Wh-questions in dynamic inquisitive semantics*

This target article presents a type-theoretic dynamic inquisitive semantics framework, extending the first-order system presented in Dotlačil and Roelofsen (2019). Within this framework, we develop a compositional treatment of wh-questions whose basic premise is that a wh-phrase introduces a discourse referent u, just like non-interrogative indefinites do, and requires the presence of an operator in the left periphery of the clause which requests a witness for u, i.e., it raises an issue whose resolution requires identifying at least one individual that has all the properties ascribed to u. In previous work, it has been argued that a dynamic semantic analysis of questions is needed to account for anaphora with wh-antecedents and for certain kinds of intervention effects (Haida, 2007). Here, we further develop the general approach and argue that it has several additional benefits. Namely, it allows for a uniform treatment of single-wh and multiple-wh questions, it derives mention-some and mention-all readings in a principled way, as well as an often neglected partial mention-some reading in multiple-wh questions, it can capture certain constraints on the availability of mention-some readings, and the effects of number marking on which-phrases across single-wh and multiple-wh questions.


A dynamic perspective on meaning
Formal semantic theories often construe the semantic value of a declarative sentence as a formal object representing the truth conditions of the sentence. While many semantic phenomena can be fruitfully analyzed within such a truth-conditional approach, it is well-known that certain phenomena are beyond its reach. A familiar example is the licensing of anaphoric pronouns. Consider (1) and (2), adapted from Heim (1982) who in turn attributes such examples to Partee.
(1) a. Normally there are ten marbles in the box, but today a u marble is missing. b. It u may be under the sofa.
(2) a. Normally there are ten marbles in the box, but today there are only nine. b. #It may be under the sofa.
Truth-conditionally, (1a) and (2a) are equivalent. Yet, only (1a) licenses anaphoric reference to the missing marble using the anaphoric pronoun it in the next sentence. This observation demonstrates that there is more to the meaning of a declarative sentence than its truth conditions, and has led to * We are grateful to Lucas Champollion, Ivano Ciardelli, Alexandre Cremers, Veneeta Dayal, Danny Fox, Haoze Li, and Yimei Xiang for helpful discussion at various stages of this project, and to Hans-Martin Gärtner for very helpful comments on a previous version of the manuscript. Finally, we gratefully acknowledge financial support from the Netherlands Organization for Scientific Research (NWO).
the development of dynamic semantics (Kamp, 1981;Heim, 1982;Groenendijk and Stokhof, 1991, a.m.o.), a framework in which the semantic value of a sentence is a formal object representing its context change potential, i.e., the way in which it changes the context in which it is uttered, rather than its truth-conditions. In this framework, the indefinite a marble in (1a) changes the context by introducing a discourse referent, which serves as a suitable antecedent for the anaphoric pronoun in (1b). By contrast, (2a) does not introduce such a discourse referent, and the anaphoric pronoun in (2b) therefore lacks a suitable antecedent. This, in dynamic semantics, explains the contrast between (1) and (2).

A dynamic perspective on questions
While dynamic semantics has led to important new insights into anaphora and several other phenomena (e.g., presuppositions, modals, conditionals, and tense), dynamic semantic theories have typically restricted their attention to declarative sentences. Semantic analyses of questions have, with very few exceptions (see below), usually been developed within a static framework. In such static analyses the semantic value of a question is typically a formal object representing its answerhood conditions (Hamblin, 1973;Karttunen, 1977) or its resolution conditions (Ciardelli et al., 2018). Yet, it is straightforward to see that, just like the meaning of a declarative sentence goes beyond its truth conditions, the meaning of a question goes beyond its answerhood/resolution conditions. Consider, for instance, the following examples modeled after Partee's marble cases.
(3) a. Which u one of her three sons will inherit the estate? b. And how did he u react when the will was disclosed?
(4) a. Which u two of her three sons will not inherit the estate? b. #And how did he u react when the will was disclosed?
These examples show that even if two questions are equivalent in terms of resolution/answerhood conditions, they may still differ in their potential to license pronominal anaphora. 1,2 The present target article contributes to the development of a dynamic framework for the analysis of questions, bringing together and extending insights from dynamic theories of declaratives and static theories of questions, in particular those developed within the inquisitive semantics framework (Ciardelli et al., 2018). The basic idea that underlies the approach can be conveyed on the basis of two simple examples. First consider the declarative in (5): (5) Someone fainted.
1 It may be useful to unpack in somewhat more detail why (4b) is infelicitous. There are two factors at play. First, the which-phrase in (4a) introduces a discourse referent for the two sons who won't inherit the estate, while it does not introduce a discourse referent for the son who will inherit the estate. Second, the pronoun he in (4b) is singular, so it needs a singular discourse referent as antecedent. But the only available discourse referent is plural. These two factors can be teased apart by considering cases like (i): (i) a. Which u one of her two sons will not inherit the estate? b.
And how did heu react when the will was disclosed?
In this case, (ia) introduces a discourse referent for the son who won't inherit the estate, which is singular, while still not introducing a discourse referent for the son who will inherit the estate. Now, the pronoun he in (ib) has a suitable singular antecedent, so (ib) is felicitous, unlike (4b). Crucially, however, it is still impossible for the pronoun to refer to the son who will inherit the house, because the which-phrase did not introduce a discourse referent for this fortunate son.
In dynamic semantics, this statement is taken to give rise to three consecutive updates of the context in which it is uttered. First, a discourse referent, let's call it u, is introduced; then the context is updated with the information that u is a person; and finally, the context is further updated with the information that u fainted. This sequence of updates can be represented as follows (the details of this notation will be reviewed later, but are not needed to understand the general idea we want to convey at this point): (6) [u]; person{u}; f ainted{u} ↑ someone fainted Now consider the question in (7).
What does this question have in common with the declarative in (5), and what is it that makes the two different? In dynamic semantics it is possible to give an answer to this question which we find particularly attractive (for reasons to be discussed shortly) and which cannot be formulated in a static semantic framework. Namely, we will analyze the question in (7) as follows. First, it introduces a discourse referent u and updates the context with the information that u is a person and the information that u fainted. This part is shared with the declarative in (5). But then a further update occurs, contributed by an operator which we assume to generally be present in the left periphery of interrogative clauses and which for now we will call C int . 3 This update raises an issue whose resolution requires identifying one or more individuals that have the properties ascribed to u, i.e., one or more persons who fainted. In short, this additional update requires identifying a witness set for u. We therefore refer to it as a witness request update and write ?u for it. The sequence of updates that (7) gives rise to can thus be represented as follows: (8) [u]; person{u}; f ainted{u}; ?u ↑ ↑ who fainted C int Note that, on this proposal, the semantic contribution of the wh-word who is exactly the same as that of the indefinite someone. Both introduce a discourse referent u and update the context with the information that u is a person. This is in line with the well-known observation that in many languages wh-words are similar, or even identical in form to non-interrogative indefinites (Ultan, 1978;Haspelmath, 1997;Bhat, 2000;Hengeveld et al., 2021, among others). For instance, somewhere and somehow are based on where and how, respectively, and the Dutch word wat can mean either 'something' or 'what'. On our proposal, the difference between wh-elements and non-interrogative indefinites is that the former stand in a particular relationship with C int . Specifically, we assume a semantic dependency between wh-elements and C int that is akin to dynamic binding: the ?u operator contributed by C int anaphorically accesses the discourse referent u introduced by the wh-element and requests a witness set for it. 4 3 In Section 4 we will make more precise assumptions about the left periphery of interrogative and declarative clauses, and we will also revisit the notation adopted here. 4 It may be worth commenting briefly on the 'witness request' terminology that we use here, to avoid a potential misunderstanding. When we say that ?u 'requests a witness set' for u, we do not mean that the role of ?u is to induce a particular kind of speech act. Rather, this terminology is meant as a description of the semantic contribution of ?u: a context update which raises an issue whose resolution requires identifying a witness set for u. One particular consequence of this is that we take ?u to be freely embeddable, something that generally does not hold of speech act The main precedent of our proposal is the work of Haida (2007). There is earlier work on questions in dynamics semantics as well (e.g., Groenendijk, 1998;van Rooij, 1998;Aloni and van Rooij, 2002). However, this earlier work does not adopt the specific approach pursued here. In particular, it does not separate the introduction of a discourse referent, which we take to be the contribution of wh-elements, from the witness request update, which we take to be the contribution of an operator in the interrogative left periphery. Nor does it assume a semantic dependency between wh-elements and the operator in the left periphery.
A more recent dynamic approach to question semantics has been developed in Li (2019Li ( , 2021a. This approach is similar to Haida's approach and the one we will pursue here in that it also takes whelements to introduce discourse referents. However, there is also a fundamental difference between Li's approach and ours. This difference concerns the way conversational contexts are modelled and the way in which questions are taken to update the context in which they are uttered. On Li's approach, a conversational context is modelled as an information state. Such an information state represents information about the world and information about the discourse referents that have been introduced so far. Crucially, however, it does not represent the issues that have been raised so far. How, then, is a question taken to update the context in which it is uttered? Li proposes that a question expresses a non-deterministic update. Technically, the semantic value of a question is a set of context updates. Rather than specifying one specific update, such a set specifies a number of possible updates. By contrast, on our approach, contexts will be modelled in such a way that they embody both the information that has been established so far and the issues that have been raised so far, and the semantic value of a question will be a deterministic update function, adding a new issue to the context. In short, Li keeps the notion of contexts structurally simple (just information, no issues) but complicates the notion of context updates (allowing both deterministic and non-deterministic updates), while on our approach the notion of contexts will be structurally more complex (representing both information and issues) but the notion of context updates will be simple (only deterministic updates).
One important reason for us to prefer the latter approach is that it allows for a uniform semantic treatment of declarative and interrogative clauses. Both types of clauses will be treated as having the same semantic type-they both express deterministic context updates. Various arguments have been made in recent work to highlight the benefits of such a uniform treatment of declaratives and interrogatives, in particular when one aims to give a semantic account of elements that interact with both types of clauses, such as connectives (Ciardelli et al., 2018), clause-embedding predicates (Theiler et al., 2018;Uegaki, 2019;Roelofsen, 2019a), and certain discourse particles (Theiler, 2021).
Turning from conceptual considerations to empirical ones, previous work on questions in dynamic semantics was motivated primarily by observations concerning anaphora with wh-antecedents. A simple example was already given in (3) above. A more complex case involving so-called dependent anaphora is given in (9). Dependent anaphora is exemplified by the pronoun it in (9b), which does not refer to a particular dress but, for each woman, to the dress that she bought (see in particular van Rooij, 1998;Li, 2021b).
(9) a. Which u 1 woman bought which u 2 dress? b. And how much did she u 1 pay for it u 2 ?
In addition to anaphora with wh-antecedents, Haida (2007) argued that a dynamic theory of questions which assumes a semantic dependency between wh-elements and a witness request operator in the left periphery (which his theory indeed does assume but earlier work on questions in dynamic semantics did not) provides a principled account of certain intervention effects in questions. operators.
Illustrating such effects and how a dynamic theory can account for them would take us too far afield here, so we refer to Haida (2007) for details. For current purposes it suffices to note that existing work has already provided some compelling empirical arguments for a dynamic approach to questions. 5 A limitation of Haida's proposal, however, is that it is based on the partition approach to questions (Groenendijk and Stokhof, 1984). The partition theory, which was initially developed within a static semantic framework, holds that the semantic value of a question is a set of propositions which (i) are mutually incompatible and (ii) together cover the set of all possible worlds. In short, these propositions form a partition of the set of all possible worlds. Haida (2007) develops a dynamic version of the partition theory, and thereby inherits its shortcomings. The main problem is that there are many questions whose basic answers/resolutions are not mutually incompatible and therefore do not form a partition of the set of all possible worlds. Consider the example in (10): Who has a bike that I could borrow for 15 minutes?
This question has a prominent mention-some reading. That is, resolving it does not necessarily involve identifying all people who have a bike that the speaker could borrow. Rather, it is sufficient to establish that, say, Amy has a bike that the speaker could borrow, or that Bob has one, etcetera. These resolutions are not mutually exclusive: it is very well possible that both Amy and Bob have a bike that the speaker could borrow. Therefore, such questions pose a problem for the partition approach to questions, a problem that Belnap (1982) referred to as the unique answer fallacy.
The recently developed inquisitive semantics framework (Ciardelli et al., 2018) overcomes this problem. In inquisitive semantics, the semantic value of a question is still a set of propositions, just like in partition semantics as well as approaches to questions based on the work of Hamblin (1973) and Karttunen (1977). However, in inquisitive semantics these propositions need not be mutually incompatible. Rather, the constraint that inquisitive semantics places on the set of propositions that make up the semantic value of a question is that it be downward closed. This means that if the set contains a certain proposition p, then it should contain every stronger proposition p ⊂ p as well. The rationale behind this constraint is simple: the propositions that make up the semantic value of a question are intended to be all and only those propositions that resolve the issue that the question expresses. If a proposition p resolves the issue expressed, then any stronger proposition p ⊂ p will resolve the issue as well.
Inquisitive semantics avoids the unique answer fallacy and has been argued to have several further advantages over previous frameworks as well (Roelofsen, 2013;Ciardelli, 2017;Ciardelli et al., 2018). However, the basic inquisitive semantics framework is static in nature. As such, it cannot capture anaphora with wh-antecedents, nor the intervention effects that Haida's dynamic theory accounts for. This motivates the development of a dynamic inquisitive semantics framework.
A first step in this direction was taken in Dotlačil and Roelofsen (2019). There, an elementary dynamic inquisitive semantics for a first-order logical language was presented, and its utility in analyzing the interaction between anaphora and questions was demonstrated. This framework, however, is still limited in a number of ways. Most importantly, it only provides a semantic interpretation of expressions in a first-order logical language, which means that it does not yet facilitate a compositional analysis of semantic interpretation in natural language, i.e., an analysis that makes precise what the semantic contribution is of every meaningful element of a sentence, and how these basic semantic building blocks are combined to form the semantic value of the sentence as a whole. To make this possible, a dynamic inquisitive semantics must be formulated for a suitable type-theoretic logical language, rather than just a first-order language.

Contributions
We aim to make two contributions. The first concerns framework building. We present a fullfledged dynamic inquisitive semantics for a suitable type-theoretic logical language, incorporating and extending insights from previous type-theoretic dynamic frameworks which were restricted to declarative sentences (Muskens, 1996;van den Berg, 1996;Brasoveanu, 2008, among others), and demonstrate how this framework can be used to formulate a compositional account of the semantic interpretation of declarative and interrogative sentences in English.
The second aim is to offer additional empirical support for a dynamic approach to questions. As discussed above, the empirical focus of previous dynamic theories of questions has been on anaphora and intervention effects. We will argue that there are other empirical phenomena which lend support to a dynamic approach as well. Specifically, we claim that a dynamic analysis of wh-questions in English, unlike existing static theories, can simultaneously satisfy all of the following desiderata: A) Non-stipulative derivation of mention-some and mention-all readings Wh-questions generally can have mention-some and mention-all readings. For instance, the question in (11) on its most prominent reading requests a specification of all people who were invited to the workshop, while (12) can be resolved by specifying just one good speaker, even if there are in principle multiple good candidates.
(11) Who has been invited to the workshop?
(12) Who would be a good speaker to invite for the workshop?
This flexibility in the interpretation of questions should be derived in a non-stipulative way, ideally relying on mechanisms that are independently needed elsewhere in the grammar.
B) Uniform treatment of single-wh and multiple-wh questions In many languages, including English, questions may involve multiple wh-expressions, as exemplified in (13).
Who ordered what?
The interpretation of single-wh and multiple-wh questions should be derived uniformly. That is, unless there is strong independent evidence to the contrary, a theory of questions should assume that single-wh and multiple-wh questions involve essentially the same basic semantic operations.
C) Partial mention-some readings in multiple-wh questions While the distinction between mention-some and mention-all readings in single-wh questions has received quite some attention, related distinctions in the interpretation of multiple-wh questions have not been discussed much at all. Usually, it is assumed that such questions only allow for a mention-all interpretation. We argue, however, building on an earlier observation by Xiang (2016), that in some cases multiple-wh questions also allow for a 'partial mentionsome reading', which requires an answer that is mention-some w.r.t. one of the wh-phrases but mention-all w.r.t. the others. Suppose, for instance, that 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: (14) Which of these herbs grows where?
On the most natural reading of this question, Mary can resolve it by mentioning just one location for each herb. She does not have to exhaustively specify all locations for every herb. A theory of questions should be able to derive such partial mention-some readings in multiple-wh questions.
D) Constraints on the availability of mention-some readings Questions with plural which-phrases, e.g., which participants, typically lack a mention-some reading. For instance, the question in (15)  A theory of questions should account for the fact that plural which-phrases generally block mention-some readings though not in the presence of existential modals. E) Uniqueness presuppositions due to singular number marking in which-phrases Questions with singular which-phrases, e.g. which girl, typically invoke a uniqueness presupposition. For instance, (18) presupposes that there is a unique girl who danced with Robin.
(18) Which girl danced with Robin? ; just one girl danced with Robin In multiple-wh questions, however, this uniqueness presupposition gets a twist. For instance, (19) does not presuppose that there is a unique girl and a unique boy who danced. It does presuppose that every boy who danced did so with a unique girl, but all in all, there could have been many boys and girls dancing (Higginbotham and May 1981, Dayal 1996, Dayal 2016. Which girl danced with which boy? ; just one girl and just one boy danced Another observation that needs to be captured is that the uniqueness requirement normally induced by a singular which-phrase can be obviated in questions involving an existential modal operator (Hirsch and Schwarz, 2019) or disjunction (Socolof et al., 2020). For instance, it is perfectly natural to ask (20) without assuming that there is a unique letter that can be inserted in fo m to make a word.
(20) Which letter can be inserted in fo m to make a word?
Similarly, (21) can be felicitously asked without assuming that there is a unique town in which Shakespeare was born or Bach died.
(21) In which town was Shakespeare born or did Bach die?
A theory of questions should therefore not just derive the uniqueness presupposition in questions like (18), but also the absence of such a presupposition in cases like (19), (20), and (21).
To our knowledge, no existing theory of questions meets all of these desiderata at once. For instance, 6 Groenendijk and Stokhof (1984) provide a uniform account of single-wh and multiple-wh questions, but their proposal does not successfully capture mention-some readings and the effects of number marking on wh-phrases. Dayal (1996) pays close attention to the effects of number marking, but her account does not successfully capture mention-some readings and does not provide a uniform treatment of singlewh and multiple-wh questions, in the sense that it takes such questions to involve different operators in the left periphery. Moreover, as pointed out by Xiang (2016Xiang ( , 2022 and discussed in more detail below, the so-called domain exhaustivity presupposition that Dayal's account predicts for multiple-wh questions is problematic. Fox (2012) provides a uniform account of single-wh and multiple-wh questions which captures the fact that singular which-phrases give rise to uniqueness presuppositions in single-wh but not in multiple-wh questions. However, Fox's account wrongly predicts that uniqueness presuppositions should also be present in cases like (20) and (21). Moreover, as discussed by Xiang (2016Xiang ( , 2022), Fox's account, just like Dayal's, derives the problematic domain exhaustivity presupposition for multiple-wh questions. In addition, while Fox (2013) derives mention-some readings for single-wh questions, it is not clear yet whether his approach could also capture partial mention-some readings in multiple-wh questions. Xiang (2016Xiang ( , 2022 captures mention-some readings and effects of number marking, but her account of single-wh and multiple-wh questions is not uniform, in the sense that it assumes special LF ingredients for multiple-wh questions which are not present in single-wh questions (this point is discussed in some more depth in Dotlačil and Roelofsen 2020, Appendix B).
We do not claim that it is impossible to further extend and refine currently available static theories to meet the above desiderata. Rather, our aim here is only to show that a dynamic theory can achieve this quite straightforwardly, and therefore has empirical advantages over existing static theories. To keep the paper of manageable length, we will restrict ourselves here to the development of a compositional dynamic inquisitive framework that does not include modal operators. Within this framework it will not be possible yet to provide a formally explicit account of the interaction between wh-elements and modal operators, such as exemplified in (17) and (20) above. While we believe that our framework, when extended with modal operators, could provide a natural account of such cases, we leave this for future work.
The paper is organised as follows. First, in Section 2 we provide a concise, informal presentation of the main ideas underlying our account. Then, in Sections 3-5 we present the account in greater detail. In Section 3, we develop a type-theoretic dynamic inquisitive semantics framework, Inq D . In Section 4, we spell out some basic assumptions about the syntax of declarative and interrogative sentences in English, and provide a compositional semantic analysis of simple declaratives and interrogatives in Inq D . Finally, in Section 5, we turn to more complex cases and show that the given dynamic account of questions satisfies the desiderata formulated above. Section 6 concludes and presents some avenues for future work.

Informal proposal
In this section we present the proposal informally. Many details are suppressed here, so as to provide direct access to the main ideas. The account rests on five key ideas, which are presented in Sections 2.1-2.5, respectively. A formal implementation will be provided in Sections 3-5.

Wh-phrases and the witness request operator
As already outlined in Section 1, at the core of our proposal is the idea that, first, the semantic contribution of a wh-phrase, just like that of indefinites in dynamic semantics, is to introduce a discourse referent u and to constrain the range of possible values of u in a certain way (e.g., to people, things, times, or places), and second, that wh-questions involve an update that raises an issue whose resolution requires identifying a set of individuals that satisfy the properties attributed to u, in short a witness set for u. This update is written as ?u and is contributed by an element in the left periphery of a wh-clause. We refer to ?u as the witness request operator.

Mention-all readings from maximization of witness sets
As is well-known, when indefinites serve as antecedents for so-called donkey anaphora, two readings can arise, strong and weak. In (22) the strong reading is most salient: the sentence conveys that whenever Bill eats one or more apples, he peels all of the apples he eats.
If Bill eats an u apple he peels it u first.
In (23) on the other hand, the weak reading is most prominent: whenever Bill parks his car and has one or more dimes in his pocket, he throws one of the dimes into the parking meter, not necessarily all of them.
(23) If Bill parks his car and has a u dime in his pocket, he throws it u into the parking meter.
This strong/weak ambiguity of donkey anaphora has been a central topic in the dynamic semantics literature (see, e.g., Heim, 1990;Kanazawa, 1994;Brasoveanu, 2008;Champollion et al., 2019). We will follow Brasoveanu (2008) here, who argues that the strong reading is derived by a witness maximization operator, max * {u}, in the interpretation of the indefinite. Intuitively, in (22) max * {u} ensures that all possible values of u (all the apples that Bill eats) are considered in the consequent. When max * {u} is absent, only some values might be considered, which captures the weak reading of (23).
If Brasoveanu (2008) is right that the weak/strong reading of donkey anaphora resides in an ambiguity of indefinites, and if the core idea behind our dynamic approach to questions-namely, that wh-phrases behave semantically just like indefinites-is correct as well, then we should expect that wh-phrases, just like indefinites, can appear with or without the max * operator as well. Recall the simple example in (7) above, repeated in (24), and the basic dynamic analysis of this question in (25).
(25) [u]; person{u}; f ainted{u}; ?u If wh-phrases are treated as optionally contributing a max * operator, then we predict that (24) cannot only be interpreted as in (25) but also as in (26).
(26) [u]; person{u}; f ainted{u}; max * {u}; ?u On this analysis, resolving the question requires identifying the maximal witness set for u. This corresponds to the mention-all reading of the question. On the other hand, the analysis in (25), without the max * {u} operator, captures the mention-some reading. 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. In other words, the ambiguity between mentionsome and mention-all readings can be derived in a principled way, relying on a mechanism that is independently needed elsewhere in the grammar. 7

Scaling up to multiple-wh questions
The account sketched so far is about single-wh questions but can be generalized in a natural way to deal with multiple-wh questions as well. We propose that in a multiple-wh question, each wh-phrase introduces a discourse referent (dref for short), and that the witness request operator asks for a functional witness in this case, i.e., a function mapping the possible values for one dref to the possible values of another. For instance, the multiple-wh question in (27a) is analyzed as in (27b), where ?u 1 u 2 is a functional witness request operator. Informally, ?u 1 u 2 raises an issue whose resolution requires determining, for every possible witness set of u 1 , the corresponding witness set of u 2 . Or, in other words, a suitable function from the witness sets of u 1 to those for u 2 . We will see in Section 3 that both ?u and ?u 1 u 2 can be seen as 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 A formulated on page 6.

Partial 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. 8 For instance, to resolve the question in (28), one must specify, for each party guest, everything that they ate (rather than just something that they ate).
Who ate what at the party last night?
We have observed, however, that in some cases multiple-wh questions also allow for a partial mention-some reading. This was exemplified in (14), repeated in (29).
(29) Which of these herbs grows where?
This question can, in a suitable context, be resolved by mentioning just one location for each 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 however that, while (29) is mention-some w.r.t. locations on the relevant reading, it is still 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.
One might think that the mention-all reading with respect to herbs is a consequence of the fact that the 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 whphrases which favor a mention-some reading in single-wh questions. For instance, while (30a) has a very salient mention-some reading, (30b) only has a reading that is mention-some w.r.t. one of the wh-phrases, not w.r.t. both.
(30) a. What is a typical French dish? b. What is a typical dish where?
In English, this appears to be the case in general: multiple-wh questions always require what we will call domain maximality-their resolution must map as many potential witnesses of the 'domain dref' as possible to corresponding witnesses of the 'target 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 witness of the 'target dref'. Xiang (2019) shows that this requirement is too strong, based on the following scenario. 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: 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. 9 However, an additional observation to make about this example 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. In our formal 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 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, our account will have to be refined. 9 Xiang's intuition that domain exhaustivity does not need to be satisfied in multiple-wh questions has been confirmed experimentally in Martin (2021). max * operator applies to the domain dref or not. Our account will derive two readings for multiplewh questions: one that is mention-all with respect to all wh-phrases, and one that is mention-all with respect to the 'domain wh-phrase(s)' but mention-some with respect to the 'range wh-phrase'. 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. On such accounts, it is not possible to derive partial mention-some readings of multiple-wh questions (at least not without substantial modifications). To our knowledge, the only previous account that derives such readings is that of Xiang (2016, §5.4.3).

Effects of number marking
Several effects of number marking need to be captured but the main puzzle is that singular whichphrases induce a uniqueness presupposition in single-wh questions but allow for multiple witnesspairs in multiple-wh questions (e.g., Dayal, 1996).
; just one girl danced with Robin (33) Which girl danced with which boy? ; just one girl and just one boy danced We will assume, following Brasoveanu (2008) and much other work on number marking in dynamic semantics, that singular number marking, e.g., on which girl, invokes an atomicity requirement. This ensures that all the possible values assigned to the dref introduced by which girl are atomic individuals. Furthermore we will assume, following Roelofsen (2015), that every interrogative clause involves a presuppositional closure operator, †, which requires that it is already known in the input context that the actual world lies in one of the alternatives that the question introduces (see also Abusch, 2010). This presupposition is trivially satisfied whenever the alternatives cover the entire logical space, but not when the alternatives cover only part of the logical space. A question like (32), repeated in (34a), is then analyzed as in (34b): As we will see in detail in Section 5, a uniqueness presupposition is derived in this case from the interplay between atom{u}, max * {u}, ?u, and †. A particularly important role is played by ?u. This operator removes from the context any possibilities where more than one girl dances. For a multiple-wh question we get: A global uniqueness requirement is not derived here because, instead of ?u, the two-place witness request operator ?u 1 u 2 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 function mapping each girl who danced with a boy to the unique boy she danced with. As a result, the account derives a 'relativized' uniqueness presupposition to the effect that no girl danced with more than one boy.
This concludes the informal presentation of the main ideas behind our approach. We will now start to work out these ideas more rigorously.

Contexts
In dynamic semantics the meaning of a sentence is viewed as its context change potential. One common way to formalize this idea is to take the semantic value of a sentence to be a function that maps any input context (the context in which the sentence is uttered) to a corresponding output context (the new context after the utterance). A central question, then, is how such contexts are to be modelled.

Contexts in basic inquisitive semantics
We start by considering the notion of context in Inq B , the basic, static inquisitive semantics framework. In Inq B , contexts are intended to represent information and issues. For instance, in a certain context it may be known that Bill is in Paris, and it may be an open issue whether Sue is in Paris too. To capture both the information that has been established and the issues that have been raised, contexts are formally modelled in Inq B as sets of information states, where each information state in turn is a set of possible worlds. The information states that make up a context c are precisely those that (i) contain enough information to resolve the issues that have been raised in the conversation so far and that (ii) do not contain any possible worlds that are already ruled out by the information established in the conversation so far (Ciardelli et al., 2018). For instance, the context informally described above consists of (a) information states s such that all worlds in s are ones in which Bill and Sue are both in Paris, as well as (b) information states s such that all worlds in s are ones in which Bill is in Paris but Sue isn't. All these information states support the information that Bill is in Paris and resolve the issue whether Sue in Paris.
The union of all the elements of a context c, ∪c, is the set of all worlds that are compatible with the information available in c. 10 This union is written as info(c). In our example, info(c) is the set of all worlds in which Bill is in Paris.
Not just any set of information states constitutes a proper context representation in Inq B . Rather, this only holds for sets of information states that are downward closed : if they contain a certain information state s, they must also contain any stronger information state s ⊂ s. This requirement follows from how contexts are construed. To see this, suppose that a context c contains an information state s. This means (i) that s contains enough information to resolve the contextual issues and (ii) that s does not contain any possible worlds that are ruled out by the information available in c. But then, the same goes for any stronger information state s ⊂ s, which in turn means that these stronger information states must also be in c.
Furthermore, it is assumed in Inq B that the inconsistent information state, ∅, trivially resolves any issue and is therefore contained in any context representation. Given the requirement that contexts are downward closed, the additional requirement that any context contains the inconsistent information state is equivalent to the requirement that any context be non-empty.
In sum, contexts are modeled in Inq B as non-empty, downward closed sets of information states, which in turn are sets of possible worlds. This allows us to represent both the information that has been established and the issues that have been raised.
However, a limitation of the context representations in Inq B is that they only capture information and issues about the world. They do not capture discourse information. In particular, they do not keep track of which discourse referents have been introduced, let alone of the information that has been established and the issues that have been raised about these discourse referents.
How should the context representations of Inq B be enriched in order to capture such discourse information? Our proposal is to keep the general Inq B notion of contexts as downward closed sets of information states intact, but to refine the notion of an information state. In Inq B , information states are simply sets of possible worlds. In dynamic semantics, however, more fine-grained models of information states have been developed, with the intention to capture both world information and discourse information. We will first discuss, in Section 3.1.2, a very simple dynamic model of information states, based on Groenendijk et al. (1996) and other work in dynamic semantics. Incorporating this model of information states into inquisitive semantics leads to a basic dynamic inquisitive framework, Inq D B , which we have presented in Dotlačil and Roelofsen (2019). In the present article, however, we will work with a more general dynamic model of information states, based on van den Berg (1996); Nouwen (2003) and Brasoveanu (2007Brasoveanu ( , 2008. This model will be presented in Section 3.1.3, and the inquisitive notion of contexts that we build on top of it in Section 3.1.4.

A simple dynamic model of information states
It is quite common in dynamic semantics to model an information state as a set of pairs w, g , where w is a possible world and g an assignment function mapping all discourse referents introduced so far to some concrete individual (Groenendijk et al., 1996, among others). Such pairs are called possibilities in Groenendijk et al. (1996). A context c formally represented as a set of possibilities captures both information about the world, as well as information about discourse referents. Finally, it also captures possible dependencies between these two types of information. To give an example of a dependency, it may be known that the first discourse referent denotes either Alice or Kim, and that the actual world is one in which the individual denoted by this discourse referent, whoever it is, is the tallest woman on earth. Further, it may be known that the individual denoted by the second discourse referent is a friend of the woman denoted by the first discourse referent (Sue for Alice and Mary for Kim). In this context, the value of the second discourse referent depends on that of the first, and the value of the first discourse referent depends on what the world is like.
Let us briefly illustrate how these dynamic information states are updated as a discourse proceeds. Consider the following mini-discourse: A u 1 woman came in. She u 1 sat down.
Suppose that the domain of discourse consists of two individuals, a and b, and that when the discourse starts our information state consists of three possibilities w 1 , g 1 , w 2 , g 2 , and w 3 , g 3 . Since no discourse referents have been introduced yet, we have that g 1 = g 2 = g 3 = ∅. Suppose that w 1 , w 2 and w 3 are as follows: w 1 : a and b are both women, they both came in, only a sat down.  The interpretation of the first sentence, A u 1 woman came in, involves two steps. The first step is the introduction of a new discourse referent u 1 . This means that each possibility w i , ∅ in our initial information state is replaced by two new possibilities w i , u 1 → a and w i , u 1 → b , whose world component is the same as before, but whose assignment function maps the new discourse referent u 1 to a or b, respectively. This first step in the interpretation of the sentence thus leads to the following information state: The second step in the interpretation of A u 1 woman came in is to eliminate all possibilities w, g from our information state which are such that the individual assigned to u 1 by g is not a woman who came in according to w. More concretely, the possibilities w 3 , u 1 → a and w 3 , u 1 → b are eliminated because according to w 3 , neither a nor b came in. This leaves us with the following information state: The second sentence, She u 1 sat down, places further constraints on the possibilities in our information state. Namely, it eliminates possibilities w, g which are such that the individual assigned to u 1 by g did not sit down according to w. This leads to the following information state: This information state captures that we have obtained the information that the actual world is either w 1 or w 2 , and also that the denotation of the discourse referent u 1 depends on what the actual world is. If we are in w 1 , then u 1 denotes a; if we are in w 2 , then u 1 denotes b. So, again, we see that information states, modelled as sets of possibilities, capture information about the world, discourse information, and dependencies between them. The example also highlights another aspect of how information states are updated. Namely, it exemplifies that there are two kinds of updates: constructive updates, which introduce new discourse referents and create new possibilities, and eliminative updates, which remove possibilities.
As noted above, an inquisitive semantics based on this simple dynamic notion of information states has been developed in Dotlačil and Roelofsen (2019). We now turn to a more general dynamic model of information states which allows us to deal with pluralities, including pluralities in questions. Initial empirical motivation for this more general model came from so-called dependent anaphora. An example of a dependent anaphoric expression is the pronoun it u 1 in (40). This pronoun does not refer to a particular paper. Rather, for every student, it refers to the paper written by that student.
(40) Every student wrote a u 1 paper. Most of the students submitted it u 1 to a journal.
As has been argued by Krifka (1996) and van den Berg (1996), dependent anaphora require a non-trivial treatment of the way plural entities are assigned to discourse referents (cf. Nouwen 2003Nouwen , 2007 for further refinements). It turned out that this analysis of dependent anaphora is also useful in accounting for donkey anaphora (Brasoveanu, 2007), adjectives such as different and same (Brasoveanu, 2011;Brasoveanu and Dotlačil, 2012), reciprocals (Dotlačil, 2012), and dependent indefinites (Henderson, 2014;Kuhn, 2017). We will argue in the course of this target article that the more sophisticated model also has significant advantages, when incorporated in inquisitive semantics, in the analysis of questions.

A more general dynamic model of information states
In the simple model above, an assignment function g specifies a simultaneous instantiation of the active drefs ('simultaneous' in the sense that it specifies an instantiation of each dref). When a new discourse referent u 1 is introduced, new possibilities are created, each involving a randomly picked instantiation of u 1 . To determine whether a given possibility w, g survives a certain eliminative update we have to look at the individuals specified by g as instantiating the active drefs and check whether they satisfy certain properties in w.
The important thing to note is that possibilities are always taken to involve a single instantiation of the active drefs. In particular, in constructive updates, newly created possibilities always involve a single instantiation of the newly introduced dref, and in eliminative updates, determining whether a given possibility survives the update only requires us to check a single instantiation of the drefs.
A natural way to generalize this model is to drop the assumption that every possibility involves a single simultaneous instantiation of the active drefs. Instead, as proposed in van den Berg (1996) and Brasoveanu (2007Brasoveanu ( , 2012) (see also Väänänen 2007), we may allow for multiple simultaneous instantiations as well. Formally, this means that a possibility is modelled as a pair w, G where w is a possible world and G a set of assignment functions, each specifying a simultaneous instantiation of the active drefs. We will refer to such sets of assignments G as assignment matrices, because a set of assignment functions can be displayed as a matrix, where each row lists the values assigned by a specific assignment function to the various drefs, and each column lists the values assigned to a specific dref by the various assignment functions. An example of a dref assignment matrix consisting of three assignment functions, g 1 , g 2 , g 3 , each assigning values to three active drefs, u 1 , u 2 , u 3 , is given in (41): Once we generalize the notion of possibilities in this way, going from pairs w, g to pairs w, G , a broader range of natural update functions becomes available. Consider constructive updates, which introduce new discourse referents. In the simple model above, it was assumed that each new possibility that such an update creates involves a single instantiation of the newly introduced drefs. Indeed, this is the only natural option we have in that simple model. In the generalized model, however, there are other natural options as well. For instance, while certain constructive updates may create possibilities whose assignment matrices only need to specify some possible instantiations of the newly introduced drefs, other constructive updates may only generate possibilities whose assignment matrices specify all possible instantiations of the newly introduced drefs. Brasoveanu (2007) makes crucial use of these two types of constructive updates, existential and universal, to account for weak and strong interpretations of donkey anaphora, respectively. We will make use of it as well in accounting for mention-some and mention-all question interpretations.

Contexts in Inq D
We define contexts in Inq D exactly as they are defined in static inquisitive semantics, Inq B , namely as non-empty downward closed sets of information states. However, the simple static notion of information states as sets of possible worlds is now replaced by the dynamic notion of information states defined above. That is, information states are modelled as sets of possibilities, where each possibility consists of a possible world and an assignment matrix. We think of these information states, just like in Inq B , as those that (i) contain enough information to resolve the contextual issues, and (ii) do not contain any possibilities that are ruled out by the contextually established information. Only now, the contextual issues and the contextually established information may concern not only what the world is like but may also pertain to the discourse referents that have been introduced. This way, our new notion of context encompasses both information and issues about the world and about discourse referents.

Formal definitions
We now provide explicit formal definitions of the notions discussed above in a type-theoretic logical framework.
Types and frames. First, let us fix the set of types that we assume. There are basic types and complex types. As usual, we assume a basic type e for entities, a basic type s for possible worlds, and a basic type t for truth values. In addition, we also assume a basic type r for discourse referents. 11 Complex types are constructed from these four basic types in two ways. First, for every two types σ and τ , (στ ) is also a type, namely the type of functions from objects of type σ to objects of type τ . Besides these functional types, which are standardly assumed in type-theoretic semantics, we will also include so-called relational types: for every two types σ and τ , (σ × τ ) is also a type, namely the type of pairs whose first element is of type σ and whose second element is of type τ (Muskens, 1995;Carpenter, 1997).

Definition 1 (Inq D types).
Inq D has four basic types: e for entities, s for possible worlds, t for truth values, and r for discourse referents; 12 The set of all Inq D types Types is the smallest set containing the four basic types, and which is such that for any two types σ ∈ Types and τ ∈ Types, we have that (στ ) ∈ Types and (σ × τ ) ∈ Types as well. We will often omit brackets around complex types.
An Inq D frame F , based on a given set of individuals I, specifies for every type τ ∈ Types a set D F τ , which is called the domain of objects of type τ in F . We will often omit the F index on the domains in F , and simply write D τ . The domain of entities D e consists of non-empty subsets of I.
Definition 2 (Inq D frames). An Inq D frame based on a set of individuals I is a set {D τ | τ ∈ Types} such that: D e , D s , D t , and D r are pairwise disjoint sets. D e is the set of all non-empty subsets of I. That is, D e = ℘(I) \ ∅. D s is a non-empty set of objects that we refer to as possible worlds.
D r is a non-empty set of objects that we refer to as discourse referents.
For any functional type στ ∈ Types, D στ is the set of all functions from D σ to D τ .
For any relational type σ × τ ∈ Types, D σ×τ is the set of all pairs in D σ × D τ .
Singleton sets in D e are called atomic entities; non-singleton sets are called plural entities. The sum of two (atomic or plural) entities a, b ∈ D e is defined as their union and is written as a⊕b. Similarly, the sum of a set of (atomic or plural) entities E ⊆ D e is defined as their union and written as ⊕E. The part-of relation, ≤, over elements of D e is the partial order induced by set inclusion, and the proper part-of relation, <, holds between non-identical entities that satisfy the part-of relation.
We refer to elements of D r as discourse referents, and to functions from discourse referents to entities in D e as dref assignment functions. A dref assignment function can be a partial function. That is, it does not necessarily map all discourse referents to an entity. The set of discourse referents δ ⊆ D r that it does map to an entity is called its domain. We refer to a set of dref assignment functions that have the same domain δ as a dref assignment matrix with domain δ. We will abbreviate the type (re)t of dref assignment matrices as m. All abbreviations used for types are summarized in Table 1.
Definition 3 (Dref assignment functions and matrices). Let F be an Inq D frame and δ ⊆ D r a set of discourse referents in F . Then: A function in D re with domain δ is called a dref assignment function with domain δ; A non-empty set of dref assignment functions with domain δ is called a dref assignment matrix with domain δ.
Note that dref assignment matrices are by definition non-empty. In the initial context of a conversation, when no drefs have been introduced yet, we assume that the dref assignment matrix is {∅}, i.e., a singleton set containing only an empty assignment function, rather than a set containing no assignment functions at all. This will make some notions introduced below easier to define in full generality.
Definition 4 (Collecting the values of a discourse referent). If G is a dref assignment matrix with domain δ and u a discourse referent in δ, then: Possibilities, information states, and contexts. Given these preliminary notions we can now formally define the notions that played a central role in the informal discussion above-possibilities, information states, and contexts. From now on, we assume a given Inq D frame F . This means, for instance, that when we say 'let δ be a set of discourse referents' we mean a set of discourse referents in F .

Definition 5 (Possibilities).
For any set of discourse referents δ, a possibility with domain δ is a pair w, G , where w is a possible world and G a dref assignment matrix with domain δ. Possibilities are thus of type s × m.
An information state is a set of possibilities.
Information states are thus of type (s × m)t. We abbreviate this type as i.
In order to define contexts we need one auxiliary notion, that of downward closed sets of information states.
Definition 7 (Downward closed). A set S of information states is downward closed just in case for every information state s ∈ S, every more informed information state s ⊂ s is also in S.

Definition 8 (Contexts).
A context is a non-empty, downward closed set of information states.
Contexts are thus of type (it). We abbreviate this type as k.
Definition 9 (The domain of an information state and a context).
The domain of an information state s is the union of the domains of the possibilities in s.
The domain of a context c is the union of the domains of the information states in c.

Informed and inquisitive contexts
As in Inq B , the union of all the information states in a context c, ∪c, is the set of possibilities that are compatible with the information available in c. We will continue to write info(c) for ∪c.
Definition 10 (The information available in a context). For any context c, info(c) := ∪c.
A context with domain δ contains non-trivial information if there is a possibility w, G with domain δ which is not in info(c). In this case, we think of the possibility w, G as having been excluded by the information that has been established in c. If no possibilities have been excluded yet, then the context does not contain any non-trivial information (yet).
A context c with domain δ is informed just in case there is a possibility w, G with domain δ such that w, G ∈ info(c).
Otherwise, c is called uninformed.
Resolving the open issues in a context c (if there are any) requires extending the contextually available information, represented by info(c), in such a way as to reach one of the information states in c. In case info(c) is itself an element of c, all the contextual issues are already resolved, i.e., there are no open issues in c. In this case, we say that c is non-inquisitive. On the other hand, if info(c) ∈ c we say that c is inquisitive.

Definition 12 (Inquisitive and non-inquisitive contexts).
A context c is inquisitive just in case info(c) ∈ c.
Otherwise, c is non-inquisitive.
Given a context c, we can always construct a context !c which contains exactly the same information as c, both about the world and about the discourse referents introduced so far, but is not inquisitive. This is achieved by letting !c consist of the information state info(c) plus all subsets thereof. We will refer to this context, again using terminology familiar from Inq B , as the non-inquisitive closure of c. In the definition below we use a downarrow to represent closure under subsets, i.e., for any set of information states S, S ↓ := {s | s ⊆ s for some s ∈ S}.
Definition 13 (Non-inquisitive closure). For any context c, !c := {info(c)} ↓ Since contexts are always downward closed they can often be represented by means of their maximal elements. 13 These elements are information states which contain just enough information to resolve the contextual issues. Non-maximal elements also contain enough information to resolve these issues, but they contain more information than is strictly needed to do so. The maximal elements of a context representation are referred to in inquisitive semantics as the alternatives that are under consideration in that context. To resolve the contextual issues, one of these alternatives must be established.
Definition 14 (Alternatives). An alternative in a context c is an information state s such that s ∈ c and there is no strictly weaker information state t ⊃ s such that t ∈ c.
It generally holds that if a context c contains more than one alternative, it is inquisitive. Vice versa, if a context is non-inquisitive, then it contains only one alternative. 14 Finally, we define trivial contexts, the initial context and the inconsistent context as follows.

Definition 15 (Trivial contexts).
A context is trivial just in case it is neither informed nor inquisitive.
Definition 16 (The initial context c ). The initial context c is the trivial context whose domain is empty.
Definition 17 (The inconsistent context c ⊥ ). The inconsistent context c ⊥ := {∅} only contains the inconsistent information state.

Context extension and subsistence
When exactly should one context be seen as an extension of another? In Inq B , where information states are plain sets of worlds, the answer to this question is straightforward, namely, c is an extension of c if and only if c ⊆ c. This guarantees not only that c contains at least as much information as c but also that the open issues in c subsume those in c, in the sense that any piece of information that resolves the open issues in c also resolves those in c.
In Inq D , however, where information states are sets of possibilities rather than sets of possible worlds, set inclusion is not the right notion of context extension. To see this, consider the following two contexts: where the domain of g is {u}, that of g is {u, u }, and g agrees with g on the value of u, i.e., g (u) = g(u). Both c and c carry the same information about the world. The only difference is that c pertains to a situation in which there is a single discourse referent, u, whereas c pertains to a situation in which there is another discourse referent, u , as well. In this case we would like to say that c is an extension of c. After all, all the information available in c is also available in c , together with additional information about the discourse referent u . Yet, c ⊆ c. This shows that set inclusion is not the right notion of context extension in Inq D . Below we provide a notion of context extension that does suit our purposes, building on Groenendijk et al. (1996). In order to define this notion, we first specify what it means to extend a dref assignment function and dref assignment matrix, a possibility, and an information state.
A dref assignment function g is an extension of another dref assignment function g if and only if g ⊇ g. Note that for this to hold it is necessary (but not sufficient) that the domain of g contains the domain of g.
A dref assignment matrix G is an extension of another dref assignment matrix G if and only if every g ∈ G is an extension of some g ∈ G and every g ∈ G is extended by some g ∈ G .
In this case we write G ≥ G.
Definition 19 (Extending possibilities). A possibility w , G is an extension of another possibility w, G if and only if w = w and G ≥ G. In this case we write w , G ≥ w, G .
Definition 20 (Extending information states). An information state s is an extension of another information state s if and only if every possibility in s is an extension of some possibility in s. In this case we write s ≥ s.

Definition 21 (Extending contexts).
A context c is an extension of another context c if and only if every information state in c is an extension of some information state in c. In this case we write c ≥ c.
It is useful to also define a specific kind of context extension, one which only involves the addition of discourse information. In Groenendijk et al. 1996 this is called subsistence and we will use the same term here.
Definition 22 (Subsistence of one information state in another). Let s, s be information states such that s ≥ s. Then we say that s subsists in s if and only if every possibility in s has an extension in s .
Definition 23 (Subsistence of an information state in a context). Let s be an information state and c a context. We say that s subsists in c if and only if there is one or more s ∈ c such that s subsists in s . We call all s in c that satisfy this condition descendants of s in c.
Definition 24 (Subsistence of one context in another). Let c, c be two contexts such that c ≥ c. Then we say that c subsists in c if and only if every state in c subsists in c .

A basic formal language and its interpretation
The central formal notion introduced so far is that of Inq D frames, type-theoretic frames in which, besides the usual basic types e, s, and t, we also have a basic type r for discourse referents. All other notions (possibilities, information states, contexts, etc.) were defined in terms of such frames.
In principle, we could now turn to natural language and develop a semantic theory which maps expressions of, say, English to objects in our type-theoretic frames. However, following Montague (1973) and many others, we will take a more indirect approach. That is, we will first introduce a logical language, and a semantics that maps expressions in this language to objects in our typetheoretic frames. Once this logical language will be in place, we will specify how expressions in a fragment of English, given a certain syntactic analysis, can be translated into our logical language.
We will introduce the logical language and its semantic interpretation in two steps. First, in the remainder of the present subsection we will specify a basic, mostly familiar, type-theoretic logical language and its interpretation in terms of Inq D frames. The main novelty at this point will be that there are individual constants of type r, denoting discourse referents, and that there are dref assignment functions which assign values to constants of type r. In Section 3.3 we will add several pieces of new vocabulary to the language. This new vocabulary will all be definable in terms of our basic vocabulary, so it does not extend the expressive power of the language, but it will allow for a more transparent mapping from English expressions to expressions in our logical language.
The basic logical language We define a type-logical language L, in which every expression is of a certain type τ ∈ Types. We write Expr τ for the set of all expressions of type τ . Expressions are built out of constants and variables, which also each have a type. We write Con τ for the set of all constants of a given type τ , and Var τ for the set of all variables of type τ .

Definition 25 (Constants and variables in L).
For any type τ ∈ Types, L contains a set of constants of type τ , Con τ , and a set of variables of type τ , Var τ . Some constants and variables in L are specified in Table 2.  Definition 26 (Expressions in L). L is the smallest set such that for any σ, τ ∈ Types: If α ∈ Expr σ and β ∈ Expr τ , then α, β ∈ Expr σ×τ 7. If α ∈ Expr σ×τ , then π 1 (α) ∈ Expr σ and π 2 (α) ∈ Expr τ 8. If R ∈ Expr s(e n t) , then *R ∈ Expr s(e n t) as well 15 9. If G ∈ Expr m and u ∈ Expr r , then G(u) ∈ Expr et 10. If α, β ∈ Expr e , then α ⊕ β ∈ Expr e and α < β ∈ Expr t 11. If α ∈ Expr et , then ⊕α ∈ Expr e 12. If g ∈ Expr re , then dom(g) ∈ Expr rt 13. If g, g ∈ Expr re and u ∈ Expr r , then g[u]g ∈ Expr t 14. If G, G ∈ Expr m and u ∈ Expr r , then G[u]G ∈ Expr t 15. If p, p ∈ Expr s×m and u ∈ Expr r , then p[u]p ∈ Expr t Thus, our basic language L is built on a standard type-theoretic language with functional and relational types (generated by the first seven rules). L also includes several additional elements that are less commonly used (rules 8 to 15) and whose interpretation will be explained in the following paragraphs. 16

Semantic interpretation
We now turn to the semantic interpretation of the expressions in L. First, we define interpretation functions and variable assignment functions in the usual way, i.e., as determining the denotation of constants and variables, respectively.

Definition 27 (Interpretation functions).
Given an Inq D frame F , an interpretation function into F is a function which, for any type τ ∈ Types, maps every constant α ∈ Con τ to some object in D F τ .
Given an Inq D frame F , a variable assignment function into F is a function which, for any type τ ∈ Types, maps every variable v ∈ Var τ to some object in D F τ .
If θ is a variable assignment function into a frame F , v ∈ Var τ a variable of some type τ ∈ Types, and d ∈ D F τ an object of type τ in F , then we write θ v/d for the assignment function which is identical to θ except that it maps v to d.
We write [[α]] F,I,θ for the semantic value of an expression α ∈ L relative to a frame F , an interpretation function I into F , and a variable assignment function θ into F . [[α]] F,I,θ is recursively defined as follows.
Definition 29 (The semantic value of an expression). Let F be an Inq D frame, I an interpretation function into F , θ a variable assignment function into F , and σ, τ ∈ Types. Then: The semantic rules in 1-11 are standard. Let us comment on those in 12-23. Rule 12 says that the semantic value of a pair α, β (e.g., a possibility) is, as expected, a pair whose first element is the semantic value of α and whose second element is the semantic value of β. Rules 13 and 14 specify that if α is a pair, π 1 (α) denotes the first element of the pair and π 2 (α) its second element.
Rule 15 says that the denotation of *R is obtained by closing the denotation of R under sums of entities, which are constructed using ⊕. That is, whenever two entities a and b are in the extension of R, then a, b, as well as a ⊕ b are in the extension of *R. Closure of relations under sum formation is common when the domain consists of both plural and atomic entities, as is the case here (see Krifka, 1989;Landman, 2000;Kratzer, 2008). The * operator is usually called the pluralization operator.
Rule 16 says that the semantic value of G(u) is the collection of all those entities that are assigned to u by some element of G. For instance, if G = {g 1 , g 2 }, g 1 maps u to a, and g 2 maps u to b, then the semantic value of G(u) is the set {a, b}.
Rules 17-19 specify the interpretation of ⊕ and < in the logical language. These are the same as in the meta-language: ⊕ is used to 'merge' two entities, or an arbitrary set of entities, into a single, plural entity. If α and β are two entity-denoting expressions then α < β is true just in case all the individuals that α consists of are also part of β.
Rule 20 defines the domain of a dref assignment function. Rules 21-23 will be used to define update functions that introduce new discourse referents. Rule 21 specifies the interpretation of g [u]g , which says that two dref assignments, g and g , differ from each other at most with respect to the value assigned to a, possibly newly introduced, dref u. Rules 22 and 23 extend Rule 21 to dref assignment matrices and possibilities, respectively. The definition in rule 22 follows Brasoveanu (2007) and subsequent dynamic systems that operate with dref assignment matrices. It is assumed that G differs from G at most w.r.t. u if and only if for all g in G there is a corresponding g in G that differs from g at most w.r.t. u, and vice versa, for all g in G there is a corresponding g in G such that g differs from g at most w.r.t. u. Finally, rule 23 says that the possibility p differs from the possibility p at most w.r.t. u if and only if they have the same world (the first projection) and the dref assignment matrix of p differs from that of p at most w.r.t. u.
It may be useful at this point to highlight the difference between variables and discourse referents in our system. Variables are expressions in our logical language, while discourse referents are objects in a frame. Discourse referents are mapped to entities in D e by dref assignment functions, which are themselves objects in D re . Variables of type e are mapped to entities in D e by variable assignment functions. These functions are not part of a frame, but specify a mapping from elements of the logical language to elements of the frame. Finally, note that our logical language contains expressions u, u , . . . whose semantic values are discourse referents. However, these expressions are not discourse referents themselves. Neither are they variables. Rather, they are constants. Their semantic value is not determined by a variable assignment function, but by an interpretation function. These notions are all depicted in Figure 1.

Extending the basic formal language with update expressions
We now extend the basic logical language introduced above with expressions that will make it easier to translate English sentences into the logical language. These additional expressions in the logical language will all be of type T, which means that they all denote context update functions. We will therefore refer to them as update expressions.

Predicational and relational update expressions
We start with expressions specifying that a certain dref has a particular property, e.g., smile{u}, or that two or more drefs stand in a particular relation to each other, e.g., see{u 1 , u 2 }. 17 Semantically, smile{u} denotes a context update function. This function takes an input context c and returns a new context, which is the set of information states s in c such that for every possibility p in s and every assignment g in G p , 18 it holds that g(u) is in the extension of *smile in w p . We adopt the common assumption that lexical predicates like smile are closed under sums, i.e., they are cumulative (Krifka, 1989;Landman, 2000;Kratzer, 2008). In (42), this is captured by the * preceding the predicate. The semantic interpretation of *R was given in Section 3.2. 19 The predicational update function smile{u} can be generalized to the n-ary relational update function R{u 1 , . . . u n } in (43): To illustrate the effects of update functions, we will use diagrams like the one in Figure 2 for smile{u}. In this diagram, the context on the left is the input context and the one on the right is the output context. Recall that a context is a downward closed set of information states. In diagrams, we always depict only the maximal information states in the context, i.e., the alternatives. In Figure 2, both the input context and the output context contain just one alternative. The grey arrow in the middle of Figure 2 represents the transition from input to output context due to the update function expressed by smile{u}. The update eliminates all states in the input context containing at least one possibility according to which one or more entities assigned to u do not smile. Note that the update does not eliminate possibilities w a,b , G which are such that all assignments in G map u to a ⊕ b (see the second column, fourth line in the output context). This is a consequence of closure under sums, which lets the plurality a ⊕ b be in the extension of *smile if a and b are in the extension of smile.

Conjunction
Update functions can be combined by means of conjunction. As usual in dynamic semantics, we take conjunction to express function composition: the input context is first updated by the first conjunct and then by the second.
(44) U T ; U T := λc k . U (U(c)) 17 We use curly brackets in these expressions rather than round ones to flag that predicates like smile and see do not directly apply to drefs (smile is not of type rt but rather of type s(et)). Context update expressions like smile{u} and see{u1, u2} are abbreviations of complex expressions in the basic logical language, as defined below.
18 Here and throughout we use wp as an abbreviation of π1(p) and Gp as an abbreviation of π2(p). 19 When u has not been introduced yet in a context c, then the result of applying R{u} to c is undefined. This is a case of unresolved anaphora. We do not provide an explicit analysis of how such undefinedness projects, i.e., of how it affects the semantic value of complex expressions containing R{u}, because this is orthogonal to our main concerns here and adding an explicit account of undefinedness projection would significantly complicate the presentation of the framework. As far as we know it is possible to combine Inq D with dynamic systems that deal with undefinedness, like partial CDRT (Haug, 2013) and dynamic semantics with stacks (Vermeulen, 1993;Nouwen, 2007;van Eijck and Unger, 2010). Figure 2: The update effect of smile{u}. In this and the following diagrams, subscripts on worlds specify which individuals smile in that world.

Introducing new discourse referents
We write [u] for the update function that introduces a new discourse referent u. Loosely speaking, [u] updates a state in the input context by 'randomly assigning' any possible values to u across the dref assignment matrices of the possibilities in the state. The formal definition, given in (45), makes use of the logical vocabulary p[u]p , which was defined in Sect. 3.2: it expresses that the dref assignment matrix in the possibility p differs from that in p at most w.r.t. u. The effect of the update function [u] is illustrated in Figure 3. In the initial context, no dref is present yet. The dref assignment matrix in each possibility is a singleton set containing an empty dref assignment function. After the update, every possibility in every state assigns some entity to u. 21 If the context is further updated with smile{u}, it is restricted to states consisting of possibilities with dref assignment matrices assigning only entities to u that smile.

Disjunction
Disjunction is defined in a similar way as in basic (static) inquisitive semantics. We assume that the result of applying a disjunction of two update functions to a context c is obtained by applying the two update functions separately to c and then forming the union of both output contexts: 20 Readers familiar with inquisitive semantics might notice that the introduction of discourse referents does not introduce inquisitiveness in Inq D , in contrast to existential quantification in Inq B .
21 Recall that we assume that there are only two atomic entities a and b in the domain of discourse. The input context assigns no value to any dref. It is assumed here that D e contains only one entity, a. w sing is a world in which a sings, w smile a world in which a smiles, w sing,smile a world in which a does both, and w ∅ a world in which a does neither.
Disjunctions can turn a non-inquisitive context into an inquisitive one. This is illustrated in Figure 4, where the input context contains a single alternative, but the output context contains two alternatives, each corresponding to one of the disjuncts. In dynamic semantics, it is often assumed that disjunction is 'internally dynamic' and 'externally static' (see, e.g., Groenendijk and Stokhof, 1991;Kamp and Reyle, 1993;Groenendijk et al., 1996). This means that discourse referents introduced by one of the disjuncts cannot be 'picked up' by anaphoric expressions outside of the disjunction. The treatment of disjunction in Inq D is more subtle. Following Stone (1992), we predict that when both disjuncts introduce the same new discourse referent, it is accessible for anaphoric expressions outside of the disjunction. This can be seen in Figure 4: every possibility in the output context assigns a value to the discourse referent introduced inside the two disjuncts, u 1 . This discourse referent is therefore predicted to be available as a possible antecedent for subsequent anaphoric reference. Stone (1992) argues that this is a good prediction, based on contrasts like the one in (47). (47) a. Bill u 1 either rented a u 2 station wagon or a u 2 van. He u 1 took it u 2 out for three days.
b. Bill u 1 either rented a u 2 car or decided to hitchhike. *He u 1 took it u 2 out for three days. Figure 5 provides a visual illustration of how this contrast is captured on our account. In the input context in this figure, there is one discourse referent, u 1 , which is mapped to b, Bill. w van is a world in which a is a van and Bill rented a, w sw is a world in which a is a station wagon and Bill rented a, in w ∅ Bill did not rent anything, but decided to hitchhike. The output context on the left, which corresponds to the first sentence in (47a), licenses a subsequent update which makes anaphoric reference to u 2 , e.g., T akeOutF orT hreeDays{u 1 , u 2 }. This is not the case for the output context on the right, which corresponds to the first sentence in (47b). In this output context, u 2 does not receive a value in all possibilities, so the result of a subsequent update with T akeOutF orT hreeDays{u 1 , u 2 } would be undefined. 22 Our analysis of disjunction also predicts that if subsequent updates resolve the issue raised by the disjunction by confirming the disjunct that introduced a new dref, then that dref becomes accessible. This captures the felicity of the following two mini-dialogues: 23

(48)
A: Bill either rented a u car or decided to hitchhike. B: The former, of course. He took it u out for three days.
(49) A: Did Bill rent a u car ↑ or did he decide to hitchhike ↓ ?
22 Note that the analysis of (47a) given here assumes that the indefinite a station wagon in the first disjunct and the indefinite a van in the second disjunct both introduce the same discourse referent, in this case u2. This is possible because, following Heim (1982) and many others, we assume that the only general constraint on the discourse referents introduced by indefinites is that they should be novel in their local context, and u2 is novel both in the local context of a station wagon and in the local context of a van. Of course it is also possible to analyze a station wagon and a van in (47a) as introducing distinct discourse referents. What is crucial for licensing the subsequent anaphoric reference, however, is that it is possible for them to be analyzed as introducing the same discourse referent.
23 Famously, anaphoric expressions in the second disjunct can also pick up discourse referents introduced in the scope of a negation operator within the first disjunct, as in the following example due to Partee: Either this house doesn't have a bathroom or it's in a funny place.
Our treatment of disjunction currently leaves such cases unaccounted for. An exploration of how existing accounts of such cases (e.g., Krahmer and Muskens, 1995;Rothschild, 2017) may be incorporated in Inq D is left for future work. Such an extension of Inq D would be needed to account for questions like (ii), which, to our knowledge, no existing theory can deal with.
(ii) Does this house not have a bathroom or is it in a funny place?   (47) is captured.

B:
The former, of course. He took it u out for three days.

Negation
Our treatment of negation, given in (50), is close in spirit to the treatment of negation in static inquisitive semantics. Recall that, informally speaking, a state t subsists in an output context c if, modulo additional discourse referents, t is still present in c. The formal definition of subsistence was given in Section 3.1.7. 24 The update function ¬ ¬U, when applied to an input context c, yields an output context consisting of those states in c whose non-empty substates do not subsist in U(c). Note in particular that states in the output context are always ones that were already present in the input context as well. This means that any discourse referents introduced within the scope of a negation operator (by U) are inaccessible outside of that scope. In other words, negation is 'externally static', as is standardly assumed in dynamic semantics (see, e.g., Groenendijk and Stokhof, 1991;Kamp and Reyle, 1993). Another property of negation as defined in (50), inherited from static inquisitive semantics, is that ¬ ¬U never raises any issues, even if U itself does. These two features of negation-that it is externally static and discharges issues raised inside its scope-are illustrated in Figure 6. Figure 6: Negation applied to an inquisitive update expression, ¬ ¬(smile{u 1 } smile{u 2 }), and to an update expression introducing a new discourse referent, [u 3 ]; smile{u 3 }. The output context is the same in both cases. Note that this output context is not inquisitive, and that u 3 is not present in the output context. 24 In Sect. 3.1.7 we defined extension and subsistence in the meta-language. These notions can also be defined in our logical language, as follows:

Double negation and removing inquisitiveness
A doubly negated update expression ¬ ¬¬ ¬U always conveys exactly the same information about the world as U itself, but it never raises any issues and it always renders the drefs introduced by U inaccessible, due to the two features of negation highlighted above. As we will see, it will be useful in the analysis of English to have another operator that only discharges issues in its scope without affecting information about the world and discourse information. Following work on static inquisitive semantics, we will refer to this operator as the issue-cancelling operator and write it as '!'. This operator will be used, for instance, in our analysis of declarative sentences, which do not raise issues but are externally dynamic.
We define the update effect of ! U as follows: 25 (51) ! U T := λc k λs i .s ∈ !(U(c)) ∧ ∃s ∈ c. s ≥ s The first conjunct says that states in the output context ! U must be in the non-inquisitive closure of U(c). This condition, however, is not sufficient since if we simply took ! U to be the output context, any issues that were present in the input context c would be discharged. What ! U is meant to do is to discharge any issues that are raised by U, but not to discharge issues that are already present in the input context. The second conjunct in (51) therefore requires that all states in the output context must be extensions of states in the input context. This means that all states in the output context still resolve all issues that were present in the input context. Figure 7 highlights the difference between ! and double negation when applied to the inquisitive update smile{u 1 } smile{u 2 } and the dref-introducing update [u 3 ]; smile{u 3 }. Both ! and double negation eliminate the issue raised by the disjunction (left column). By contrast, when ! applies to an update that introduces a discourse referent such as [u 3 ]; smile{u 3 } it does not make the discourse referent inaccessible for subsequent anaphoric expressions, unlike double negation.

Introducing inquisitiveness
While the ! operator removes inquisitiveness, static inquisitive semantics also comes with the ? operator, which introduces inquisitiveness by taking the disjunction of a sentence and its own negation. In our dynamic setting, this operator can be defined as follows: ? U T := U ¬ ¬U Applying ? to a given update U always results in an inquisitive update, except when U is tautological or contradictory. However, it is not always the most conservative way of turning U into an inquisitive update. In particular, if U is already inquisitive itself, forming the disjunction with ¬ ¬U is of course not needed to obtain an inquisitive update; we can simply leave U untouched. Following work on 25 Recall that for any context c, !c is the non-inquisitive closure of c as defined in the meta-language in Sect. 3.1.6. The definition in the logical language is given here in two steps. First we provide auxiliary definitions for c ↓ and info(c): Next, we define !c: (iii) !c := (λs.s = info(c)) ↓ Recall that s ≥ s means that s is an extension of s , see Sect. 3.1.7 and fn. 24.

Input context
?
In the work cited above it has been proposed that ensuring inquisitiveness in this more conservative way is one of the semantic contributions of interrogative clause type markers in languages like English. We will incorporate this proposal, which means that the ? operator will play an important role in our formal analysis of interrogatives.

Presupposing non-informativeness
Another operator which will play an important role in our treatment of interrogatives is the presuppositional closure operator, † (Roelofsen, 2015;Champollion et al., 2017). This operator, when applied to an update U, contributes the presupposition that any information that U provides about the world must already be present in the input context c. If this presupposition is not met, i.e., if U provides new information about the world that is not available yet in c, then U(c) is undefined.
This operator is relevant for the semantics of questions because we will assume, following e.g. Roelofsen (2015) and Champollion et al. (2017), that questions never provide any at-issue information about the world. If they have non-trivial informative content, then this information is always presupposed to be already available in the input context. Figure 8 displays three contexts to illustrate the workings of †. Suppose that U is an update function such that U(c 1 ) = c 3 and U(c 2 ) = c 3 as well. Then † U(c 1 ) is defined but † U(c 2 ) is undefined, because w a,b is considered possible in c 2 but not anymore in U(c 2 ). So U provides information about the world that is not available yet in c 2 . This means that c 2 does not subsist in U(c 2 ), and therefore †U(c 2 ) is undefined.

Atomicity
We now define the update function atom{u}, which provides the information that the value of u is atomic. The effect of this update function is to eliminate all states from the context containing at least one possibility whose dref assignment matrix includes one or more assignments assigning a non-atomic entity to u. As in earlier work on dynamic semantics (e.g., Brasoveanu, 2008) we will use this update function to capture the semantic contribution of singular number morphology.
(55) atom{u} := λc k λs i . s ∈ c ∧ ∀p ∈ s. ∀g ∈ G p . ¬∃y. y < g(u) The effect of atom is illustrated in Figure 9. Note that atom{u} only removes states containing possibilities in which some specific dref assignment function assigns a plural entity to u. It does not remove states with possibilities in which multiple values are assigned to u by different dref assignment functions in the dref assignment matrix, as long as all these values are atomic. For instance, in Figure 9, the row u/a, u/b is still present in the output context.

Maximality
Next, we define the update function max{u}, which requires that the cumulative value of u in a given possibility p is maximal, compared to other possibilities in the input context with the same world parameter as p. 26 As in previous work on dynamic semantics (e.g., Brasoveanu, 2008) we will use this update function, among other things, to derive strong readings of donkey anaphora.
A graphical illustration of this update function is given in Figure 10. There are two things to notice. First, concentrating on the second column (possibilities with the world parameter w a,b ), note that only those states that include either of the possibilities in the top two rows are removed. Why are no other states removed from the second column? This is because max{u} checks whether the cumulative value of u is maximal, and the cumulative value ⊕G(u) for any possibility in the third row and lower is the same, namely, a ⊕ b. Second, note that states containing the possibility in the leftmost column (the possibility with the world parameter w a ) are not removed by max{u} even though the cumulative value of u in that possibility is just a. The reason that states with this possibility are not removed is that max{u} is sensitive to the world parameter: in checking whether the cumulative value of u is maximal in a given possibility, we do not compare against possibilities with a different world parameter. Figure 10: Workings of max{u}. Now, the max{u} update function as defined in (56) suffices to derive strong readings of donkey anaphora involving a single anaphoric expression and a single indefinite antecedent. However, the operator is not sufficient to derive strong readings of donkey anaphora involving multiple anaphoric expressions and indefinite antecedents, and therefore needs to be refined. Consider the following example: (57) If a student sees a professor, he greets her.
We did not provide a compositional procedure yet to translate English sentences into Inq D expressions, but it should not be hard to see that when we just focus on the antecedent, the clause should be translated as follows: (58) [u 1 ]; student{u 1 }; atom{u 1 }; [u 2 ]; professor{u 2 }; atom{u 2 }; see{u 1 , u 2 }; max{u 1 }; max{u 2 } Let us first consider a scenario in which max{u} correctly filters out states containing possibilities that assign non-maximal cumulative values to {u}. Suppose that there are two students, a and b, student a sees professors c and d, and student b sees professors e and f . Then we correctly derive that any state containing a possibility in which the cumulative value of u 1 is strictly contained in a ⊕ b or the cumulative value of u 2 is strictly contained in c ⊕ d ⊕ e ⊕ f is removed since in these possibilities the cumulative value of either u 1 or u 2 is non-maximal. But now let us consider a second, problematic, scenario. We again assume that there are two students, a and b. Furthermore, we assume that a sees professors c and d, and b sees professors c and e. In this case, max would yield an output context that is intuitively incorrect. In particular, max would not be able to distinguish possibilities with either of the two dref assignment matrices in (59), since for both matrices the cumulative value of u 1 is a ⊕ b and the cumulative value of u 2 is c ⊕ d ⊕ e, which are maximal. However, we do want to distinguish these two cases and keep only those possibilites in which the dref assignment matrix is maximal for u 2 with respect to any value of u 1 . In (59), this only holds for the dref assignment matrix on the right.

(59)
Two dref matrices indistinguishable for max: This example shows that we need a refined version of the max operator. This refined version, which we will write as max * , restricts a context c to states consisting of possibilities whose dref assignment functions cumulatively assign a maximal set of individuals to u with respect to any value of other drefs. A formal definition is given in (60). In this definition, G u\un → x (u n ) is the set of values assigned to u n by assignments in G which pointwise assign the values x to u \ u n , all drefs except u n . If u n is the only dref, then max * {u n } just amounts to max{u n }. But if there are other drefs around, the two operators may come apart.
The effect of max * is illustrated in Figure 11. There are two discourse referents in this example and max * applies to u 2 . The output context contains only those possibilities in which u 2 is maximal for any value of u 1 . Concretely, states containing the top possibility in the input context are eliminated because u 2 is not maximal there for any value of u 1 compared to the two other possibilities. Figure 11: Workings of max * . There are two discourse referents in this example, u 1 and u 2 . Sets of assignment functions G belonging to the same possibility are separated from other assignment functions by a horizontal line. For example, the top possibility in the left diagram has a set of assignments G such that some assignments assign a to u 1 and b to u 2 and the remaining assignments assign b to u 1 and a to u 2 .

Requesting a witness
We now turn to the final update function that we will define, ?u. This update function will play a crucial role in our analysis of wh-questions. Let us first illustrate this informally on the basis of (61).
This question first introduces a discourse referent, u, and updates the context with the information that u is a person and u smiled. These updates are shared with the declarative statement someone smiled. In contrast to the declarative, however, we assume that a wh-question induces an additional update, ?u. The ?u update raises an issue whose resolution requires identifying one or more individuals that have the properties ascribed to u, i.e., it requires identifying one or more 'witnesses' for u. For this reason, we call ?u the witness request operator.
A simple formal definition of the witness request operator is given in (62). As we will see, this definition will have to be further refined, but it is still instructive to consider in some detail what it amounts to.
?u := λcλs. s ∈ c ∧ ∃x e . ∀p ∈ s. ∀g ∈ G p . g(u) = x (to be modified) According to (62), a state s in the input context survives an update with ?u just in case all the dref assignment functions in all the possibilities in s assign the same entity x to u. In such a state, we know that x satisfies all the properties that have been ascribed to u, so x is a witness for u. A graphical illustration of the update effect of ?u as defined in (62) is given in Figure 12.
To see why this simple definition of ?u needs to be further refined, suppose that the output context in Figure 12 obtains after an utterance of the question in (61), Who smiled?. Intuitively, this question can be resolved by specifying a person who smiled, for example: ?u 1 Figure 12: The update effect of ?u as defined in (62), to be refined below.
However, such an answer does not resolve the issue present in the output context in Figure 12. For the sake of illustration, let us assume that (63) induces the following sequence of updates:  (62), this update is wrongly predicted not to resolve the given question. If we remove all states containing w b -possibilities from the output context in Figure 12, we still have a context with an unresolved issue, because, even though we have identified a person who smiled, we have not determined yet what the value of u is. The problem is that ?u, as defined in (62), raises an issue that can only be resolved by establishing two things. First, one must establish certain information about the world, i.e., sufficient information to identify an individual who has the properties ascribed to u. But this alone is not enough according to the definition in (62). Rather, one must also establish the specific value of u. This second requirement is too demanding, because in fact, resolving a question only ever requires establishing information about the world, not about the specific value of a certain discourse referent.
This insight leads us to the refined definition of ?u in (65). According to this refined definition, ?u selects states s in the input context c for which we can find a witness x such that the world parameter of each possibility in s is compatible with assigning x to u. That is, for every possibility p in s, there is a possibility p in some state in c with the same world parameter as p whose dref assignment functions all assign x to u. Whether the dref assignment functions in p itself assign x to u doesn't matter. What matters is whether the world parameter of p is compatible with assigning x to u.
?u := λc k λs i . s ∈ c ∧ ∃x e . ∀p ∈ s. ∃p ∈ c. (w p = w p ∧ ∀g ∈ G p . g(u) = x) A graphical illustration of the workings of ?u as defined in (65) is given in Figure 13. Note that the output context in this figure contains the same information as the input context, but is inquisitive: two alternatives have been introduced. That is, ?u raises an issue, which could be paraphrased as 'What is an individual that has the properties ascribed to u? Is a such an individual (the alternative extending to the left), or is b such an individual (the alternative extending to the right)?' This issue would be resolved by (64), which would yield a context with only the former alternative, by eliminating states containing the w b -possibility. So the problematic prediction arising from the initial definition of ?u no longer obtains. Our analysis of single-wh questions in English will rely heavily on the ?u operator. However, for multiple-wh questions, we need a generalized witness request operator, which does not make reference to just one discourse referent u but rather to two or more discourse referents, u 1 . . . u n . The update effect of this operator, ?u 1 . . . u n , is defined in (66). The update selects states in the input context which contain enough information about the world to guarantee the existence of a witness function, which maps any tuple of entities satisfying the properties ascribed to u 1 . . . u n−1 to a corresponding entity satisfying the properties ascribed to u n . (66) ?u 1 . . . u n−1 u n := λc k λs i . s ∈ c ∧ ∃f (e n−1 ,e) . ∀p ∈ s. ∃p ∈ c.
(w p = w p ∧ ∀g ∈ G p . g(u n ) = f (g(u 1 ), . . . , g(u n−1 ))) Crucially, when n = 1, ?u 1 . . . u n just boils down to ?u. In other words, ?u is just a specific instance of ?u 1 . . . u n . 27 Because of this, single-wh and multiple-wh questions can be treated in a uniform way in our account, as we will demonstrate in Section 5.  which regulate the interplay between informative and inquisitive content, come from inquisitive semantics. On the other hand, the operators [u], atom{u}, and max * {u}, which regulate discourse referents, come from dynamics semantics. The witness request operator, ?u 1 . . . u n , is the only operator that does not have a precedent either in inquisitive semantics or in dynamic semantics. This is because the update it expresses involves both discourse referents, which are present in dynamic semantics but not in static inquisitive semantics, and the raising of an issue, which is modeled in static inquisitive semantics but not in basic (non-inquisitive) dynamic semantics. It is exactly this new dynamic-inquisitive witness request operator that will play a crucial role in our analysis of wh-questions.

Syntax-semantics interface for a fragment of English
We now specify some basic assumptions about the syntax-semantics interface for a fragment of English, in order to provide a compositional semantic analysis of declaratives, polar questions and wh-questions in Inq D . Of course, what we present here is not the only way to analyse these sentence types using Inq D . The assumptions we make about the syntax of English and the syntax-semantics mapping are independent of the semantic framework, and other researchers may well want to use Inq D in conjunction with different assumptions about the syntax of English and a different mapping between syntax and semantics. The fragment of English that we consider here can be divided into two parts. On the one hand, we consider certain lexical and morphological elements that are merged inside the TP, and on the other hand, we consider functional heads that are assumed to be part of the left periphery, external to the TP. Among the elements that are merged within the TP we first consider verbs, nouns, and number morphology (Section 4.1), and then turn to indefinites and wh-words (Section 4.2). As for the left periphery, we concentrate on three functional heads, one that indicates the clause type, declarative or interrogative, one that triggers wh-fronting in interrogative clauses, and one which marks the clause as being either finite or non-finite (Section 4.3). Specifying the semantic contribution of these elements will be sufficient to obtain a rather general account of declarative and various kinds of interrogative (polar, single-wh, multiple-wh) clauses in English, demonstrating what we consider to be some of the main virtues of the Inq D framework.

Verbs, nouns, and number morphology
In (67), we provide an Inq D -translation for three verbs: the intransitive verb sleep, the transitive verb see, and the ditransitive verb give. We restrict our attention here to distributive predicates. A detailed analysis of collective predicates in Inq D is left for future work. 28  Q(λv .Q (λv .give{v, v , v }) In (68), we provide Inq D -translations for the singular noun boy and the plural noun boys. We assume here that plural number morphology is semantically vacuous, while singular number morphology contributes an atomicity requirement, following Zweig (2008Zweig ( , 2009  This does not mean that the use of a plural noun does not communicate any cardinality information. Plurals can (and typically do) give rise to a more-than-one inference. We assume, however, that this inference is not semantically encoded, but rather results from pragmatic reasoning (Sauerland 2003;Sauerland et al. 2005;Spector 2007;Zweig 2008;Ivlieva 2013;Mayr 2015, among others). Roughly, when a speaker uses the plural boys, the fact that she did not use the singular boy, which would have been semantically more specific, can give rise to the inference that she is referring to multiple boys. We will return to this point in more depth in Section 5.

Indefinites and wh-words
The Inq D -translation of the indefinite article a is given in (69). We treat the indefinite article a and the determiner some as generalized determiners that introduce a discourse referent. This treatment of indefinites is standard in compositional dynamic semantics (see, e.g., Muskens 1996). As anticipated in Section 2.2, we will follow Brasoveanu (2007) in assuming that indefinites have a weak interpretation, given in (69a), and a strong interpretation, given in (69b). The only difference is that under a strong reading they place a maximality requirement on the values that are assigned to the discourse referent that is introduced. Under the weak reading, this maximality requirement is not present. 29  Brasoveanu (2007) argues for this weak/strong ambiguity in the semantics of indefinites based on constructions involving donkey anaphora. Such constructions are well-known to allow for weak and strong readings, as illustrated in (70). Under its most prominent reading, (70a) implies that whenever Joe sees one or more donkeys he beats all of them. This is called a strong reading of donkey anaphora. By contrast, (70b) is most naturally interpreted as saying that whenever Joe has one or more dimes in his pocket, he throws one of them in the parking meter. This is a weak reading.
(70) a. If Joe u 1 sees a u 2 donkey, he u 1 beats it u 2 . b. If Joe u 1 has a u 2 dime in his pocket, he u 1 throws it u 2 in the parking meter.
We refer to Brasoveanu (2007Brasoveanu ( , 2008 for a detailed analysis of this phenomenon arguing that the ambiguity of such constructions is due to a weak/strong ambiguity in the lexical semantics of indefinite determiners, as we assume here. T, denoting an update function. 29 The type rT that is used in (69) is the type of dynamic properties in Inq D . Such properties apply to a discourse referent and yield an update function.
We now turn to wh-words. We assume that they are interpreted just like indefinites: their main function is to introduce a discourse referent. This means in particular that they are not inquisitive; they do not raise an issue. We further assume that simplex wh-phrases (who, where, what etc.) have both a weak and a strong reading, just like indefinites. On the other hand, we assume that the wh-determiner which only has a strong reading in English: This distinction between simplex wh-phrases and which-phrases is motivated by the fact that the former can generate both mention-some and mention-all readings while the latter forces a mentionall reading at least when there is no modal or disjunction in the question. When a modal or a disjunction is present, which-questions may trigger a mention-some reading. This, however, will be derived on our approach without postulating ambiguity in the lexical entry of which.

Left periphery
The left periphery of a clause contains functional heads which syntactically c-command the TP and semantically operate on the semantic content of the TP. Early work in linguistics often assumed that the left periphery involves a single functional head C, referred to as the complementizer of the clause. However, Rizzi (1997) and much subsequent work has argued that the left periphery actually involves multiple functional heads, each playing a distinct syntactic and semantic role (Rizzi, 2001;Aboh, 2004;Haegeman, 2012, among many others). There is no general agreement in the literature yet on what the complete inventory of relevant functional heads is and to what extent this inventory varies across languages, but there seems to be general consensus that at least in English and related languages: 1. All clauses involve a functional head which encodes the clause type. The two clause types relevant for us here are declarative and interrogative. We will write Dec for the declarative clause type marker and Int for the interrogative clause type marker. 30 2. All clauses involve a functional head which triggers fronting of a wh-phrase in interrogative root clauses, and may trigger fronting of a focused element in declarative root clauses. This head is generally written as Foc (short for Focus). We will adopt this notational convention, although we will be mainly interested here in interrogative root clauses-where Foc associates specifically with wh-phrases and not with focused elements in general. 31 3. All clauses involve a functional head Fin which determines whether the clause is finite or non-finite.
4. The scopal relationship between the various heads is: Dec/Int > Foc > Fin 30 We intend Dec and Int to determine clause type, not illocutionary force. We therefore write TypeP for the maximal projection headed by Dec or Int, not ForceP, as is done in some other work (e.g., Rizzi, 1997). The interrogative clause type marker Int is sometimes also written as Inter (Aboh, 2004;Aboh and Pfau, 2011). We use Int here for brevity. 31 We limit ourselves here to specifying an explicit syntax and syntax-semantics interface for root clauses. The syntax of embedded clauses is different (e.g., in embedded interrogative clauses there is no subject-auxiliary inversion, and embedded non-wh-interrogatives involve whether ). The syntax-semantics interface will therefore also have to be somewhat different. Spelling this out is beyond the scope of this paper. We will assume that Fin does not have a specific semantic contribution, at least not one that is relevant for our purposes here. However, we are not ignoring this functional head altogether, because we do take it to play a syntactic role that is relevant for our purposes. Namely, we assume that in English interrogative root clauses, it triggers T-to-Fin movement, which results in subject-auxiliary inversion. 33 Below we propose a semantic treatment of Dec, Int, and Foc in Inq D . The semantic contribution of Foc is discussed in Section 4.3.1, that of Int in Section 4.3.2, and finally that of Dec in Section 4.3.3.

Semantic contribution of Foc
We assume, following Rizzi (1997) and subsequent work, that every argumental wh-phrase must be associated with a Foc head in the left periphery of a clause in which it is contained. In some cases this syntactic association manifests itself as obligatory movement of the wh-phrase to the specifier of the associated Foc head. In other cases, wh-phrases may remain in-situ. In every case, however, a wh-phrase must be associated with a Foc head in the left periphery of a clause in which it is contained. 34 We assume that wh-movement to the left periphery is not driven by semantic considerations and that the wh-element is interpreted in its lower position. To illustrate what we have in mind, consider the following example from Baker (1970): (73) Who remembers where we bought which book?
32 Besides the functional heads taken into consideration here, there is also general consensus that the left periphery of a clause may include one or multiple functional heads which host preposed topical constituents. Such functional heads are not relevant for our purposes here and will therefore be ignored. 33 Haegeman (2012) instead assumes that subject-auxiliary inversion in interrogative root clauses in English involves movement of the auxiliary from T to Foc (rather than to Fin). These two syntactic hypotheses cannot, as far as we can tell, be teased apart on purely syntactic empirical grounds, because it is claimed that in English there is never any material between Foc and Fin (see, e.g., Haegeman, 2012). For almost all cases that we will discuss, the choice between the two syntactic hypotheses has no repercussions for what needs to be assumed about the syntax-semantics interface either. In one case, however, namely example (106) on page 55, simpler assumptions about the syntax-semantics interface can be made if we assume that subject-auxiliary inversion involves movement from T to Fin rather than to Foc. We return to this point when we discuss example (106), see footnote 42. In any case, our proposal does not strongly hinge on the assumption that subject-auxiliary inversion involves movement to Fin. Different syntactic assumptions would require some adaptations of the proposed syntax-semantics interface, but we expect that such adaptations would not raise any substantial issues. 34 We intentionally remain neutral here as to what the syntactic relationship between a wh-phrase and the associated Foc head amounts to exactly. This can be made concrete in several ways, depending on which syntactic framework one assumes. Our proposal is intended to be compatible with a broad range of syntactic frameworks.
Both the matrix clause and the embedded clause are assumed to involve a Foc head. The matrix subject wh-phrase who associates with the matrix Foc head and has moved to its specifier position. In the embedded clause there are two wh-phrases, where and which book. The former associates with the embedded Foc head and has moved to its specifier position. On the other hand, which book appears in-situ. The assumption is that, even though it has not undergone overt movement, it associates either with the matrix Foc head or with the embedded Foc head. These two possible associations result in two distinct readings, as Baker (1970) already noted. Namely, if which book associates with the embedded Foc head, (73) raises an issue whose resolution requires an answer like (74): (74) John and Martha remember where we bought which book.
On the other hand, if which book associates with the matrix Foc head, (73) raises an issue whose resolution requires an answer like (75): (75) John remembers where we bought the physics book and Martha remembers where we bought the biology book.
This example also illustrates that, while every wh-phrase must be associated with one, and only one, Foc head, the converse does not hold. A Foc head may be associated with one wh-phrase, but it may also be associated with multiple wh-phrases or with no wh-phrases at all. We will write Foc u if Foc is associated with a single wh-phrase introducing the discourse referent u. Similarly, we will write Foc u 1 ...un if Foc is associated with multiple wh-phrases introducing discourse referents u 1 . . . u n , respectively. And finally, we simply write Foc if there is no associated wh-phrase.
For reasons that will become clear shortly, if Foc is associated with one or more wh-phrases introducing discourse referents u 1 . . . u n , respectively, we will refer to u 1 . . . u n−1 as its domain discourse referents and to u n as its target discourse referent.
We propose that Foc u 1 ...un generally makes three semantic contributions, even though, as we will see right below, if n = 1 one of these contributions is always vacuous, and if n = 0 two of these contributions are vacuous. First, Foc u 1 ...un discharges the inquisitiveness of the TP that it combines with, by applying the ! operator to the semantic value of the TP. Second, Foc u 1 ...un places a maximality requirement on all its domain discourse referents. And finally, Foc u 1 ...un requests a (functional) witness for its target discourse referent by applying the witness request operator ?u 1 . . . u n . The proposal is summarized in the following lexical entry: where u 1 , . . . u n are the discourse referents introduced by the associated wh-phrases As anticipated above, this general entry reduces to something simpler when n ≤ 1, i.e., if Foc is associated with just one wh-phrase or with no wh-phrases at all. If Foc is associated with a single wh-phrase, i.e., in the case of a simple wh-question, its semantic contribution amounts to (77), and if it is not associated with any wh-phrase whatsoever, which is the case in a polar question, an alternative question or a declarative clause, its semantic contribution reduces to (78). 35 35 While we assume here that Foc generally has three semantic tasks-applying !, applying max * , and applying ?u1 . . . un-another way to articulate the account would be to assume that Foc is only responsible for placing a maximality requirement on u1, . . . , un−1 and requesting a witness for un, i.e., only those tasks that directly concern the discourse referents introduced by the wh-phrases associated with Foc. The third task, applying ! to the semantic content of the TP, would then have to fall under the responsibility of another functional head in the left periphery, scoping below Foc (such a functional head is assumed, for instance, in Roelofsen 2015). In the present setting, we may indeed assume that it is Fin, rather than Foc, that applies ! to the content of the TP. This would leave most of (77)

Semantic contribution of Int
We now turn to the interrogative clause type marker, Int. Following Roelofsen (2015), Roelofsen and Farkas (2015), and Ciardelli et al. (2018), we assume that interrogative clause typing has two semantic effects. First, Int ensures that the clause it heads is inquisitive (provided that the clause is non-tautological). This is done by applying the ? operator. Second, Int ensures that the clause is not informative w.r.t. the input context. This is done by applying the † operator, which requires that all information conveyed by the TP about the world, if any, is already established in the input context. Thus, we assume the following treatment of Int: Recall that interrogative root clauses in English characteristically involve subject-auxiliary inversion and wh-movement (the latter only in the case of wh-interrogatives, not in the case of polar or alternative interrogatives). We assume that subject-auxiliary inversion and wh-movement are conditioned on the presence of Int, but only indirectly, because subject-auxiliary inversion is assumed to involve movement to Fin, and wh-phrases move to the specifier of FocP. Neither of these, then, involves movement to Int or to the specifier of TypeP. To ensure the indirect connection between subject-auxiliary inversion, wh-movement, and the presence of Int, we assume that (i) a Foc head can only associate with wh-phrases if it is dominated by Int, and (ii) a Fin head only triggers Tto-Fin movement if it is in a root clause headed by Int. The other clause type marker, Dec, which appears in declarative clauses, will be discussed below. We will assume that Foc cannot associate with wh-phrases when dominated by Dec, and that Fin does not trigger inversion when dominated by Dec, or when part of a subordinate clause. These syntactic dependencies could be implemented with a suitable feature system, which we leave implicit here. We will go through four examples to illustrate the interaction between Int and Foc in the interpretation of questions: a simple wh-question, a polar question, an alternative question and a wh-question with a TP-internal disjunction.
Example 1: simple wh-question First, consider the simple wh-question in (80) with a weakly interpreted wh-phrase, which should yield a mention-some reading.
Where u weak do they sell pens?
Under the assumptions laid out above, the sentence has the syntactic structure shown in Figure 15, and it is translated into Inq D as shown in (81) where Int Figure 15: The left periphery of the sentence (80).
As seen in (81a), the TP introduces a discourse referent, u, and further updates the context with the information that u is a location at which pens are sold. Foc u enters the stage in (81b), applying the ! operator to discharge any inquisitive content that the TP might carry, and the ?u operator to request a witness for u. Note that ! is vacuous in this case because the TP is not inquisitive. Finally, Int comes into play in (81c), applying ? and † to ensure that the clause is inquisitive and non-informative w.r.t. the input context. ? is vacuous in this case, since FocP is already inquisitive. Figure 16 graphically represents the update effect of [u]; location{u}; sellP ens{u}; ?u on an initial context in which no information and no discourse referents are present yet. Note that the output context has no states containing possibilities that have w ∅ as their world parameter, i.e., in which no place sells pens. † requires that such possibilities are already ruled out in the input context as well. Overall, then, the question is predicted to raise an issue whose resolution requires identifying at least one place that sells pens, and to presuppose that it is already established in the input context that such a place exists.
Example 2: polar question We now turn to a second example, the polar question in (82). This case serves to illustrate in particular why ? , which was vacuous in (80), is a necessary component of the semantic contribution of Int. We treat raining as a 0-place predicate here. Since neither the TP nor the FocP are inquisitive, the ?u Figure 16: The update effect of (80).
? operator contributed by Int in (83c) is crucial here to derive the correct interpretation. It ensures that the sentence is interpreted as a polar question, whose resolution requires establishing whether it is raining or not. The † operator is vacuous in this case, because when ? applies non-trivially, it always ensures that the question is non-informative w.r.t. the input context.
Example 3: alternative question Next, we consider the alternative question in (84). This question is to be read with rising intonation on the first disjunct and falling intonation on the second disjunct, as indicated by the ↑ and ↓ arrows. Other intonation patterns give rise to different interpretations (see Bartels 1999, Pruitt and Roelofsen 2013, Roelofsen and Farkas 2015, Westera 2017, Meertens 2021, among others).
(84) Is it raining ↑ or is it snowing ↓ ?
We assume that alternative questions involve clausal disjunction (see Han and Romero 2004, Haida 2010, Pruitt and Roelofsen 2011, Uegaki 2014, Roelofsen 2015, Gračanin-Yuksek 2016, Hoeks and Roelofsen 2019. More specifically, we assume that in alternative questions the disjunction operates on two FocPs. This yields the interpretation in (85). 36 The ! operators contributed by the two Foc heads are vacuous in this case, as in the previous examples. The ? operator contributed by Int is also vacuous again because the disjunction in its scope already ensures inquisitiveness. The † operator contributed by Int is not vacuous in this case-36 If we take a disjunction of two TypePs rather than two FocPs we derive an interpretation that is not correct for (84) and which is in fact not expressible in English without resorting to rather artificial queries such as 'Please answer at least one of the following two questions either positively or negatively: (a) Is it raining? (b) Is it snowing?. We refer to Hoeks and Roelofsen (2019) and references therein for discussion of syntactic, semantic, and pragmatic mechanisms that may block TypeP disjunctions. Here, we simply take it as a given that disjunction applies at the FocP level in alternative questions.
it requires that it is already established in the input context that it is either raining of snowing. Overall, then, the question is predicted to presuppose that it is either raining or snowing and to request new information that establishes which of the two is the case.
Example 4: wh-question with TP-internal disjunction Finally, we consider the wh-question in (86) which involves a TP-internal disjunction. This example serves to illustrate in particular why we assume that Foc always discharges inquisitive content generated within the TP by applying !.
Who u weak has a pen or a pencil that I could borrow?
Given the assumptions we have made, this question is translated into Inq D as in (87). As seen in (87a), the TP/FinP is inquisitive in this case because of the disjunction: it introduces an issue as to whether u has a pen or whether u has a pencil that the speaker could borrow. In (87b), we see that Foc discharges this issue by applying the ! operator. In addition, it raises a different issue by applying the ?u operator, requesting a witness for u. Finally, as seen in (87c), Int ensures inquisitiveness by applying the ? operator, which is vacuous in this case because the FocP is already inquisitive. Int also ensures non-informativity w.r.t. the input context by applying the † operator. Overall, then, the question is predicted to presuppose that someone has a pen or a pencil to borrow, and to raise an issues whose resolution requires identifying at least one such a person. Importantly, it is predicted that the issue does not require establishing whether the person in question has a pen or whether she has a pencil. This is a result of Foc discharging the issue introduced by the disjunction. This example thus demonstrates why the ! operator needs to be invoked in questions and why the operator must take scope over the entire TP (to discharge issues raised by disjunctions inside the TP) but below the witness request operator contributed by Foc (otherwise the issue raised by this operator would also be discharged). As already mentioned in footnote 35 above, while we assume here that ! is part of the semantic contribution of Foc itself, one may in principle also assume an additional functional head in the left periphery, below Foc, whose semantic role is specifically to discharge inquisitiveness introduced within the TP by applying !. The role of Foc would then be restricted to requesting a (functional) witness for its target discourse referent, and placing a maximality requirement on its domain discourse referents. 37 37 The proposed analysis also makes correct predictions for cases in which two wh-questions are disjoined, as in (i), and cases in which a wh-question is disjoined with a polar question, as in (ii), assuming that the disjunction operates at the FocP level in such cases. Such cases are problematic for many other theories of questions (see Ciardelli et al., 2018;Hirsch, 2018;Hoeks and Roelofsen, 2019, for discussion).

(i)
Where can we rent a car or who has one that we could borrow?
(ii) Where can we rent a car or do you have one that we could borrow?
In particular, our analysis correctly predicts that 'I don't have a car that you could borrow' fails to resolve the issue raised by (ii), even though it would succeed in resolving the issue raised by the polar question in the second disjunct when that would appear on its own (not as part of a disjunction). For reasons of space, we do not go through the derivations for these examples in detail-they are analogous to the ones discussed in the main text.

Semantic contribution of Dec
We now turn to the declarative clause type marker, Dec. We assume that Dec simply discharges any inquisitiveness that may have been introduced within its scope, by applying the ! operator. 38 We will consider two examples to illustrate the effect of Dec. The first is a disjunctive declarative sentence: (89) It is raining or snowing.
The disjunction may be operating at different levels. Suppose first that it operates TP-internally.
Then we obtain the following derivation: The correct interpretation is derived, but the semantic contribution of Dec is vacuous here, because Foc already discharges the inquisitiveness introduced by the disjunction. The same is true if the disjunction operates at the TP or FinP level, i.e., if both disjuncts are TPs or FinPs, because in this case, disjunction still scopes below Foc. This is not true, however, if disjunction operates at the FocP level, i.e., if both disjuncts are FocPs. We have seen above that we must allow FocP disjunction in the grammar in order to derive the interpretation of alternative questions-see example (84). If we assume that disjunction operates at the FocP level in (89), we obtain the following derivation. The end result is the same as when disjunction is taken to operate TP-internally, but now the semantic contribution of Dec is not vacuous because the FocP disjunction is inquisitive and there is no operator other than Dec which discharges this inquisitiveness. Now, what do we derive if we assume that the disjunction applies at the TypeP level, i.e., that it operates on two full TypePs? In that case we derive the interpretation !!raining !!snowing, which is equivalent to raining snowing. This interpretation is not available for (89). That is, (89) cannot be interpreted as a question asking whether it is raining or snowing. Thus, we have to assume that disjunction cannot take scope over the declarative clause type marker, just as we assumed above that it cannot take scope over the interrogative clause type marker. This assumption has also been made in previous work on inquisitive semantics (e.g., Roelofsen and Farkas, 2015;Ciardelli et al., 2018). However, we should note that, while there is work suggesting that the inability of disjunction 38 We restrict ourselves here to declarative sentences with falling intonation, which, in English, are never inquisitive. Declarative sentences with rising intonation can be inquisitive. To account for such cases, our analysis would have to be extended. For instance, we could assume an additional functional head in the left periphery encoding whether the clause is open or closed in the sense of Roelofsen and Farkas (2015) and Ciardelli et al. (2018). Another option would be to derive the inquisitive nature of rising declaratives at a pragmatic level (Westera, 2017;Rudin, 2022). For reasons of space, we do not explicitly pursue such extensions here.
to scope over Int has a deeper explanation (see Hoeks and Roelofsen, 2019, and references therein), there is no work as of yet providing a deeper explanation for the inability of disjunction to scope over Dec. This assumption, then, must be stipulated at this point. Now consider a second example, the declarative sentence in (92), which involves an indefinite.
This sentence is translated into Inq D as follows. The example underlines the fact that indefinites are treated on our account as introducing a discourse referent, as in previous work on dynamic semantics, but not as introducing inquisitiveness, unlike in several analyses based on static inquisitive semantics (e.g., AnderBois, 2014;Onea, 2016;. Therefore, in a declarative sentence like (92) the ! operators contributed by Foc and Dec are both vacuous, since the TP/FinP is already non-inquisitive.

Detailed predictions for wh-questions
In Section 4.3.2 we already discussed the predictions of our account for a simple mention-some whquestion, Where do they sell pens?, and a mention-some wh-question with TP-internal disjunction, Who has a pen or a pencil that I could borrow?
. These examples were meant as a first illustration of the interaction between Int, Foc, wh-phrases, and disjunction. We now turn to a more detailed discussion of the predictions of our account for wh-questions. To what extent does the account meet the desiderata described in Section 1.3? Besides mention-some readings, can it also derive mention-all readings in a non-stipulative way? Does it provide a uniform treatment of single-wh and multiple-wh questions? Does it capture the effects of singular and plural number marking in whichquestions? In particular, does it derive uniqueness presuppositions for singular which-questions without additional stipulations? And does it predict that such uniqueness presuppositions can be obviated by disjunction? We will demonstrate that, indeed, all these desiderata are met. The assumptions we have made about Int, Foc, wh-phrases, indefinites, disjunction, and number morphology do not only derive correct results in the simple cases we have seen already, but also in many more complex cases. We will show this by going through a number of representative examples, presented in order of increasing complexity. The first example illustrates the derivation of mention-all readings.

Deriving mention-all readings through maximization
We assumed in Section 4.2 that simplex wh-phrases (who, where, what etc.) have both a weak and a strong reading, and that the difference between these two readings is that, on a strong reading, a maximality requirement is placed on the values of the discourse referent that these wh-phrases introduce. The entry for who is repeated in (94)  Importantly, the assumed ambiguity in the interpretation of simplex wh-phrases parallels an ambiguity in the semantics of indefinites, which, as mentioned above, was argued for by Brasoveanu (2008) on the basis of donkey anaphora. That is, the assumed ambiguity is not stipulated ad-hoc, but has been motivated independently, based on an empirical phenomenon that lies outside the domain of questions.
We have already seen in Section 4.3.2 that, when who is given a weak interpretation, we derive a mention-some reading. We will now show that when who is given a strong interpretation, placing a maximality requirement on the discourse referent that it introduces, a mention-all reading is derived. No additional assumptions are needed.
To see this, consider (95) and its Inq D translation in (96). The predicted update effect of (95) is represented graphically in Figure 17 (modulo †). It is instructive to compare this figure with Figure 16. The only difference is that in Figure 17, the max * {u} operator applies before ?u does. This removes all states containing possibilities whose assignments cumulatively assign a non-maximal plural individual to u, compared to other possibilities with the same world-parameter (i.e., other possibilities in the same column of the diagram). Concretely, the operator excludes all states containing either or both of the top two possibilities in the second column.
The output context in Figure 17 contains three mutually exclusive alternatives, while the output context in Figure 16 contained two overlapping alternatives. The alternatives Figure 17 correspond to the three possible exhaustive answers: (i) only a called, (ii) only b called, and (iii) both a and b called. The two overlapping alternatives in Figure 16 on the other hand, correspond to the two non-exhaustive answers: (i) a called (and perhaps b did as well), and (ii) b called (and perhaps a did as well). Thus, the assumed ambiguity in the semantics of simplex wh-phrases indeed derives mention-some and mention-all readings. Comorovski (1996), Dayal (1996) and others have observed that, unlike simplex wh-phrases like who and what, plural which-phrases force mention-all readings and singular which-phrases invoke a uniqueness requirement, at least in basic cases. This is illustrated in (97a-c), variants of examples that were already briefly discussed in Section 1.3. 39 (97) a. Who has a pen that the chair of the committee can borrow to sign the report? b. Which committee members have a pen that the chair can borrow to sign the report? c. Which committee member has a pen that the chair can borrow to sign the report?

Which-questions
While (97a) has a salient mention-some reading, (97b) only permits a mention-all reading. Finally, in (97c), it is presupposed that there is exactly one committee member who has a pen that the chair can borrow. These basic empirical characteristics of plural and singular which-questions are derived on our account, under the assumption that the determiner which, unlike who and what, only has a strong interpretation. That is, it always places a maximality requirement on the values of the discourse referent that it introduces (see the lexical entry for which in (72)). Let us demonstrate this. The Inq D translation of (98) is the same as that of (95) above, so the context update that (98) induces is the one graphically represented in Figure 17. As we discussed above, the issue in the output context captures the mention-all reading of the question.
Singular which-questions For singular which-questions our assumption that singular number marking invokes an atomicity requirement is crucial. To see the effect of this assumption, consider (99). The atomicity requirement invoked by singular number marking is underlined. The update effect of (99) is graphically represented in Figure 18 (again modulo †). The update is broken down into three steps. In the first step, atom{u} removes from the context all states containing possibilities in which u is assigned a non-atomic value by some assignment. In the sec-ond step, max * {u} eliminates states containing possibilities that cumulatively assign non-maximal plural individuals to u compared to other possibilities with the same world-parameter. Finally, in the third step, the witness request operator ?u raises an issue whose resolution requires identifying an individual which has all the properties assigned to u. Crucially, ?u removes states containing the possibility w a,b , {{u → a}, {u → b}} , because there is no possibility in the input context with the same world parameter whose assignments all agree on the value of u. Thus, after the ?u update, all states that remain in the output context consist exclusively of possibilities in which exactly one person called, either a or b.
Since the entire sequence of updates is in the scope of a † operator, we correctly derive that the question triggers a uniqueness presupposition. That is, for the update to be well-defined, it should already be established in the input context that there is exactly one person who called. On our account, this uniqueness presupposition is not triggered by a single operator but rather results from the interaction between atom, max * , ?u and †. If any of these operators were left out of the derivation, the uniqueness presupposition would not be predicted. In this regard, our account differs from other accounts of uniqueness presuppositions in singular which-questions (e.g., Dayal, 1996;Hirsch and Schwarz, 2019), a difference we will return to in Section 5.3. Pragmatic derivation of more-than-one inferences A plural which-question like (98) can be taken to suggest that more than one person called. We assume that this inference can arise, in certain contexts, through pragmatic reasoning. 40 The pragmatically strengthened interpretation is given in (100), where the inferred update is underlined.
(100) Which u people called? ; † ([u]; person{u}; ¬ ¬atom{u}; called{u}; max * {u}; ?u) Support for such a pragmatic view on more-than-one inferences in plural which-questions comes from a parallel between the behavior of plural which-phrases and that of plural non-wh-indefinites when occurring in questions. While plural non-wh-indefinites typically give rise to a more-than-one inference in declarative sentences, they typically don't in questions. For instance, the question posed by A in (101) must be answered affirmatively when the addressee has just one child. This suggests that plural non-wh-indefinites can range over both atomic and non-atomic entities, and we are assuming that the same holds for plural which-phrases. However, there are cases of non-wh-indefinites in questions that do favour a strengthened interpretation, ranging only over non-atomic entities. This is illustrated in (102) from Sauerland et al. (2005). It is odd to respond to (102) with 'yes' if one knows that a dog has only one tail. Indeed, Sauerland et al. (2005) show experimentally that adults predominantly respond to questions like (102) with 'no'. This indicates that tails in (102) is interpreted as ranging only over non-atomic entities.
(102) Does a dog have tails? Sauerland et al. (2005) suggest that the crucial difference between questions like (101) and ones like (102) is that the former are naturally taken to be genuine information seeking questions, whose answer is not yet known to the speaker, while the latter are more naturally taken to be 'exam questions', whose answer is already known to the speaker. In the latter case the more-than-one inference is more likely to arise, because the pragmatic reasoning that it is based on only goes through if the speaker can be assumed to have sufficient knowledge to make an informed choice between singular and plural number marking. We submit that a similar contrast can be observed with plural which-phrases. For instance, suppose that Professor Baker always brings a cake if one or more of his students get a paper accepted at a conference. When his colleague sees him come into work with a cake she asks him: (103) Which students of yours got a paper accepted?
In this case, a more-than-one inference seems unlikely to arise. This would be explained by the fact that (103) is a genuine information seeking question. Professor Baker's colleague cannot be assumed to have sufficient knowledge to use singular rather than plural number marking. Now suppose that the following question is posed as part of a history exam: In which centuries did the eighty years war take place?
In this case, a more-than-one inference is very likely to arise. The explanation for this, we suggest, is that the person who formulated the question can be assumed to know whether the eighty years war took place in just one century or across two centuries. If she had known that it took place in just one century, she would have used singular number morphology in the which-phrase. Since she didn't, we can infer that the war spanned more than one century.

Obviation of uniqueness effects
Recall from Section 1 that the uniqueness requirement normally induced by a singular which-phrase can be obviated in questions involving an existential modal operator (Hirsch and Schwarz, 2019) or disjunction (Socolof et al., 2020). For instance, it is perfectly natural to ask (105) without assuming that there is a unique letter that can be inserted in fo m to make a word.
Which letter can be inserted in fo m to make a word?
Similarly, (106) can be felicitously asked without assuming that there is a unique town in which Shakespeare was born or Bach died.
(106) In which town was Shakespeare born or did Bach die?
As Hirsch and Schwarz (2019) and Socolof et al. (2020) point out, however, it would be mistaken to conclude that singular which-phrases in questions with modals or disjunctions do not contribute a uniqueness requirement at all. To see this, consider (107).
(107) #Which letter can be inserted in f m to make a word?
Based on the observation that this question is infelicitous, Hirsch and Schwarz (2019) argue that, while the question does not presuppose that there is exactly one letter that can be inserted in f m to make a word, it does presuppose that it is possible to make a word by inserting exactly one letter in f m. We will refer to this presupposition as a 'local uniqueness requirement'. 41 The question is infelicitous, because one can obviously not fill two slots by inserting exactly one letter. The local uniqueness requirement also arises in singular which-questions with disjunction (Socolof et al., 2020). To see this, consider (108): (108) a. Which letter is missing in pill w or was replaced with a dollar sign in ketch$p? b. #Which letter is missing in p ll w or was replaced with a dollar sign in ketch$p? c. #Which letter is missing in pill w or was replaced with a dollar sign in k$tch$p?
The fact that (108b) and (108c) are infelicitous shows that a local uniqueness requirement pertains to both disjuncts. As Hirsch and Schwarz (2019) argue, the fact that non-local uniqueness requirements can be obviated by modals and disjunction is problematic for what they call globalist accounts, which derive a uniqueness presuppostion in singular which-questions through an 'answer operator' ANS that is located at the highest position in the logical form of the question (Dayal, 1996(Dayal, , 2016Fox, 2013Fox, , 2018Xiang, 2016). The problem for globalist accounts is that they always predict a uniqueness presupposition at the highest level. They do not derive a local uniqueness presupposition, within the scope of a modal operator or disjunction, because ANS always takes scope above everything else in the question. Hirsch and Schwarz (2019) contrast globalist accounts with what they call localist accounts, in which the uniqueness requirement is triggered by the which-phrase rather than an answer operator (Champollion et al., 2017;Uegaki, 2018Uegaki, , 2020Hirsch and Schwarz, 2019). Such accounts can deal with the obviation of a global uniqueness requirement in examples like (126) and (106), and can also correctly derive a local uniqueness requirement in such cases.
What about our analysis? Is it a globalist or a localist account? Certainly, it is not a globalist account, since it derives uniqueness presuppositions without recourse to an answer operator scoping over the entire question. However, it also differs significantly from the localist account of Hirsch and Schwarz (2019). While Hirsch and Schwarz assume that the uniqueness presupposition of Which student danced? is triggered by a single question-internal element, namely the which-phrase, on our account there is no single element that contributes a uniqueness presupposition. Rather, the presupposition results from an interplay between atom, max * , ?u, and †, where atom and max * are contributed by the which-phrase, ?u is contributed by Foc u , and † is contributed by Int. We will therefore refer to our account as a decompositional account.
How does the decompositional account fare with respect to the cases discussed above in which uniqueness effects are obviated? As already remarked in Section 1, we will not provide a formal analysis of questions with modals here, since it is a non-trivial task to add modals to the Inq D framework, one that is beyond the scope of this paper. We will therefore concentrate on singular which-questions with disjunctions, such as (106). We will show that, for such cases, the decompositional account correctly predicts obviation of a global uniqueness requirement and instead derives the observed local uniqueness requirements pertaining to each individual disjunct.
The assumed syntactic structure for (106) is given in Figure 19. 42 Notice that, just as Socolof et al. (2020), we assume that the disjunction scopes below Foc. We refer to Socolof et al. (2020) for evidence supporting this syntactic assumption.  Figure 19: Assumed syntactic structure of (106).
The Inq D -translation of the question is given in (109). We assume here that the which-phrase is interpreted inside each disjunct via reconstruction after Across-the-Board movement. Any localist account has to assume such a reconstruction mechanism as well to derive local uniqueness requirements. 42 We assume here, as we do in general, that subject-auxiliary inversion involves movement of the auxiliary from T to Fin. Recall from footnote 33 that Haegeman (2012) instead assumes that subject-auxiliary inversion involves movement from T to Foc. The choice between these two syntactic assumptions has no repercussions for any of the examples we have seen so far. For example (106), however, the choice does make a difference. Under the T-to-Fin assumption that we are making, (106) can simply be taken to involve FinP disjunction, which means that the ! operator, contributed by Foc, takes scope over the disjunction without further ado, as desired. On the alternative T-to-Foc assumption, (106) would, at least as face value, involve Foc disjunction (there is an inverted auxiliary in both disjuncts, so each disjunct must contain a Foc head, but the which-phrase, assumed to be in SpecFocP, is outside of the disjunction). This would mean that the ! operator does not scope over the disjunction, which would yield undesirable predictions. There are various other syntactic assumptions that one may try to adapt in order to avoid this. For instance, one might assume that ! and the witness request operator are in SpecFocP rather than in Foc. Or one might assume that cases like (106) involve FinP disjunction after all and the auxiliaries in both disjuncts move only to Fin rather than Foc, as a last resort, because across-the-board movement to Foc is impossible if the two auxiliaries are not identical in form. We do not consider such options further here, and simply assume, unlike Haegeman (2012), that subject-auxiliary inversion always involves movement to Fin. This yields a straightforward analysis of cases like (106) The predicted update effect of the question is graphically represented in Figure 20. For simplicity, the diagrams in this figure involve only three worlds: w actual : the actual world, in which Shakespeare was born in Stratford upon Avon and Bach died in Leipzig.
w poss : a non-actual but possible world in which Shakespeare was born in London and Bach died in Berlin.
w imposs : a non-actual and impossible world in which Shakespeare was born in Stratford upon Avon and in London, and Bach died in Leipzig and in Berlin.
The third world is included to show that our account derives a local uniqueness requirement. The first update displayed in Figure 20 is the one induced by the disjunction. This update results in two alternatives, one consisting of possibilities in which u gets assigned atomic values which are, according to the world parameter of that possibility, the place(s) where Shakespeare was born, and one alternative consisting of possibilities in which u gets assigned atomic values which are, according to the world parameter of that possibility, the place(s) where Bach died. In the second update, induced by !, these two alternatives are joined into one big alternative consisting of all possibilities in which u gets assigned atomic values which are, according to the world parameter of that possibility, either the place(s) where Shakespeare was born or the place(s) where Bach died. Finally, in the third update, ?u applies. This again introduces two alternatives, but different ones. Namely, the first alternative consists of all possibilities in which Shakespeare was born in Stratford (and not in London) and Bach died in Leipzig (and not in Berlin), and the second alternative consists of all possibilities in which Shakespeare was born in London (and not in Stratford) and Bach died in Leipzig (and not in Berlin). Note that states containing possibilities in which Shakespeare was born in two places and Bach died in two places do not survive this update.
Crucially, as desired, we do not derive a global uniqueness requirement. That is, we do not predict that the question presupposes the existence of a unique town in which Shakespeare was born and Bach died. This is because atom{u} and max * {u} are interpreted inside the two disjuncts. At the same time, we do derive a local uniqueness requirement, that is, we predict that the question presupposes that there is a unique town in which Shakespeare was born and a unique town in which Bach died.
So, just like the localist account of Hirsch and Schwarz (2019) and Socolof et al. (2020), our decompositional account successfully derives a local uniqueness requirement in singular whichquestions with disjunction. As we will see next, our decompositional account also overcomes what Hirsch and Schwarz (2019, p.765) identify as the main problem that the localist account faces, namely the fact that a multiple-wh question like Which student talked to which professor?  does not presuppose that exactly one student talked to exactly one professor. A localist account would, without further stipulations, wrongly derive such a presupposition, since it assumes that each which-phrase contributes a uniqueness requirement all by itself. As we will see in Section 5.4, our decompositional account does not run into this problem since the operators contributed by a singular which-phrase, atom{u} and max * {u}, do not induce a uniqueness presupposition by themselves, but only through interaction with ?u and †, which are contributed by functional heads in the left periphery.

Multiple-wh questions with singular which-phrases
We now turn to multiple-wh questions. In this and the next two subsections we will discuss several variants. We start with multiple-wh questions containing two singular which-phrases, exemplified in (110).
(110) Which u 1 student read which u 2 book?
This question has a so-called pair-list interpretation. That is, to resolve the issue that the question raises, one has to provide a complete list of pairs x, y such that x is a student, y a book, and x read y. For instance, if the relevant set of students is {a, b, c, d}, the relevant set of books is {a , b , c , d }, a read a , b read b , c read c , and d read d , then the list of pairs that needs to be established is: Such a list of pairs encodes a certain relation between students and books. Dayal (1996Dayal ( , 2016 argues that multiple-wh questions with singular which-phrases generally presuppose that this relation has two specific properties. First, Dayal proposes that the relation must be domain exhaustive, which means that every student in the relevant domain must be related to at least one book. And second, the relation must be functional, which means that every student must be related to at most one book. The pair-list in (111)  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. Still, the question is felicitous. So, Xiang argues, domain exhaustivity is not a necessary component of the pair-list interpretation of multiple-wh questions, and we endorse this conclusion. We further observe, however, that in order to resolve the question in (114) one must still specify for as many candidates as possible (in this case three) which jobs they got. We refer to this requirement as domain maximality. For instance, if {c 1 , . . . , c 100 } are the relevant candidates, {j 1 , j 2 , j 3 } the relevant jobs, c 23 got j 1 , c 45 got j 2 , and c 87 got j 3 , then to fully resolve (114) one must establish the pair-list in (115), and not some subset thereof.
We will now demonstrate that our account captures these empirical properties of multiplewh questions with singular which-phrases, i.e., functionality and domain maximality, without any further stipulations. Given the assumptions we have made about the syntax of English and how logical forms of English sentences are mapped to Inq D expressions, the question in (110) is translated as in (116): Which u 1 student read which u 2 book? † ( [u 1 ]; student{u 1 }; atom{u 1 }; [u 2 ]; book{u 2 }; atom{u 2 }; read{u 1 , u 2 }; max * {u 1 }; max * {u 2 }; ?u 1 u 2 ) TypeP [u 1 ]; student{u 1 }; atom{u 1 }; [u 2 ]; book{u 2 }; atom{u 2 }; read{u 1 , u 2 }; max * {u 1 }; max * {u 2 }; ?u 1 u 2 FocP [u 1 ]; student{u 1 }; atom{u 1 }; [u 2 ]; book{u 2 }; atom{u 2 }; read{u 1 , u 2 }; max * {u 1 }; max * {u 2 } FinP which u 1 student read which u 2 book Foc u 1 ,u 2 Int Figure 21: Deriving the translation of (110). In this tree structure, we do not show the movement of the wh-phrase which student to SpecFocP. Rather the wh-phrase is shown in its base position, where it is interpreted. † The main steps in the derivation of this translation are given in Figure 21. Note that the ! operator contributed by Foc u 1 ,u 2 is vacuous in this case and has therefore been omitted from the translation.
The update effect of the question is graphically represented in Figure 22 and Figure 23. In these figures, dref assignment functions belonging to the same possibility are separated from other dref assignment functions by horizontal lines. Super-and subscripts on a world specify who read what in that world. For intance, w a;c b;c is a world in which a read c and b read c and no other reading happened. For reasons of space, we assume here that it is already known in the input context that a read c, d, or nothing at all, and that b read c, d, or both of them. We furthermore assume that a and b are students and c and d are books in all worlds.
Let us now zoom in on Figure 22, which graphically represents the update induced by the TP, i.e., before ?u 1 u 2 applies. This initial update yields a context with just one alternative, consisting of all possibilities in which (i) u 1 is a student, u 2 a book, u 1 read u 2 , and (ii) the values cumulatively assigned to u 1 and u 2 are maximal in comparison to other possibilities with the same world parameter. For instance, states containing the possibility w a;c b;c , {{u 1 → a, u 2 → d}} are eliminated because this possibility does not satisfy (i), and states containing the possibility w a;c b;c , {{u 1 → a, u 2 → c}} are eliminated because this possibility does not satisfy (ii). Figure 23 displays the result of subsequently applying the witness request operator ?u 1 u 2 contributed by Foc u 1 ,u 2 . This operator, as defined in (66), raises an issue whose resolution requires identifying a function from individuals satisfying all properties ascribed to u 1 to individuals satisfying all properties ascribed to u 2 . There are six alternatives in the output context in Figure 23. The leftmost alternative corresponds to the function {b → c}, the next corresponds to the function {b → d}, the rightmost alternative corresponds to the function {a → d, b → d}. Note that the two leftmost alternatives correspond to the functions that do not satisfy domain exhaustivity because student a is not mapped to any book. Also note that all states containing the possibility w b;c b;d , {{u 1 → b, u 2 → c}, {u 1 → b, u 2 → d}} (fifth row, third column) have been removed by ?u 1 u 2 . This is a desirable result, since the mapping from students to books that they read is not functional in w b;c b;d . After all, student b read two books, c and d, in this world. This example illustrates three general features of our account. First, it successfully captures the fact that a pair-list answer to a multiple-wh question with singular which-phrases must establish a functional mapping between individuals in the domain set (in our example, the students) and individuals in the target set (in our example, the books). Second, in line with Xiang's (2019) empirical observations contra an earlier hypothesis by Dayal (1996Dayal ( , 2016, our account does not require that the mapping satisfies domain exhaustivity (for instance, in our example it is predicted that the question can be resolved by establishing that student b read book c and student a did not read any book). And third, in line with our own empirical observations, the account does require that the mapping satisfies domain maximality-it must map as many individuals in the domain set as possible to suitable individuals in the target set. The account thus captures what we identified, building on the work of Dayal (1996Dayal ( , 2016, Xiang (2019) and others, as the main empirical properties of multiple-wh question with singular which-phrases.
Moving now from empirical adequacy to theoretical/explanatory adequacy, two additional features of the account are worth highlighting. First, the semantic relation between wh-phrases and Foc is akin to the semantic relation between anaphoric expressions and their antecedents: wh-phrases introduce discourse referents and the witness request operator contributed by Foc targets these discourse referents. This means that, for semantic purposes, it is not necessary for wh-phrases to move either overtly or covertly to the left periphery to take scope there. In particular, in multiple-wh questions in English, the wh-phrase that remains in-situ at the surface can also be interpreted in situ-covert movement to the left periphery is not necessary for interpretation. The same holds for languages in which wh-phrases more generally remain in-situ at the surface. This is a desirable feature of the account given that in situ wh-words can appear inside islands for movement in English and other languages, as exemplified in Reinhart's (1997) famous example in (117). (117) Which linguist will be offended if we invite which philosopher?
We assume that, when wh-phrases do move to the left periphery, this is entirely for syntactic reasons. A second feature of the account that speaks to its theoretical/explanatory adequacy is the fact that single-wh and multiple-wh questions are treated in a uniform way. In both cases, all the relevant elements are treated exactly the same: wh-phrases introduce discourse referents, possibly with an atomicity requirement (contributed by number marking) and/or a maximality requirement (encoded in the lexical semantics of which). Foc contributes a witness request operator, and Int contributes the † operator. Crucially, while in a single-wh question the witness request operator targets just one discourse referent and in multiple-wh questions it targets multiple discourse referents, we have given a single, general definition of ?u 1 . . . u n which applies to all cases at once. The example we've discussed above involved two wh-phrases, but cases with three or more wh-phrases, exemplified in (118) and (119), can be dealt with in exactly the same way. 43 (118) Which patient will receive which treatment on which day?
(119) [The library maintains a digital record of. . . ] which book was taken out by which member on which date for how many days.
Unlike other accounts (e.g., Dayal, 1996Dayal, , 2016Xiang, 2019), our account of single-wh questions scales up to multiple-wh questions without the need to stipulate additional semantic mechanisms (see Dotlačil and Roelofsen 2020, Appendix B, for further elaboration of this point).

Multiple-wh questions with plural which-phrases
We now briefly turn to multiple-wh questions with plural which-phrases, exemplified in (120).
(120) Which u 1 students read which u 2 books? † The only difference between the Inq D translation of (120) and that of (110) is that in (120) no atomicity requirement is placed on the values of u 1 and u 2 . As a consequence, we correctly predict that (120), unlike (110), is felicitous in a situation in which it is known that one or more students read multiple books. We also predict that domain maximality still needs to be obeyed. That is, the response needs to pair as many students as possible with the books they read. On the other hand, as in the case above with singular which-phrases, domain exhaustivity need not be obeyed. That is, the question can be felicitously asked even if the speaker knows that some students did not read any books.

Multiple-wh questions with partial mention-some readings
So far, we have focused on multiple-wh questions with which-phrases. Such phrases always place a maximality requirement on the values of the discourse referents that they introduce. We now turn to a case in which the target wh-phrase is a simplex wh-phrase rather than a complex which-phrase.
We have assumed that such simplex wh-phrases, just like indefinites, allow for a strong and a weak interpretation. On their strong interpretation, their behaviour in multiple wh-questions is predicted to be similar to that of plural which-phrases, which we have already examined above. However, on their weak interpretation they are predicted to behave differently and give rise to a reading that is impossible to get with which-phrases. Consider the example in (121).
43 Note in particular that examples like (118) can be asked in a situation in which each patient receives multiple treatments, each on a particular day. For instance, patient p1 receives treatment t1 on Monday and treatment t2 on Wednesday, while patient p2 receives treatment t1 on Tuesday and treatment t2 on Thursday. In such a situation, the type of treatment does not functionally depend on the patient, because there are two treatments per patient. However, the day of the treatment does functionally depend on the patient and the type of treatment. This kind of dependency is permitted by our account. Situations in which both the type of treatment and the day of the treatment functionally depend on the patient are special cases of this. As we observed in Section 2, this question has a partial mention-some reading. On this reading, its resolution requires specifying, for each herb (mention-all), at least one place where it grows (mentionsome). Such a partial mention-some reading is impossible to derive for most theories of questions, because the mention-all vs mention-some distinction is generally seen as a distinction that pertains to the interpretation of a question as a whole, rather than to the interpretation of individual whphrases (see, e.g., Beck and Rullmann, 1999;George, 2011;Theiler, 2014;Champollion et al., 2015). On our account, the mention-all vs mention-some distinction does pertain to the interpretation of individual wh-phrases. More specifically, whether a given wh-phrase receives a mention-all or a mention-some interpretation depends on whether a maximality requirement is placed on the values of the discourse referent that it introduces. As a result, our account straightforwardly derives partial mention-some readings for questions like (121). The update induced by (121) is graphically represented in Figure 24. Note that the issue in the output context is mention-all w.r.t. herbs but mention-some w.r.t. places. To resolve the issue, we have to specify at least one place (c, d, e or f ) for every herb (a and b). For example, establishing that a grows at c and b grows at e (without excluding that a and b grow in other places as well) is enough to remove all states containing the possibility w a;d b;f , {{a → d, b → f }} in the rightmost column, which leads to a new output context with only one alternative, i.e., a context which is no longer inquisitive.
The multiple-wh question in (121) involves two types of wh-phrases: the subject is a complex which-phrase, which only has a strong interpretation requiring maximality, while the adverb is a simplex wh-phrase, where, which permits both a strong and a weak interpretation, with or without maximality. What if we consider a case with two simplex wh-phrases, neither of which necessarily requires maximality? Is it possible to construct a multiple-wh question in this way which permits a 'pure' mention-some reading, i.e., a reading that is mention-some w.r.t. both wh-phrases?
It seems that, in English, this is impossible. To see this, first consider the single-wh questions in (122).
(122) a. What is a typical dish in France?
b. Context: Susan is throwing a party and I want to go. I can't drive, but my friends, who can drive, also want to go. So I ask my friends: Who is driving there tonight?
Note that each of these questions involves a simplex wh-phrase and has a very salient mention-some reading. However, if we add another simplex wh-phrase, as in (123), we do not obtain questions with pure mention-some readings.
(123) a. What is a typical dish where?
b. Context: Sue and Kim are both throwing a party, in different locations. I want to go to one of the parties, I don't care which. In either case I need a lift. So I ask my friends: Who is driving where tonight? This is predicted by our account, since we assume that Foc applies max * to all its domain discourse referents. However, this assumption is otherwise inessential for the account. It is only motivated by the absence of pure mention-some readings in questions like (123a-b) in English. Interestingly, Lucas Champollion (p.c.) has noted that in German, multiple-wh questions with two simplex wh-phrases do, at least in some cases, permit a pure mention-some reading. Champollion provides the following example: 44 This cross-linguistic variation can be captured on our approach if we assume that in German, unlike in English, Foc does not apply max * to all its domain discourse referents. This would only affect the predictions of the account concerning questions like (123), all other predictions would remain intact. We leave open here whether this difference between German and English can be tied to other differences between the two languages.

Conclusion
We have developed a type-theoretic dynamic inquisitive semantics framework, Inq D , and have presented a compositional dynamic analysis of wh-questions within this framework. The proposed analysis is based on the assumption that wh-phrases introduce a discourse referent, and an associ-ated operator in the left periphery raises an issue whose resolution requires identifying a witness for this discourse referent, i.e., an individual satisfying all the properties that have been ascribed to it.
We have argued that the analysis improves on previous proposals both in terms of empirical predictions and in terms of theoretical parsimony. In particular, the analysis derives mentionsome and mention-all readings from independently motivated assumptions about wh-phrases, it provides a uniform treatment of single-wh and multiple-wh questions, it derives partial mentionsome readings in multiple-wh questions, it captures the effect of singular and plural number marking in which-questions, and it predicts a local uniqueness requirement in singular which-questions with disjunction.
There are several natural avenues for further work. One salient open issue concerns the treatment of modal operators in Inq D . This issue is pertinent because, as we have seen, wh-questions interact with modal operators in interesting ways. A further extension of the present account should explain, for instance, why, as discussed in Section 1.3, questions like (125) permit a mention-some reading (Dayal, 2016) and why questions like (126) lack a global uniqueness presupposition (Hirsch and Schwarz, 2019).

(125)
Context: A researcher needs a few people with AB blood type to test a new drug. The study requires her to test multiple patients but not necessarily all the patients in the hospital with AB blood type. The researcher has a list of all patients in the hospital, but not their blood types. The administrator does have information mapping patients to blood types. The researcher asks the administator: Which patients can I approach for this test?
(126) Which letter can be inserted in fo m to make a word?
Other questions that need to be addressed in future work concern the behaviour of various kinds of quantifiers, and relatedly, various types of verbal predicates (collective vs distributive) in whquestions. For instance, a further extension of the present proposal should ideally leverage insights from earlier work on quantification and predication in dynamic semantics to explain why (127a) permits a reading under which the quantifier takes scope over the question, while (127b) does not (Szabolcsi, 1997, among others), and to derive the intricate resolution conditions of wh-questions with collective predicates like (128) (Xiang, 2021).
(127) a. Which book did every student read? b. Which book did fewer than three students read?
(128) Which children formed a team?