A Dynamic Semantics of Single and Multiple Wh-Questions

A dynamic semantics of single and multiple wh-questions

Jakub Dotlacilˇ Floris Roelofsen Utrecht University University of Amsterdam SALT, August 20, 2020

1 Desiderata

The goal of this paper is to develop an analysis of single and multiple wh-questions • in English satisfying the following desiderata:

(i) Uniformity. The interpretation of single and multiple wh-questions is derived uniformly. The same mechanism is operative in both types of questions. (ii) Mention-some vs mention-all. Mention-some and mention-all readings are derived in a principled way, and contraints on the availability of mention-some readings in single and multiple wh-questions are captured. (iii) Number marking. The eﬀects of number marking on wh-phrases is captured. In particular, singular which-phrases induce a uniqueness requirement in single wh-questions but do not block pair-list readings in multiple wh-questions.

To our knowledge, no existing account fully satisfies these three desiderata. • For instance: (we cannot do justice here to the rich literature on the topic) • – Groenendijk and Stokhof(1984) provide a uniform account of single and multiple wh-questions, but their proposal does not successfully capture mention- some readings and the effects of number marking. – Dayal(1996) captures e ffects of number marking in single and multiple wh- questions, but her account does not succesfully capture mention-some readings and is not uniform, in the sense that it involves different question operators for single and multiple wh-questions, respectively. Moreover, as pointed out by Xiang(2016, 2020b), the so-called domain exhaustivity presupposition that Dayal predicts for multiple wh-questions is not borne out (examples will be given later).

1 – Fox(2012) provides a uniform account of single and multiple wh-questions which captures the fact that singular which-phrases give rise to uniqueness presuppositions in single but not in multiple wh-questions. However, as discussed by Xiang(2016, 2020b), Fox’s account, just like Dayal’s, derives the problematic domain exhaustivity presupposition for multiple wh-questions. Moreover, while Fox(2013) derives mention-some readings for single wh- questions, it is not clear yet whether his approach could also capture mention- some readings in multiple wh-questions (which we will show do exist).1 – Xiang(2016, 2020b) captures mention-some readings and e ﬀects of number marking, but her account is not uniform, in the sense that it assumes special LF ingredients for multiple wh-questions which are not present in single wh- questions (see Appendix II for some more details).

Our account is crucially dynamic, combining and extending insights from: • – Dynamic inquisitive semantics (Dotlacilˇ and Roelofsen 2019).2 – PCDRT (Brasoveanu, 2008).

2 Informal proposal

We ﬁrst present the proposal informally. • Some details will be suppressed in this section, so as to provide easy access to the • main ideas.

2.1 Wh-phrases and the witness request operator In dynamic semantics a sentence like (1) gives rise to two consecutive updates of the • context.

(1) Someone smiled.

– First a discourse referent u is introduced by the indeﬁnite; – Then the context is updated with the information that u smiled.

This sequence of updates is represented in (2); the exact semantics of this represen- • tation will be given further below.

(2)[ u]; smiled u { } 1Fox (p.c.) tells us that, while he was developing the material in Fox(2013) he also noted that mention- some readings are possible in multiple wh-questions, based on examples similar to the ones we will give below. He also had some initial ideas for how to derive such readings, but did not get to develop those ideas at the time. 2Dynamic inquisitive semantics in turn strongly builds on the dynamic partition semantics of Haida (2007). For another recent dynamic approach to questions see Li(2019).

2 Now consider the wh-question in (3). • (3) Who smiled?

We analyze wh-phrases just like indeﬁnites: as introducing a discourse referent. • However, we propose that wh-questions involve an additional update, ?u, raising • an issue whose resolution requires identifying one or more individuals that have the properties ascribed to u—in short, a witness set for u.

Thus, (3) is analyzed as in (4). • (4)[ u]; smiled u ;?u { } We refer to ?u as the witness request operator. • 2.2 Mention-all readings from maximization of witness sets As is well-known, when indeﬁnites serve as antecedents for donkey anaphora, two • readings can arise, strong and weak.

In (5-a) the strong reading is most salient, while in (5-b) the weak reading is. • u (5) a. If Bill eats an apple he peels itu ﬁrst. u b. If Bill has a dime he throws itu in the parking meter.

Brasoveanu(2008) argues that the strong reading is derived by value maximization, • max u , in the interpretation of the indeﬁnite. { } Intuitively, in (5-a) max u ensures that all possible values of u (all the apples that • Bill eats) are considered{ in} the consequent.

Since we treat wh-phrases just like existential indeﬁnites, we expect that they can • also involve max.

Adding max u to (4) results in (6). • { } (6)[ u]; smiled u ; max u ;?u { } { } In order to resolve this question, we have to identify a maximal witness set for u. This • corresponds to the mention-all reading.

The mention-some reading is captured by (4), without max u . • { } So, our dynamic semantics of wh-questions makes it possible to connect the mention- • some/mention-all ambiguity in wh-questions to the weak/strong ambiguity in donkey anaphora.

3 In other words, the ambiguity between mention-some and mention-all readings is • derived in a principled way, relying on a mechanism that is independently needed elsewhere in the grammar.3

We will assume that which-phrases, unlike who/where/what, always involve maxi- • mization, so in principle they block mention-some readings.

Our account will predict, however, that interaction with possibility modals can give • rise to mention-some readings even in the presence of maximization. For reasons of time, we will not discuss this part of the account in detail today.

2.3 Scaling up to multiple wh-questions The account so far can be generalized in a natural way to deal with single and • multiple wh-questions in a uniform way.

We assume that in a multiple wh-question, each wh-phrase introduces a discourse • referent.

We propose that in this case the witness request operator asks for a functional • witness, i.e., a function mapping the possible values for one dref to the possible values of another.

Consider: • (7) Who ate what?

This multiple wh-question is analyzed as in (8), where ?u1u2 is a functional witness • request operator.

(8)[ u ]; [u ]; ate u , u ;?u u 1 2 { 1 2} 1 2 Informally, ?u u raises an issue whose resolution requires determining, for every • 1 2 possible witness set of u1, the corresponding witness set of u2. . . . or, in other words, a suitable function from the witness sets of u to those for u . • 1 2

We will see below that both ?u and ?u1u2 are special instances of a general n-ary • witness request operator, where n = 1 or n = 2, respectively. This means that the account is uniform, satisfying desideratum (i). 3Champollion et al. (2019) have recently proposed an alternative strategy to derive strong and weak readings in donkey anaphora, which does not make use of maximization. We leave open here whether this strategy may also be applied in the domain of questions to derive mention-some and mention-all readings.

4 2.4 Mention-some readings in multiple wh-questions It is typically assumed in the literature that multiple wh-questions only have a • mention-all pair-list reading.4

Consider: • (9) Who ate what at the party last night?

To resolve the issue expressed by this question, one must specify, for each party • guest, everything that they ate (rather than just something that they ate).

We observe, however, that multiple wh-questions, at least in some cases, also allow • for a mention-some reading.

Suppose Mary has a large herb garden. The herbs that Mary has in her garden each • grow in several places. Susan has a list of ﬁve herbs that she needs to collect for a certain recipe and asks Mary the following question:

(10) Which of these herbs grows where?

On the most natural reading of this question, Mary can resolve it by mentioning one • location for each herb.

Importantly, the question does not require that Mary exhaustively speciﬁes all loca- • tions of every herb.

Our account can straightforwardly derive this reading. Just like in single wh- • questions, mention-all readings arise through maximization, and mention-some readings arise in the absence of maximization.

Note that, while (10) is mention-some w.r.t. locations, it is mention-all w.r.t. herbs. • That is, the question requests information about all herbs on the list: it does not • suﬃce to specify just for one herb where it grows.5

4It is sometimes argued that multiple wh-questions can have a so-called ‘single-pair reading’ which is separate from their ‘pair-list’ reading. In our view, cases in which multiple wh-questions have been reported to have a ‘single-pair’ reading are just cases where the (most likely) context is one in which only one pair of objects stands in the relation that the question inquires about (see also Dayal, 2016, p.95). If this is correct, an account of multiple wh-questions does not need to derive single-pair readings separately from pair-list readings. In any case, we will concentrate here on pair-list readings. If it turns out that there is a need to derive ‘single-pair’ readings separately, the account will have to be refined. 5One might think that this is a consequence of the fact that the first wh-phrase which of these herbs is a which-phrase whose domain of quantification is clearly given in the context (it is so-called ‘D-linked’). Such wh-phrases are known to favour a mention-all reading in single wh-questions as well. However, the same effect arises with non-D-linked wh-phrases which favor a mention-some reading in single wh-questions. For instance, while (i) has a very salient mention-some reading, (ii) only has a reading that is mention-some w.r.t. one of the wh-phrases, not w.r.t. both.

5 This is generally the case: multiple wh-questions always require what we will call • domain maximality—their resolution must match as many potential witnesses of the ‘domain dref’ as possible with corresponding witnesses of the ‘range dref’.

It is important to note that domain maximality as we understand it is diﬀerent from • domain exhaustivity in the sense of Dayal(1996).

Domain exhaustivity, adapted to our setting, would require that every potential • witness of the ‘domain dref’ be matched with a corresponding witnesses of the ‘range dref’.

Xiang(2019) shows that this requirement is too strong, based on the following sce- • nario. Suppose that 100 candidates have applied for 3 jobs. Mary knows the outcome of the search procedure, but Bill doesn’t. Then it is natural for Bill to ask:

(11) Which candidate got which job?

Xiang observes that resolving this question does not require specifying for every • candidate which job they got, something that would be impossible to do because there were fewer jobs than candidates.

An additional observation to make, however, is that a resolution of the question must • still specify for as many candidates as possible (in this case three) which jobs they got.

This requirement is what we call domain maximality. • On our account, domain maximality will be captured by an operator that is always • present in the left periphery of an interrogative clause.

In particular, it will be derived no matter whether a max operator applies to the • domain dref or not.

So our account will derive two readings for multiple wh-questions: • – One that is mention-all with respect to all wh-phrases. – One that is mixed: maximal with respect to the ‘domain dref’ but mention-some with respect to the ‘range dref’.

For this to be possible, it is crucial that on our account mention-some is not a property • connected to a question as a whole but rather to individual wh-phrases.

In this respect our account diﬀers from many others (e.g., Beck and Rullmann 1999, • Chapter 2 of George 2011, Theiler 2014, Champollion et al. 2015), which take mention- some to be a property connected to questions as a whole.

(i) What is a typical French dish?

(ii) What is a typical dish where?

6 On such accounts, it is not possible to derive mixed readings of multiple wh-questions • (at least not without substantial modiﬁcations).

To our knowledge, the only previous account that derives mixed readings of multiple • wh-questions is that of Xiang(2016, 5.4.3). § 2.5 Eﬀects of number marking So far we have outlined an account that satisﬁes Desiderata (i) uniformity and (ii) • mention-all vs mention-some.

We now turn to Desideratum (iii), i.e., capturing the eﬀects of number marking. • Several eﬀects need to be captured but the main puzzle is that singular which- • phrases induce a uniqueness presupposition in single wh-questions but allow for multiple witness-pairs in multiple wh-questions (Dayal, 1996, among others).

(12) Which girl danced with Robin? { a single girl danced with Robin (13) Which girl danced with which boy? { there was a single girl-boy pair who danced 6

Interestingly, it has recently been observed that in single wh-questions with an exis- • tential modal operator, the uniqueness requirement normally induced by a singular which-phrase can be obviated (Hirsch and Schwarz, 2019).

(14) Which letter can be inserted in fo m to make a word?

It is perfectly natural for a speaker to ask this question without assuming that there • is a unique letter that can be inserted in fo m to make a word.

For instance, a teacher in elementary school might ask (14) in class, knowing very • well that there are in fact two letters that may be inserted, a and r.

So the uniqueness presupposition normally carried by singular which-questions is • somehow obviated in this case.

Existing accounts of (12) have diﬃculties predicting that a global uniqueness pre- • supposition is absent both in (13) and in (14).6

6Both in multiple wh-questions and in single wh-questions with modals, the uniqueness requirement normally invoked by singular which-phrases is in fact not complete absent. However, in both cases it is relativized. For instance, (13) allows for multiple dance couples but presupposes that every girl who danced did so with only one boy. Similarly, (14) does not presuppose that there is a unique letter which can be inserted in the given frame to make a word, but it does presuppose that it is possible make a word by

7 Hirsch and Schwarz(2019) make an insightful distinction between globalist and • localist accounts of uniqueness presuppositions in singular which-questions.7 – Globalist accounts (Dayal, 1996, 2016; Fox, 2013, 2018; Xiang, 2016) derive uniqueness presuppositions by means of an operator that applies to the question as a whole and which is separate from question internal elements such as the which-phrase. – Localist accounts (Champollion et al., 2017; Uegaki, 2018, 2020; Hirsch and Schwarz, 2019) instead assume that the uniqueness presupposition of singular which-questions is not due to a global operator applying to the question as a whole but rather to the which-phrase itself.

Hirsch and Schwarz(2019) point out that globalist accounts cannot derive the lack • of a uniqueness presupposition in cases like (14).

On the other hand, localist accounts wrongly predict (at least in the absence of further • stipulations) uniqueness eﬀects in multiple wh-questions like (13).

So we have: •

Globalist Localist Obviation by modals X Obviation in multiple wh-questions X

Our own dynamic account will be neither exclusively globalist not exclusively local- • ist.

Rather, it will be hybrid in the sense that it derives uniqueness presuppositions from • the interplay between several elements, including both the which-phrase and global operators that apply to the question as a whole.8,9 Specifically, the following elements all play a role: • – the max operator contributed by which, – the singular number marking on the nominal restrictor of which, inserting a unique letter into the frame. For ease of exposition, discussion of these facts are suppressed here. The account we propose does capture them. 7In a very recent, unpublished manuscript Xiang(2020c) develops an account of uniqueness presuppositions which can be thought of as intermediate between globalist and localist accounts, similar to our own proposal. A detailed comparison must be left for a future occasion. 8A difference to note is that the relevant ‘global’ operators on our account are still part of the question, while on a globalist account such as that of Dayal(1996) the relevant operator is a so-called answer operator, which is question-external. 9Another hybrid account, developed completely independently, has been presented on the first day of the conference by Kobayashi and Rouillard(2020).

8 – the witness request operator, and – a presuppositional closure operator that is always present in the left periphery of interrogative clauses.

We will see that our hybrid account predicts obviation by modals as well as in • multiple wh-questions.10

Globalist Localist Hybrid Obviation by modals XX Obviation in multiple wh-questions X X

So how does the account work? (only a very informal sketch here, more details later) • There are two assumptions we need to make, in addition to what is in place already: • 1. Singular number marking, e.g., on which girl, invokes an atomicity requirement (following Brasoveanu, 2008, and much other work on number in dynamic semantics). This ensures that all the possible values assigned to the dref introduced by which girl are atomic individuals. 2. Every interrogative clause involves a presuppositional closure operator, which we denote as , requiring that the input context already contains the information that the actual† world lies in one of the alternatives that the question introduces (Roelofsen, 2015). Note that this presupposition is trivially satisﬁed whenever the alternatives cover the entire logical space.

A question like (12), repeated in (15-a), is then translated as (15-b): • (15) a. Whichu girl danced? b. ([u]; atom u ; girl u ; danced u ; max u ;?u) † { } { } { } { } A uniqueness presupposition is derived from the interplay between atom u , max u , • ?u, and . { } { } † 10In his invited talk yesterday, Bernard Schwarz presented joint work with Aron Hirsch and Michaela Socolof which identifies another case in which a global uniqueness effect is obviated. This is exemplified in (i):

(i) In which town was Shakespeare born or did Bach die?

It seems that the observations made concerning such disjunctive wh-questions in Schwarz’s talk are derived on our hybrid account without any further stipulations. But we will leave a worked out treatment of such cases for the full paper.

9 A particularly important role is played by ?u. This operator removes any possibilities • from the context where more than one girl dances. For a multiple wh-question we get: • (16) a. Whichu1 girl danced with whichu2 boy?  [u ]; atom u ; girl u ;   1 1 1   u atom{u } boy{u }   [ 2]; 2 ; 2 ;   { } { }  b.  danced with u1, u2 ;  †  { }   max u1 ; max u2 ;   { } { }  ?u1u2

A global uniqueness requirement is not derived here because, instead of ?u, the • functional witness request operator ?u1u2 is at play in this case. This does not remove possibilities from the input context where more than one girl • or more than one boy danced, but only ones where there is no functional dance- dependency between girls and boys, i.e., where there are girls who danced with more than one boy.

Finally, consider a single wh-question with a modal possibility operator. • We assume that a modal possibility operator introduces a world dref, [v], that • the world assigned to this dref must be accessible from the actual world, ^v, and that predicates like dance and operators like max may either be evaluated relative to the actual world @ or relative to the worlds introduced by modal operators. Then (17-a) can be analyzed as in (17-b):11 • (17) a. Which letter can be inserted to make a word?

b. ([v]; ^v;[u]; letterv u ; atom u ; insert-to-make-a-wordv u ; maxv u ;?u) † { } { } { } { } Crucially, the maximality requirement is evaluated relative to the world intro- • duced by the modal under this analysis. Thus, the question is predicted to ask: • (18) ‘what is a letter such that it would be possible for that letter to be the maximal entity that we insert in fo m to make a word’

rather than:

(19) ‘what is the maximal atomic entity such that it is possible to insert that entity in fo m to make a word’.

Thus, a global uniqueness presupposition is obviated. •

10 3 Implementation

Dynamic inquisitive semantics for single and multiple-wh questions • We will not discuss modals • 3.1 Type-theoretic framework We construct a type-theoretic framework with four basic types:

e for atomic and plural entities (e.g., a, a b) • ⊕ s for possible worlds • t for truth values • r for discourse referents (cf. Muskens, 1996, Brasoveanu, 2008) • 3.2 Contexts, information states, and possibilities As is standard in inquisitive semantics, we take a context to be a downward closed • set of information states. Every information state consists of possibilities and each possibility is a pair w, G , where w is a world and G a set of assignments mapping discourse referents toh (atomici or plural) entities. For contexts, we use the variable c, for information states, we use the variable s.

Throughout, we will illustrate contexts graphically. Each black dot is a possibility. • Each row represents G, an assignment set, each column is a world. Each shaded area is the maximal information state in a context (also called an alternative).

wa wa,b wb w u/a ∅

u/b

u/a, u/b

u/a b ⊕ For ﬁnite cases, if the context contains one alternative, it is non-inquisitive. • Otherwise, the context is inquisitive (it contains an unresolved issue). • 11We assume here, following Hirsch and Schwarz(2019), that the which-phrase can be interpreted within the scope of the modal operator.

11 Non-inquisitive Inquisitive wa wa,b wb w wa wa,b wb w u/a ∅ u/a ∅

u/b u/b

u/a, u/b u/a, u/b

u/a b u/a b ⊕ ⊕ 3.3 Context updates Update functions are functions from contexts to contexts. We consider the update • functions involved in the examples above:

1. smile u { } 2.[ u] (dref introduction) 3.? u (a witness request operator)

4.? u1u2 (a functional witness request operator) 5. max u (domain maximality) { } 6. atom u (atomicity requirement) { } 7. (the presuppositional closure) † 1. smile u takes an input context c and returns a new context, which is the set of { } information states s in c such that for every assignment g in every possibility w, G in s, it holds that g(u) smiles in w. h i

(20) smile u B λc s rt t t λs s rt t . s c w, G s. g G. smile(w)(g(u)) { } hhh × i i i hh × i i ∈ ∧ ∀h i ∈ ∀ ∈ ∗ wa wa,b wb w wa wa,b wb w u/a ∅ u/a ∅ smile u u/b { } u/b

u/a, u/b u/a, u/b

u/a b u/a b ⊕ ⊕ u/a b, u/a u/a b, u/a ⊕ ⊕ u/a b, u/b u/a b, u/b ⊕ ⊕ u/a b, u/a, u/b u/a b, u/a, u/b ⊕ ⊕

2. [u], the introduction of the dref u, is a pointwise generalization of dref introduction in Brasoveanu(2008) to contexts in dynamic inquisitive semantics.

12 ! w, G s. w, G0 s0.(G0[u]G) (21)[ u] B λc s rt t t λs s rt t . s0 c. ∀h i ∈ ∃h i ∈ ∧ hhh × i i i hh × i i ∃ ∈ w, G0 s0. w, G s. (G0[u]G) ∀h i ∈ ∃h i ∈

G0[u]G in (21) is deﬁned in the same way as dref introduction in Brasoveanu (2008). •

(22) G[u]G0 B g G. g0 G0.g[u]g g0 G0. g G. g[u]g0 ∀ ∈ ∃ ∈ ∧ ∀ ∈ ∃ ∈ Intuitively,[u] updates a state in the input context by randomly assigning any possible • values to u across the assignments of the possibilities in the state.

wa wa,b wb w wa wa,b wb w ∅ u/a ∅ ∅ u/b wa wa,b wb w ∅ u/a u/a, u/b [u] u/b u/a b ⊕ u/a, u/b u/a b, u/a ⊕ u/a b u/a b, u/b ⊕ ⊕ u/a b, u/a u/a b, u/a, u/b ⊕ ⊕ u/a b, u/b smile u ⊕ { } u/a b, u/a, u/b ⊕

3. The witness request operator ?u, deﬁned in (23), selects those states s in the input context which contain enough information about the world to ensure the existence of a witness (atomic or plural entity) for u.

Formally, a state s is selected if there is an entity x (the witness) such that for every • S possibility w, G s there is a G0 such that w, G0 is in c and such that every h i ∈ h i assignment g0 G0 assigns x to u. ∈ S (23)? u B λc s rt t t λs s rt t .s c xe. w, G s. w, G0 c. g0 G0. g0(u) = x hhh × i i i hh × i i ∈ ∧ ∃ ∀h i ∈ ∃h i ∈ ∀ ∈

13 [u]; smile u ?u wa wa,b wb w { } wa wa,b wb w wa wa,b wb w ∅ u/a ∅ u/a ∅ ∅ u/b u/b

u/a, u/b u/a, u/b

u/a b u/a b ⊕ ⊕ u/a b, u/a u/a b, u/a ⊕ ⊕ u/a b, u/b u/a b, u/b ⊕ ⊕ u/a b, u/a, u/b u/a b, u/a, u/b ⊕ ⊕ 4. The more general functional witness request operator, deﬁned in (25), selects those states s in the input context which contain enough information about the world to ensure the existence of a functional witness, mapping values of u1 ... un 1 to values − of un.

?u is a speciﬁc instance of ?u1 ... un 1un, with n = 1. • − S (24)? u1 ... un 1un B λc s rt t t λs s rt t . s c f(en 1,e). w, G s. w, G0 c. − hhh × i i i hh × i i ∈ ∧ ∃ − ∀h i ∈ ∃h i ∈ g0 G0. (g0(un) = f (g0(u1),..., g0(un 1)) ∀ ∈ − u u wa{c wa{c,d wa{d wa{c wa{c,d wa{d 1 2 b{e b{e, f b{ f b{e b{e, f b{ f a c ?u u b e 1 2

a d

b f

a c, d

b e, f

5. max u restricts the context to states in which u’s cumulative value is maximal. { } S (25) max u B λc s rt t t λs s rt t . s c w, G s. w, G0 c. G0(u) G(u) { } S hhh × i i i hh × i i ∈ ∧ ∀h i ∈ ∀h i ∈ ⊕ ⊆ ⊕ (26) G(u) B G(u) ⊕

14 wa wa,b wb w u/a ∅ max u { } u/b wa wa,b wb w ∅ u/a, u/b u/a, u/b

u/a b u/a b ⊕ ⊕ u/a b, u/a u/a b, u/a ⊕ ⊕ u/a b, u/b u/a b, u/b ⊕ ⊕ u/a b, u/a, u/b u/a b, u/a, u/b ⊕ ⊕ We need to work with a generalized version of max u , given in (27). It restricts • a context c to states consisting of possibilities p whose{ } dref assignment functions collectively assign a maximal set of individuals to u independently of the values assigned to other drefs. S (27) max∗ un B λc s rt t t λs s rt t . s c w, G s. w, G0 c. hhh × i i i hh × i i { } ∈ u~∧un ∀h~x i ∈ ∀hu~ un ~x i ∈ G0 \ 7→ (un) G \ 7→ (un) ⊕ ⊆ ⊕

u~ un ~x In (27), G \ 7→ (un) is the set of values assigned to un by assignments in G which • pointwise assign the values ~x to u~ un, all drefs except un. If un is the only dref, then \ max∗ un just amounts to max un . But if there are other drefs around, the two may come{ apart.} { }

u1 u2 w

a a

b b max∗ u { 2} a b

b a u1 u2 w

a a a a

a b a b

b a b a

b b b b

a a b a a b ⊕ ⊕ b a b b a b ⊕ ⊕

15 6. We assume that singular morphology in which-phrases contributes an atomicity requirement on assignments (see Brasoveanu, 2008, for non-wh-indeﬁnites).

(28) atom u B λc s rt t t λs s rt t . s c w, G s. g G. y(y < g(u)) { } hhh × i i i hh × i i ∈ ∧ ∀h i ∈ ∀ ∈ ¬∃ wa wa,b wb w u/a ∅

u/b wa wa,b wb w ∅ u/a, u/b atom u u/a { } u/a b u/b ⊕ u/a b, u/a u/a, u/b ⊕ u/a b, u/b ⊕ u/a b, u/a, u/b ⊕ 7. The dagger ( ) operator is defined in (29). † ( S S A(c) if c subsists in A(c) (29) AT B λck. † undefined otherwise Output context Defined input Undefined input wa wb wa,b w ∅ wa wb wa,b w wa wb wa,b w u/a ∅ ∅ ∅ ∅ u/b

4 Syntax-semantics mapping in wh-questions

The assumed syntactic structure of the left periphery in questions TypeP

Type FocP

Foc TP

The lower head, Foc, introduces the witness request operator and ensures the maxi- • mality of all but one drefs introduced by wh-phrases.

The higher head, Type, adds the presuppositional closure ( ) • †

(30) [[Focu ,...un ]] B λAT.A; max∗ u1 ; ... ; max∗ un 1 ;?u1 ... un, 1 { } { − } where u1,... un – drefs introduced by wh-phrases

16 (31) A single wh-question: [[Focu1 ]] B λAT.A;?u1

(32) [[Type]] = λAT. A † which-phrase has the maximality requirement •

whichsg has the atomicity requirement • This speciﬁcation of the syntax-semantics interface is suﬃcient for our goal. The • details of the left periphery can be expanded to provide an account that also covers declarative sentences, polar and alternative questions.

5 Illustrations

(33) Someone smiles. { [u]; smile u { } wa wa,b wb w wa wa,b wb w ∅ u/a ∅ ∅ u/b wa wa,b wb w ∅ u/a u/a, u/b [u] u/b u/a b ⊕ u/a, u/b u/a b, u/a ⊕ u/a b u/a b, u/b ⊕ ⊕ u/a b, u/a u/a b, u/a, u/b ⊕ ⊕ u/a b, u/b smile u ⊕ { } u/a b, u/a, u/b ⊕

17 (34) Whoweak smiles? { ([u]; smile u ;?u) † { } wa wa,b wb w wa wa,b wb w ∅ u/a ∅ ∅ u/b

u/a, u/b

u/a b ⊕ u/a b, u/a ⊕ u/a b, u/b ⊕ [u]; smile u u/a b, u/a, u/b { } ⊕

wa wa,b wb w u/a ∅

u/b

u/a, u/b

u/a b ⊕ u/a b, u/a ⊕ u/a b, u/b ⊕ u/a b, u/a, u/b ⊕

18 (35) Whostrong smiles? { ([u]; smile u ; max∗ u ;?u) † { } { }

wa wa,b wb w wa wa,b wb w ∅ u/a ∅ ∅ u/b

u/a, u/b

u/a b ⊕ u/a b, u/a ⊕ [u]; smile u u/a b, u/b { } ⊕ u/a b, u/a, u/b ⊕

wa wa,b wb w wa wa,b wb w u/a ∅ u/a ∅

u/b u/b max∗ u { } u/a, u/b u/a, u/b

u/a b u/a b ⊕ ⊕ u/a b, u/a u/a b, u/a ⊕ ⊕ u/a b, u/b u/a b, u/b ⊕ ⊕ u/a b, u/a, u/b u/a b, u/a, u/b ⊕ ⊕

19 (36) Which person smiles? { ([u]; atom u ; smile u ; max∗ u ;?u) † { } { } { }

wa wa,b wb w wa wa,b wb w ∅ u/a ∅ ∅ [u]; atom u ; u/b person u ;{ } smile u{ } ?u { }

wa wa,b wb w wa wa,b wb w u/a ∅ u/a ∅

u/b u/b max∗ u { } u/a, u/b u/a, u/b

atom u ensures that we do not have a plural entity (e.g., a b) as the argument of • smile {(see} 1st update) ⊕

max∗ u ensures that we remove possibilities non-maximal wrt. u (see 2nd update) • { } ?u removes possibilities in which u is assigned a non-unique value • So, (36) carries the uniqueness presupposition •

20 (37) Whichu1 student talks to whichu2 professor?

{ ([u1]; student u1 ;[u2]; pro f essor u2 ; atom u1 ; atom u2 ; † { } { } { } { } talk to u , u ; max∗ u ; max∗ u ;?u u ) { 1 2} { 1} { 2} 1 2

wa{c wa{c wa{d wa{d b{c b{d wb{c b{c b{d b{d b{c w w b{d

∅

[u1]; student u1 ;[u2]; pro f essor u2 ; atom u ; atom{ } u ; { } { 1} { 2} talk to u , u ; max∗ u ; max∗ u { 1 2} { 1} { 2}

a{c a{c a{d a{d b{c u1 u2 w w w w b{c b{d w b{c b{d b{d b{c w w b{d a c

a d

b c

b d

b c

b d

a c

b c

a c

b d

a d

b d

a d

b c

21 wa{c wa{c wa{d wa{d b{c b{d wb{c b{c b{d b{d b{c w w b{d

∅

[u1]; student u1 ;[u2]; pro f essor u2 ; atom u ; atom{ } u ; { } { 1} { 2} talk to u , u ; max∗ u ; max∗ u ;?u u { 1 2} { 1} { 2} 1 2

a{c a{c a{d a{d b{c u1 u2 w w w w b{c b{d w b{c b{d b{d b{c w w b{d a c a d b c b d b c b d a c b c a c b d a d b d a d b c

22 (38) Whostrong talks to whomstrong?

{ ([u1]; [u2]; talk to u1, u2 ; max∗ u1 ; max∗ u2 ;?u1u2) † { } { } { }

wa{c wa{c wa{d wa{d b{c b{d wb{c b{c b{d b{d b{c w w b{d

∅

[u ]; [u ]; talk to u , u ; 1 2 { 1 2} max∗ u ; max∗ u ;?u u { 1} { 2} 1 2

a{c a{c a{d a{d b{c u1 u2 w w w w b{c b{d w b{c b{d b{d b{c w w b{d a c

a d

b c

b d

b c d ⊕ b c

b d

a c

b c

a c

b d

a d

b d

a d

b c

23 (39) Which herb grows where?

u u wa{c wa{c,d wa{d wa{c wa{c,d wa{d 1 2 b{e b{e, f b{ f b{e b{e, f b{ f a c

a d

a c, d

wa{c wa{c,d wa{d b{e b{e, f b{ f b d, e a c ∅ ?u u b e 1 2 [u1]; herb u1 ;[u2]; place u2 ; atom u {; } { } a d { 1} grow at u1, u2 ; max∗ u1 { } { } b f

a c, d

b e, f

6 Conclusion and outlook

We developed an account of single and multiple-wh questions that is uniform, gen- • erates mention-some and mention-all and captures the eﬀects of number-marking.

The analysis is couched in a dynamic semantic framework. Such a framework is • useful here because the semantic values it assigns to sentences are more ﬁne-grained than the semantic values in plain propositional frameworks.

This higher level of ﬁne-grainedness is very useful (perhaps even necessary?) to deal • with multiple wh-questions in a satisfactory way.

This paper is part of a larger eﬀort to develop a dynamic inquisitive framework, • which we think is needed to capture various other aspects of questions as well.12

Besides its theoretical contribution, the paper also makes two new empirical obser- • vations concerning multiple wh-questions:

12In particular, further support for a dynamic approach to questions comes from anaphora, intervention eﬀects, and wh-conditionals (Groenendijk, 1998; van Rooij, 1998; Aloni and van Rooij, 2002; Haida, 2007; Dotlacilˇ and Roelofsen, 2019; Li, 2019). The work of Xiang(2016, 2020a) is very much in the same spirit, in that it also develops a framework for question semantics which is more ﬁne-grained than propositional frameworks (Hamblin/Karttunen, partition semantics, inquisitive semantics). Her framework is not dynamic, however, but more in the structured meaning tradition. A detailed comparison between the two approaches would be of great interest, but must be left for another occasion.

24 – Multiple wh-questions can have mention-some readings (as observed by Xiang, 2016) but only w.r.t. one of the wh-phrases. We called these mixed readings. Such readings are problematic for many previous accounts.

(40) Which herb grows where?

– Domain exhaustivity is not required in multiple wh-questions (as observed by Xiang, 2016) but domain maximality is.

(41) Which candidate got which job?

Thank you for listening / reading! If you have any questions, comments, ideas, or pointers to relevant literature, please get in touch! Jakub: [email protected] Floris: ﬂ[email protected]

25 Appendix I: Formal deﬁnitions of extension and subsistence

Deﬁnition 1 (Extending dref assignment functions and matrices).

One dref assignment function g0 is an extension of another dref assignment function • g if and only if g0 g. Note that for this to hold it is necessary (but not suﬃcient) ⊇ that the domain of g0 contains the domain of g.

One dref assignment matrix G0 is an extension of another dref assignment matrix G • if and only if every g0 G0 is an extension of some g G and every g G is extended ∈ ∈ ∈ by some g0 G0. In this case we write G0 G. ∈ ≥ Definition 2 (Extending possibilities). A possibility w0, G0 is an extension of another possibility w, G if and only if w0 = w and h i h i G0 G. In this case we write w0, G0 w, G . ≥ h i ≥ h i Definition 3 (Extending information states). An information state s0 is an extension of another information state s if and only if every possibility in s0 is an extension of some possibility in s. In this case we write s0 s. ≥ Definition 4 (Subsistence of one information state in another). Let s, s0 be information states such that s0 s. Then we say that s subsists in s0 if and only ≥ if every possibility in s has an extension in s0.

Appendix II: Some details about Xiang’s proposal

While we will not pursue a comprehensive comparison here between our proposal • and that of Xiang(2016, 2019, 2020b), we will say a bit more as to why we do not take the latter to provide a uniform account of single and multiple wh-questions.

What we mean by this is that the account invokes LF ingredients in the analysis of • multiple wh-questions with pair-list readings which are not present in the analysis of single wh-questions.

Consider the following two examples: • (42) Which boy came?

(43) Which boy watched which movie?

The LFs that Xiang assumes for (42) and for (43), the latter under a pair-list reading, • are displayed in Figures1 and2, respectively.

The LF for (43) includes various elements that are not present in the LF for (42): • T – The operator; – The Id operator;

26 Figure 1: The LF that Xiang assumes for (42).

Figure 2: The LF that Xiang assumes for (43).

27 – The node that introduces a free propositional variable p and the one that lambda- abstracts over this variable.

There are two further diﬀerences between the two LFs as well: • – In (42), the type-shifting operator BeDom applies to the translation of the wh- phrase which boy. In (43) this only happens with one of the wh-phrases, which movie, but not with the other, which boy. – In (42) the trace of the wh-phrase which boy is treated as a variable of type e. In (43) this also holds for the trace of which boy. However, the trace of which movie is treated as a variable of type (e, e).

Finally, note that the denotations derived in (42) and (43) diﬀer in type. To derive • resolution/answerhood conditions from these denotations, a polymorphic answer set operator is needed.

For comparison, on our account: • – All wh-phrases are treated equally, no matter whether they appear in single or in multiple wh-questions, as introducing discourse referents that are bound by a witness request operator. – The LFs of single and multiple wh-questions involve exactly the same two left periphery heads, one contributing a witness request operator and the other contributing a presuppositional closure operator.

No additional LF operations need to be invoked. • References

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