution of d

We assume that each occurrence of D needs to be licensed by[plural]-feature (cf. Winter 2001, De Vries 2015).

∙ Nominal conjunction ‘and’ introduces [plural] within the DP(Sauerland 2003, 2008) and licenses D.

∙ [plural] on the auxiliary/verb licenses D, but can only license one D(∴ *Double D).

NB: [plural] does not imply D. The LFs of (46) do not involve D.

(46) a. John and Mary live together.b. A man and a woman live together.c. Which man and which woman live together?

— John and Mary live together.

33

non-crisp judgments

The list reading of (47) is judged less good than the single tuple reading.


The single tuple reading requires only ER and MIN, while the listreading additionally requires two instances of Ds.

(48) a.q[ER MIN which boy and which girl] [live together]?

yw

b.q[ER D MIN which boy and which girl] [D live together]?

yw

We assume that in such a situation, the simpler LF is preferred.Speakers might differ in how willing they are to complicate the LF,hence inter-speaker variation.

For (49), there is only one coherent LF, so no degradation.

(49) Which boys sneezed?q[which D boy] [D sneezed]?

yw

34

conclusions

conclusions

Observation

Conjoined singular ‘which’-phrases can receive a list answer withcollective predicates but not with distributive predicates.

We combined Winter’s (2001) theory of conjunction and plurality andDayal’s (1996) ANS-operator to derive list readings for conjoinedsingular ‘which’-phrases with collective predicates.

But the resulting theory overgenerates. We postulated constraintson the distribution of D.

∙ D needs to be licensed by [plural] (De Vries 2015).∙ Simpler LFs are preferred.

36

Thanks!!We thank Danny Fox and Greg Scontras for helpful comments.

The experimental work was funded by the German research council(DFG project SSI (SA 925/) within SPP 1727 XPrag.de.

37

references

∙ Champollion (2015) Ten men and women got married today. Journalof Semantics.

∙ Dayal (1996) Locality in WH Quantification.∙ Sauerland (2003) A new semantics for number. SALT 13.∙ De Vries (2015) Shifting Sets, Hidden Atoms. Ph.D. dissertation,Universiteit Utrecht.

∙ Winter (2001) Flexibility Principles in Boolean Semantics.

38

appendix

list readings are not the same as pair-list readings

Multiple singular ‘which’-questions can receive pair-list answers(Dayal 1996, Fox 2012, Kotek 2014, a.o.).

(50) Which girl hugged which boy?a. Sue hugged Frank, Mary hugged Bill, and Jill hugged John.b. Sue hugged Frank.

It might be tempting to try to provide a common explanation for listreadings of (i) multiple singular ‘which’ questions like (50), and (ii)conjoined singular ‘which’ questions.

But there are reasons to doubt that such a uniform analysis isdesirable due to the following:

∙ Inter-speaker variation∙ Predicate sensitivity∙ Non-Exhaustivity

40

which np and np

Our analysis predicts that ‘which NP and which NP’ and ‘which NPand NP’ should behave similarly.

∙ Jman and womanKw = ∅∙

qER(man) and ER(woman)

yw

= λPet. ∃x, y[x ∈ MANw ∧ y ∈ WOMANw ∧ x, y ∈ P]= {Pet | ∃x, y[x ∈ MANw ∧ y ∈ WOMANw ∧ x, y ∈ P] }

∙qMIN(ER(man) and ER(woman))

yw

= { { x, y } | x ∈ MANw ∧ y ∈ WOMANw }

Applying ‘which’ to this, we derive the single-tuple reading. If Dapplies before ‘which’ (and on the predicate), we derive the listreading.

(See Champollion 2015 for how to deal with plural nouns)

41

two routes to distributive readings

The following two parses derive the same distributive reading.

(51) a. A man and a woman sneezed.b. ER MIN (a man and a woman) D sneezed.

The second strategy is necessary for cases like (52).

(52) A man and a woman went to a bar and had many beers.

According to our logic, (51b) should be dispreferred, because itinvolves more optional operations.

42

experiment: design

Inferential task: Does sincerely uttering S1 necessarily commit you toassume S2 is true?

S1: Ann knows which girl and which boy hugged each other.S2: Only one girl and one boy hugged each other.

Conditions:

∙ Conjoined-Collective (CC): Ann knows which girl and which boyhugged each other.

∙ Conjoined-Distributive (CD): Tami knows which laywer and whichjudge studied.

∙ Non-Conjoined: Rhonda knows which kid received which present.∙ Pragmatically fored SP (PS): Pam knows which spy killed the presentwith which weapon.

6 items per condition. Each subject saw 2 items from each condition,and 16 filler items.

43

experiment: results

CC CD NC PS0

20

40

60

80

100

65.2

87

10.9

87

Sentence Type

%SP

Inferenc

e

23 native speakers of English on Amazon Mechanical Turk

∙ CC vs. CD (p=0.03)∙ CC vs. PS (p=0.02)

44

list readings of conjoined singular...

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