Wh-questions in dynamic inquisitive semantics

1.1 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.

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 the development of dynamic semantics (Groenendijk and Stokhof 1991; Heim 1982; Kamp 1981, 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).

1.2 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 modelled after Partee’s marble cases.

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):

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):

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:

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 (Bhat 2000; Haspelmath 1997; Hengeveld et al. 2021; Ultan 1978, 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]

The main precedent of our proposal is the work of Haida (2007). There is earlier work on questions in dynamic semantics as well (e.g., Aloni and van Rooij 2002; Groenendijk 1998; van Rooij 1998). 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 (2019, 2021a, 2021b). This approach is similar to Haida’s approach and the one we will pursue here in that it also takes wh-elements 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 (Roelofsen 2019a; Theiler et al. 2018; Uegaki 2019), 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 Li 2021b; van Rooij 1998).

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. 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):

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 (Ciardelli 2017; Ciardelli and Roelofsen 2017; Ciardelli et al. 2017, 2018; Roelofsen 2013). 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.

1.3 Contributions

We aim to make two contributions. The first concerns framework building. We present a full-fledged 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 (Brasoveanu 2008; Muskens 1996; van den Berg 1996, 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:

1. 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.

2. Uniform treatment of single-wh and multiple-wh questions

   In many languages, including English, questions may involve multiple wh-expressions, as exemplified in (13).

     (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.

3. 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 mention-some 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.

4. 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) does not have a mention-some interpretation, unlike its variant in (16) which involves a number-neutral wh-phrase.

     (15)

   Participant of workshop in Amsterdam to the workshop organiser:

   Which local participants have a bike that I could borrow for 15 min?

     (16)

   Who has a bike that I could borrow for 15 min?

   This constraint can be obviated, however, in the presence of an existential modal operator, as illustrated by the following example from Dayal (2016):

     (17)

   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?

   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.

5. 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 (Dayal 1996, 2016; Higginbotham and May 1981).

     (19)

   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 single-wh 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 (2016, 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 (2016, 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 (2016, 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, I n q 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 I n q 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.

2.1 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.

2.2 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.

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.

This strong/weak ambiguity of donkey anaphora has been a central topic in the dynamic semantics literature (see, e.g., Brasoveanu 2008; Champollion et al. 2019; Heim 1990; Kanazawa 1994). 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).

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).

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 mention-some and mention-all readings can be derived in a principled way, relying on a mechanism that is independently needed elsewhere in the grammar.^[7]

2.3 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 ?u₁u₂ is a functional witness request operator.

Informally, ?u₁u₂ raises an issue whose resolution requires determining, for every possible witness set of u₁, the corresponding witness set of u₂. Or, in other words, a suitable function from the witness sets of u₁ to those for u₂. We will see in Section 3 that both ?u and ?u₁u₂ 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.

2.4 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).

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).

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 wh-phrases 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.

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:

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 max* operator applies to the domain dref or not. Our account will derive two readings for multiple-wh 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 Champollion et al. 2015; George 2011; Theiler 2014), 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).

2.5 Effects of number marking

Several effects of number marking need to be captured but the main puzzle is that singular which-phrases induce a uniqueness presupposition in single-wh questions but allow for multiple witness-pairs in multiple-wh questions (e.g., Dayal 1996).

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 ?u₁u₂ is at play in this case. This does not remove possibilities from the input context where more than one girl or more than one boy danced, but only ones where there is no 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.

3.1 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.

3.1.3 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 u₁ is introduced, new possibilities are created, each involving a randomly picked instantiation of u₁. 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 (2007, 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, g₁, g₂, g₃, each assigning values to three active drefs, u₁, u₂, u₃, is given in (41):

(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.

3.2 A basic formal language and its interpretation

The central formal notion introduced so far is that of I n q 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 type-theoretic 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 I n q 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 ∈ T y p e s . We write E x p r 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 C o n τ for the set of all constants of a given type τ, and V a r τ for the set of all variables of type τ.

Figure 1:

e-type variables, r-type constants, discourse referents, entities, and the various mappings between them.

3.3 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.

3.3.1 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{u₁, u₂}.^[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 (Kratzer 2008; Krifka 1989; Landman 2000). In (42), this is captured by the * preceding the predicate. The semantic interpretation of *R was given in Section 3.2.^[19]

(42)

smile{u} ≔ λc k λs i . s ∈ c ∧ ∀p ∈ s. ∀g ∈ G p . *smile(w)(g(u))

The predicational update function smile{u} can be generalized to the n-ary relational update function R{u₁, …u _n } in (43):

(43)

R{u1, …u n } ≔ λc k λs i . s ∈ c ∧ ∀p ∈ s. ∀g ∈ G p . *R(w)(g(u1)) … (g(u n ))

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.

Figure 2:

The update effect of smile{u}. In this and the following diagrams, subscripts on worlds specify which individuals smile in that world.

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.

3.3.3 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 Section 3.2: it expresses that the dref assignment matrix in the possibility p′ differs from that in p at most w.r.t. u.

(45)

[ u ] ≔ λ c k λ s i . ∃ s ′ ∈ c . ∀ p ∈ s . ∃ p ′ ∈ s ′ . ( p ′ [ u ] p ) ∧ ∀ p ′ ∈ s ′ . ∃ p ∈ s . ( p ′ [ u ] p )

Procedurally speaking, we create a state s in the output context of [u] by taking a state s′ from the input context c and requiring that every possibility p in s differs from some possibility p′ in s′ only w.r.t. u, and that for every possibility p′ in s′ there is some possibility p in s which only differs from p′ w.r.t. u.^[20]

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.

Figure 3:

An illustration of a sequence of updates starting with the introduction of a dref.

3.3.4 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:

(46)

U T ⊔ U ′ T ≔ λ c k . U ( c ) ∪ U ′ ( c )

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.

Figure 4:

The update effect of ([u₁]; sing{u₁}) ⊔ ([u₁]; smile{u₁}). 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.

In dynamic semantics, it is often assumed that disjunction is ‘internally dynamic’ and ‘externally static’ (see, e.g., Groenendijk and Stokhof 1991; Groenendijk et al. 1996; Kamp and Reyle 1993). 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 I n q 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, u₁. 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.	B i l l u 1 either rented a u 2 station wagon or a u 2 van. H e u 1 took i t u 2 out for three days.
b.	B i l l u 1 either rented a u 2 car or decided to hitchhike. * H e u 1 took i t 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, u₁, 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 u₂, e.g., TakeOutForThreeDays{u₁, u₂}. This is not the case for the output context on the right, which corresponds to the first sentence in (47b). In this output context, u₂ does not receive a value in all possibilities, so the result of a subsequent update with TakeOutForThreeDays{u₁, u₂} would be undefined.^[22]

Figure 5:

A simplified visual illustration of how the contrast in (47) is captured.

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 ↓ ?

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

3.3.5 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]

(50)

¬ ¬ U T ≔ λ c k λ s i . s ∈ c ∧ ∀ t ⊆ s . ( t ≠ ∅ → t  does not subsist in  U ( c ) )

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{u₁} ⊔ smile{u₂}), and to an update expression introducing a new discourse referent, [u₃]; smile{u₃}. The output context is the same in both cases. Note that this output context is not inquisitive, and that u₃ is not present in the output context.

3.3.6 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 ≔ λ c k λ s 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{u₁} ⊔ smile{u₂} and the dref-introducing update [u₃]; smile{u₃}. 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 [u₃]; smile{u₃} it does not make the discourse referent inaccessible for subsequent anaphoric expressions, unlike double negation.

Figure 7:

Comparison of ! and double negation. Each subfigure shows the result of updating the input context that is given at the top of the figure with the update function presented in the box. For simplicity, we assume in the first and second figure on the right that u₃ can only be mapped to a or b (no plural individuals are considered here).

3.3.8 Presupposing non-informativeness

Another operator which will play an important role in our treatment of interrogatives is the presuppositional closure operator, † (Champollion et al. 2017; Roelofsen 2015). 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.

(54)

† U T ≔ λ c k . U ( c )   if  ⋃ c  subsists in  ⋃ U ( c ) undefined   otherwise

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 c₂ but not anymore in U ( c 2 ) . So U provides information about the world that is not available yet in c₂. This means that ⋃c₂ does not subsist in ⋃ U ( c 2 ) , and therefore † U ( c 2 ) is undefined.

Figure 8:

Two input contexts and one output context to illustrate the workings of †.

3.3.9 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.

Figure 9:

Illustration of the workings of atom{u}.

3.3.10 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.

(56)

m a x { u } ≔ λ c k λ s i . s ∈ c ∧ ∀ p ∈ s . ∀ p ′ ∈ ⋃ c . w p = w p ′ → ⊕ G p ′ ( u ) ⊆ ⊕ G p ( u )

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.

Figure 10:

Workings of max{u}.

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.

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 I n q 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)

[u1]; student{u1}; atom{u1};

[u2]; professor{u2}; atom{u2};

see{u1, u2};

max{u1}; max{u2}

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 u₁ is strictly contained in a ⊕ b or the cumulative value of u₂ is strictly contained in c ⊕ d ⊕ e ⊕ f is removed since in these possibilities the cumulative value of either u₁ or u₂ 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 u₁ is a ⊕ b and the cumulative value of u₂ 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 u₂ with respect to any value of u₁. In (59), this only holds for the dref assignment matrix on the right.

(59)

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 ⃗ \ u n ↦ 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.

(60)

m a x * { u n } ≔ λ c k λ s i . s ∈ c ∧ ∀ p ∈ s . ∀ p ′ ∈ ∪ c . w p = w p ′ → ∀ x ⃗ . ⊕ G p ′ u ⃗ \ u n ↦ x ⃗ ( u n )   ⊆   ⊕ G p u ⃗ \ u n ↦ x ⃗ ( u n )

The effect of max* is illustrated in Figure 11. There are two discourse referents in this example and max* applies to u₂. The output context contains only those possibilities in which u₂ is maximal for any value of u₁. Concretely, states containing the top possibility in the input context are eliminated because u₂ is not maximal there for any value of u₁ compared to the two other possibilities.

Figure 11:

Workings of max*. There are two discourse referents in this example, u₁ and u₂. 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 u₁ and b to u₂ and the remaining assignments assign b to u₁ and a to u₂.