# Comparison to alternative approaches

# Comparison to alternative approaches

# Abstract and Keywords

Chapter 9 compares inquisitive semantics with some other frameworks which have been proposed for the analysis of questions, in particular with alternative semantics (developed by Hamblin and Karttunen in the 1970s), partition semantics (developed by Groenendijk and Stokhof in the 1980s), and inquisitive indifference semantics (developed by Groenendijk and Mascarenhas in the 2000s). It is argued that inquisitive semantics preserves the essential merits of these previous approaches, while overcoming their main shortcomings. The chapter is also concerned with the division of labor between question semantics and other components of a general theory of interpretation, including a theory of speech acts and discourse pragmatics. It discusses the received view on what the role of a compositional semantic theory of questions should be within such a larger theory of interpretation, and compares it to the one taken in inquisitive semantics, which is argued to be more parsimonious.

*Keywords:*
theories of questions, alternative semantics, partition semantics, speech act theory, discourse pragmatics

As we have seen in previous chapters, one of the main purposes of inquisitive semantics is to serve as a framework for the semantic analysis of questions in natural languages. In this chapter we will compare inquisitive semantics with some other frameworks which have been proposed for this purpose, and which have been used widely in the literature. In doing so, we will restrict our attention to those previous proposals that are most closely related to our own. That is, we will consider the *alternative semantics* framework proposed by Hamblin (1973) and Karttunen (1977), the *partition semantics* developed by Groenendijk and Stokhof (1984) and later cast in a dynamic framework by Jäger (1996), Hulstijn (1997), and Groenendijk (1999), and the *inquisitive indifference semantics* proposed by Groenendijk (2009) and Mascarenhas (2009). We will argue that the framework presented here preserves the essential insights that have emerged from these previous approaches, while overcoming their main shortcomings.^{1}

Figure 9.1 provides a global overview of the different approaches. In this figure, the proposed frameworks are ordered from left to right in terms of restrictiveness—i.e., the constraints they impose on what qualifies as a suitable question meaning—and the development through time is indicated by the bent arrows. Alternative semantics, developed in the 1970s, is the least restrictive among these framework, i.e., the one that imposes the least constraints on what qualifies as a suitable question meaning. This, as we will see below, leads to problems (p.164) of overgeneration. Partition semantics on the other hand, originally proposed in the 1980s and further developed in a dynamic setting in the 1990s, is the most restrictive framework—leading to problems of undergeneration. More recent work has tried to strike an optimal balance between these two extremes, first leading to inquisitive indifference semantics (Groenendijk, 2009; Mascarenhas, 2009) and then to the present inquisitive semantics framework.

Alternative semantics will be discussed in Section 9.1, partition semantics in Section 9.2, and inquisitive indifference semantics in Section 9.3, drawing comparisons in each case with the inquisitive semantics framework presented in this book. After having examined and compared these different frameworks for question semantics in some detail, we shift our attention in Section 9.4 to another fundamental issue, concerning the division of labor between question semantics and other components of a general theory of the interpretation of questions, including a theory of speech acts and discourse pragmatics. What exactly should the role of a compositional semantic theory of questions be within such a larger theory of interpretation? We will first consider the received view on this issue, and then compare it to the one taken in inquisitive semantics, which we argue to be more parsimonious.

# 9.1 Alternative semantics

Alternative semantics was first proposed by Hamblin (1973), driven by the following idea:

Questions set up a choice-situation between a set of propositions, namely those propositions that count as answers to it. (Hamblin, 1973, p. 48)

Thus, Hamblin takes questions to denote sets of classical propositions. These propositions are often referred to as *alternatives*, hence the name
(p.165)
of the framework. Karttunen (1977) independently proposed a very similar view on question meanings: he also took questions to denote sets of classical propositions, though he restricted the denotation of a question in a particular world to propositions that correspond to answers that are *true* in that world. In both systems, the *meaning* of a question, i.e., its intension, is a function from worlds to sets of classical propositions. In Hamblin’s system, this function maps every possible world to the same set of propositions, corresponding to the set of all possible answers; in Karttunen’s system, every world is mapped to a subset of all possible answers, namely those that are true in the given world. As noted by Karttunen (1977, p. 10), this difference is inessential. In both cases, the meaning of a question is fully determined by—and could be identified with—the set of all classical propositions that correspond to a possible answer.^{2}

This classical view on question meanings faces some fundamental problems. We will discuss these, and show that they no longer arise in inquisitive semantics.

## 9.1.1 First problem: Possible answers

The first problem is that the framework’s core notion—that of a *possible answer*—is difficult to pin down. Surely, Hamblin and Karttunen provide a compositional semantics for a fragment of English, and thereby specify what they take to be the possible answers to the questions in that fragment. But in order to assess such a compositional theory, or even to properly understand what its predictions amount to, we first need to have a pre-theoretical notion of possible answers, one that the theoretical predictions can be evaluated against. The problem is that such a pre-theoretical notion is difficult, if not impossible to identify. To illustrate this, consider the question in (1) and the responses in (2):

(1)

(2)

(p.166) In principle, each of the classical propositions expressed by the declaratives in (2) could be seen as a possible answer to (1). For Hamblin and Karttunen, only (2a) counts as such. However, it is not clear what the precise criteria are for being considered a possible answer, and on which grounds (2a) is to be distinguished from (2b–e).

In inquisitive semantics, question meanings are also sets of classical propositions, just like in alternative semantics. However, in inquisitive semantics these classical propositions are not thought of as the ‘possible answers’ to the question. Rather, they are thought of as the information states—or equivalently, the pieces of information—that *resolve* the issue that the question expresses. As a consequence, in inquisitive semantics question meanings cannot be defined as arbitrary sets of classical propositions, which is what Hamblin and Karttunen take them to be. Rather, they have to be *downward closed*. After all, if an information state *s* resolves the issue expressed by a given question *Q*, then any stronger information state $t\subset s$ will also resolve the issue expressed by *Q*. As a consequence, inquisitive semantics is more restrictive than alternative semantics.

Unlike the notion of a possible answer, the notion of a resolving information state has a clear pre-theoretical significance. For instance, here are two concrete ways to assess empirically whether an information state *s* should count as one in which the issue expressed by a question *Q* is resolved. Imagine an agent *a* whose information state is *s*, and suppose that the information available in *s* is true:^{3}

•

*Knowledge test*: would we say that*a*knows*Q*?– If the answer is

*yes*,*s*should count as a state in which the issue expressed by*Q*is resolved.– Otherwise,

*s*should not count as such a state.

•

*Wondering test*: is it possible for*a*to be wondering about*Q*?– If the answer is

*no*,*s*should count as a state in which the issue expressed by*Q*is resolved.– Otherwise,

*s*should not count as such a state.

(p.167)
If we apply these tests to the above example, it is clear that the information states corresponding to (2a–b) qualify as resolving states for (1), while those corresponding to (2c–e) do not.^{4} Thus, theories of questions formulated in inquisitive semantics can be assessed empirically in a way that theories that yield sets of ‘possible answers’ cannot—at least in the absence of a more precise characterization of what ‘possible answers’ are supposed to be.

Even though inquisitive semantics deliberately does not rely on the notion of ‘possible answers’ as a primitive notion, the framework does of course allow us to define various notions of answerhood. For instance, we may characterize a *minimal resolving answer* to a question *Q* as a piece of information that:

(i) resolves the issue expressed by

*Q*, and(ii) does not provide more information than necessary to do so, i.e., is not strictly stronger than any other piece of information that also resolves the issue expressed by

*Q*.

Under this definition, the minimal resolving answers to *Q* correspond precisely to what we called the *alternatives* in the proposition expressed by *Q*.^{5} In the above example, (2a) would count as a minimal resolving answer to the question in (1) under this definition, while (2b–e) would not. This is the same distinction that Hamblin and Karttunen made. Now, however, it is clear on which grounds the distinction is made.^{6}
(p.168)

Besides characterizing minimal resolving answers along these lines, we may also define notions of *partial answerhood* and *subquestionhood* (see Groenendijk and Roelofsen, 2009), which are important for the analysis of discourse structure and information structure (see, e.g., Ginzburg, 1996; Roberts, 1996; Büring, 2003). What is crucial is that in inquisitive semantics the meaning of a question is not characterized *in terms of* possible/minimal/complete/partial answers. Rather, as depicted in Figure 9.2, it is the other way around: question meanings, i.e., issues, are defined in terms of what it takes to resolve them, and the possible/minimal/complete/partial answers to a question are defined in terms of these resolution conditions. As a consequence, whichever notion of answerhood we choose to adopt, there will be no need for such a notion to correspond directly to some pre-theoretical concept. Rather, it will be grounded, in a precisely circumscribed way, in the pre-theoretical notion of what it takes for a given issue to be resolved.^{7}

## 9.1.2 Second problem: Entailment

A second fundamental problem for alternative semantics, which was pointed out and discussed at length in Groenendijk and Stokhof (1984), is that it is difficult to define a suitable notion of *entailment* in this framework that determines when one question is more demanding than
(p.169)
another. One consequence of this is that it is hard, if not impossible, to give a principled account of the interaction between questions and logical connectives and quantifiers. For instance, it proves problematic to give a satisfactory treatment of the conjunction of two questions. Without a suitable notion of entailment, conjunction can certainly no longer be treated as a *meet* operator.^{8}

This problem does not arise in inquisitive semantics, which comes with a natural and well-behaved notion of entailment. As discussed in Chapter 3, the space of propositions in inquisitive semantics, ordered by entailment, has a familiar algebraic structure, and a natural treatment of the logical connectives is obtained by associating them with the basic operations in this algebra. Thus, as we have seen, the classical treatment of conjunction as a *meet* operation can be preserved in inquisitive semantics to apply to informative and inquisitive sentences in a uniform way, and the same goes for the other operations.

The two problems that we just discussed for alternative semantics are closely related. After all, if it were possible to ground the notion of ‘possible answers’ in some pre-theoretical notion, then it would most likely also become clear how to characterize entailment. That is, if there were clear criteria for what it takes to count as a possible answer, we would also know better on which grounds two sets of possible answers should be compared, and under which conditions one set should be seen as entailing another.^{9}

Compare the situation with the one we have in classical logic. There, the proposition expressed by a sentence is a set of possible worlds. These worlds are intended to correspond to situations that are compatible with the information that the sentence conveys. In this case, there is a clear pre-theoretical intuition to build on, as to whether a certain situation is or is not compatible with a given piece of information. As a consequence, it is also clear when one sentence should be taken to entail another, namely if it conveys at least as much information, (p.170) meaning that the proposition it expresses is a subset of the proposition expressed by the other sentence. In alternative semantics, the meaning of a question is a set of classical propositions which are intended to correspond to its possible answers. However, since it is not clear when exactly a proposition should count as a possible answer, it is also difficult to say when one question should entail another.

In inquisitive semantics, the proposition expressed by a question is a set of information states, which are intended to be those information states that resolve the issue that the question expresses. It is also clear, then, when one question is more demanding than another, namely if every information state that resolves the former also resolves the latter. This immediately delivers the desired notion of entailment, as well as the algebraic operations that are characterized in terms of it.

## 9.1.3 Third problem: Overgeneration

A third problem, which is again connected to the other two, is that there are question meanings in alternative semantics which seem impossible to express in natural languages. These are question meanings containing two alternatives *α* and *β* such that one is strictly contained in the other, $\alpha \subset \beta $.

One may think that such meanings may be expressed by disjunctive questions, where each disjunct contributes one of the two alternatives. However, in order to get that $\alpha \subset \beta $, we would have to construct the question in such a way that one disjunct entails the other. As illustrated in (3) and (4) below, such questions are infelicitous (Ciardelli and Roelofsen, 2017a).

(3)

(4)

It has been well-known since Hurford (1974) that disjunctive declaratives where one disjunct entails the other are generally infelicitous as well.

(5)

(6)

This phenomenon, known as *Hurford’s constraint*, has been given an appealing explanation in terms of *redundancy*. More specifically, Katzir and Singh (2013) propose the following principle (see also Simons, 2001; Meyer, 2014, for closely related proposals):
(p.171)

Local redundancy:a sentence is deviant if its logical form contains a binary operator ∘ applying to two argumentsAandB, and the outcomeA∘Bis semantically equivalent to one of the arguments.^{10}

Let us briefly consider how this principle predicts Hurford’s constraint. In classical semantics, the meaning of a sentence *A* is a classical proposition |*A*|, the set of worlds where the sentence is true. *A* entails *B* just in case $|A|\subseteq |B|$. Moreover, sentential disjunction yields the union of two propositions, that is, $|A\vee B|=|A|\cup |B|$.

Now, suppose that the logical form of a sentence contains a sentential disjunction operator applying to two arguments *A* and *B* such that $|A|\subseteq |B|$, as in examples (5) and (6). Then we have that $|A\vee B|=|A|\cup |B|=|B|$. So, the output is semantically equivalent to one of the inputs. Thus, the given logical form exhibits local redundancy and is therefore predicted to be deviant.

Now, one would of course hope that this explanation of Hurford’s constraint in terms of redundancy would apply not only to declaratives like (5) and (6), but also to questions like (3) and (4). But this is not the case in alternative semantics, where the disjuncts express singleton sets, {|*A*|} and {|*B*|}, respectively, and disjunction yields the set {|*A*|, |*B*|}. Since the output of the disjunction operator is different from any of its inputs, the local redundancy condition is not violated, and no deviance is therefore predicted.

In inquisitive semantics, the explanation of Hurford’s constraint in terms of redundancy does naturally apply to questions like (3) and (4). Assuming that each of the disjuncts expresses a proposition containing all states that consist exclusively of worlds where that disjunct is true (just like atomic sentences in InqB), and that or is analysed by means of the inquisitive disjunction operator, we have [*A*] = *℘*(|*A*|), [*B*] = *℘*(|*B*|), and $[A\vee B]\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}[A]\cup [B]\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\wp (|A|)\cup \wp (|B|)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\wp (|B|)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}[B]$. Thus, the output of the disjunction operator is identical to one of its inputs, and redundancy is predicted just as for declarative Hurford disjunctions.^{11}

(p.172) Let us try to better understand this contrast between inquisitive semantics and alternative semantics by considering the notion of ‘alternatives’ that plays a role in the two frameworks. We have seen that both frameworks associate questions with sets of alternatives, but that the status of these alternatives crucially differs from one framework to the other.

In inquisitive semantics, the alternatives in the proposition expressed by a question are characterized as those pieces of information that resolve the issue that the question raises in a minimal way. This implies that sets of alternatives have to be of a particular form: two alternatives are always logically independent, that is, one is never contained in the other.

In alternative semantics on the other hand, there is no such constraint on sets of alternatives: any set will do. This is connected, of course, to the fact that the notion of an alternative is a primitive notion in this framework, not defined in terms of resolution conditions or any other more elementary notion.

Let us say that a set of classical propositions is *non-nested* if no proposition in the set is included in another. In inquisitive semantics, then, unlike in alternative semantics, only non-nested sets of classical propositions are regarded as proper sets of alternatives. Thus, certain meanings in alternative semantics do not have a counterpart in inquisitive semantics. It is precisely these additional meanings, i.e., nested sets of alternatives, which seem impossible to express in natural languages. In principle, a Hurford disjunction would be exactly the right kind of construction to express a nested set of alternatives. But we have seen that such disjunctions are infelicitous. This seems to indicate that there is something wrong with nested sets of alternatives as meanings, which is puzzling from the perspective of alternative semantics, since in this framework nested sets of alternatives have exactly the same status as non-nested sets.

In inquisitive semantics, the puzzle does not arise, because nested sets of alternatives simply do not exist. Importantly, such sets are not ruled out by some special purpose constraint: rather, it just follows from the (p.173) way alternatives are construed that they are never nested. This means that from the perspective of inquisitive semantics, what is special about Hurford disjunctions is not that they express some distinguished class of meanings, but rather that they involve redundant disjuncts, which fail to contribute an alternative to the meaning of the disjunction. As we have seen, this is precisely what explains their infelicity.

# 9.2 Partition semantics

Departing from Hamblin and Karttunen’s work, Groenendijk and Stokhof (1984) propose that a question does not denote a set of classical propositions at each world, but rather a single classical proposition embodying the *true exhaustive* answer to the question in that world. For instance, if *w* is a world in which Paul and Nina are coming for dinner, and nobody else is coming, then the denotation of (7) in *w* is the classical proposition expressed by (8).

(7)

(8)

The *meaning* of a question, i.e., its intension, then amounts to a function from worlds to classical propositions. In Groenendijk and Stokhof’s framework these classical propositions are required to have two special properties: they have to cover the entire logical space (since we must have a true exhaustive answer at every world), and they have to be mutually exclusive (since at each world, only one exhaustive answer can be true). So, in Groenendijk and Stokhof’s theory the meaning of a question is determined by a set *ρ* of classical propositions that together form a *partition* of the logical space.

## 9.2.1 Problem: Undergeneration

Partitions correspond to a specific kind of issues. Indeed, if a given question *Q* has a true exhaustive answer at each world, then resolving the question amounts to providing an exhaustive answer. This means that if *Q* is associated with a partition *ρ*, then *Q* is resolved at a state *s* if and only if *s* is included within some complete answer *t* ∈ *ρ*. This shows that each partition *ρ* determines an issue *I*_{ρ} consisting of all states that are contained in one of the cells of the partition:

(p.174) However, not every issue corresponds to a partition. More importantly, not all issues expressed by natural language questions correspond to partitions. This is so only for those questions that have an exhaustive answer at every world, in the following sense (for a precise statement of this fact, see Exercise 9.5).

(9)

There are two different reasons why a question may fail to have an exhaustive answer at a world. First, it may not be possible to truthfully resolve the question at some possible worlds. For instance, consider (10): as soon as our logical space includes worlds where Alice does not have a husband, (10) cannot have an exhaustive answer at these worlds.

(10)

However, this limitation can be overcome quite straightforwardly: it suffices to take a question to express a *partial* function from worlds to exhaustive answers, which corresponds to a partition of a *subset* of the logical space.

However, the partition theory cannot be patched up in a similar way to deal with questions that allow for various minimal resolving propositions that are all true at some world. An important class of questions with this feature is that of *mention-some* questions like those in (11), which we discussed in Section 5.4.2.

(11)

In order to resolve (11a) it is necessary and sufficient to establish of some *x* that *x* is something that Alice really likes. If Alice really likes string quartets as well as scuba diving, then the proposition that she likes string quartets and the proposition that she likes scuba diving are both true at the actual world, and both resolve (11a) in a minimal way. Thus, at the actual world, there is no single proposition that counts as the exhaustive answer to (11a). This implies that the issue expressed by (11a) cannot be represented as a partition.

(p.175) In addition to mention-some questions, the class of questions which express non-partition issues also includes conditional questions like (12a), disjoined questions like (12b), open disjunctive questions like (12c), and approximate-value questions like (12d) (about the latter type of question, see exercise 9.4).

(12)

Thus, while partition semantics gives an attractive analysis of questions, which avoids the problems that we highlighted above for alternative semantics, it does so at the cost of significant restrictions on its empirical scope.^{12}

## 9.2.2 A possible concern: disjunctions of questions

Inquisitive semantics provides a notion of question meaning that is richer than the one assumed in partition semantics, and we have just seen that this is crucial in order to accommodate several classes of questions which express issues that do not correspond to partitions of the logical space. However, this greater generality may also raise a certain concern.

Consider the following sentence from Szabolcsi (1997, p. 325), a disjunction of two *wh*-questions, which is decidedly odd.

(13)

Szabolcsi (1997, 2015a) has argued that the oddness of this sentence can be explained in partition semantics. For a partition may be identified with an equivalence relation on the space of possible worlds, and while the intersection of two equivalence relations is itself again an equivalence relation, the same is not true of the union of two equivalence relations. If conjunction and disjunction are taken to express intersection and union, respectively, it is to be expected that conjunction, but not disjunction, can apply to two questions to form a new question in natural languages.

On the other hand, in inquisitive semantics the oddness of (13) cannot be explained on purely semantic grounds, because if we take (p.176) disjunction to express the join operator it delivers a perfectly sensible issue, one that can be resolved either by establishing whom the addressee married or by establishing where the addressee lives. This issue does not correspond to a partition, but it is an issue nonetheless in our framework. Thus, while the inquisitive notion of meaning has important advantages with regard to partition semantics, it may also seem to have a certain disadvantage.

However, note that the prediction arising from partition semantics is a very strong one: it implies that questions cannot be directly disjoined in natural languages *at all*. Szabolcsi (1997, 2015a) claims that this general prediction is indeed borne out,^{13} but we are convinced by examples like (14), repeated from Chapter 1, that it is too strong: disjunctions of questions are not always infelicitous.

(14)

We should note that Szabolcsi remarks that a sentence like (13) may be marginally acceptable if regarded as a case in which the speaker first asks the question *who did you marry*, but then reconsiders and proposes to replace this first question by the second, *where do you live*. In such cases, Szabolcsi suggests, disjunction does not play its usual role but is rather used as a corrective device.

Our example (14), however, can be uttered by someone without any reconsideration halfway, and it can be addressed by an addressee as a single question, to which both disjuncts contribute. So, (14) seems to be a genuine disjunction of questions.^{14}

Szabolcsi (1997) does not base her empirical claim merely on cases like (13) but also on observations about embedded questions in Hungarian. Hungarian complement clauses, whether declarative or interrogative, are always headed by the subordinating complementizer *hogy*. Szabolcsi argues that this subordinating complementizer expresses a *lifting* operation that needs to be invoked before two
(p.177)
interrogative complement clauses can be disjoined (just like proper names have to be lifted into generalized quantifiers when they are conjoined or disjoined with a quantificational noun phrase). Support for this idea comes from examples like (15) and (16) below, which indicate that (i) conjoined interrogative complement clauses can have either two occurences of *hogy*, applying to both individual conjuncts, or a single occurence of *hogy*, applying to the conjunction as a whole, but (ii) disjoined interrogative complement clauses must have two occurences of *hogy*, each applying to one of the individual disjuncts.

(15)

(16)

Szabolcsi concludes from this observation that disjunction cannot directly apply to interrogative complement clauses, but always requires intervention of a lifting operation, expressed overtly in Hungarian by *hogy*.

However, there are counterexamples to the generalization. The Hungarian counterpart of our example (14) is a case in point. When embedded, it may come either with one or with two occurences of *hogy*, no matter whether the embedding verb is extensional (e.g., *find out*) or intensional (e.g., *investigate*).^{15}

(p.178)(17)

(18)

In (18), a single occurence of *hogy* favors a reading on which disjunction takes narrow scope with regard to the verb, while two occurences of *hogy* favor a reading on which disjunction takes wide scope (Peter is investigating where we can rent a car or he is investigating who has one we could borrow), a pattern that is in line with Szabolcsi’s idea that *hogy* expresses a lifting operation.

It thus seems that, at least in some cases, disjunction *can* apply directly to questions, both in English and in Hungarian. A question that naturally arises, then, is whether the general disjunction operation that inquisitive semantics makes available allows us to derive the correct meaning for those disjunctions of questions which *are* felicitous. For disjunctions of non-*wh*-questions, we have already argued this to be the case in Chapter 6. The predictions for disjunctions of *wh*-questions also seem to be correct. For instance, assuming that (14) is an open interrogative list and that the two interrogative clauses each receive a mention-some interpretation, the sentence is predicted to express an issue which can be resolved either by identifying a place where the speaker can rent a car, or by identifying a person who might have a car that the speaker can borrow, or by establishing that there is no such place and no such person. These are indeed the resolution conditions we expect for (14). Notice that this prediction is obtained simply by applying inquisitive disjunction to the propositions expressed by the two interrogative clauses—the same disjunction operation that, in Chapter 6, we took to be at work in disjunctive non-*wh*-questions, as well as disjunctive declaratives.

Thus, after all, disjunctive questions seem to provide a strong argument in favor of inquisitive semantics over partition semantics, where examples such as (14) can only be handled at the cost of a significant complication of the framework (and one that gives up some of its most attractive features, such as the general account of entailment and coordination among interrogatives; see Groenendijk and Stokhof, 1989).

(p.179) Of course, an interesting question that remains to be addressed is why our example (14) behaves so differently from Szabolcsi’s example (13), both as a standalone question and when embedded. We think that the difference may be explained pragmatically. A disjunction of two questions expresses an issue that may be resolved equally well by providing information resolving the first disjunct, or by providing information resolving the second disjunct. Now, it is difficult to see what kind of motivation (or what kind of decision problem, to follow van Rooij 2003) a speaker could have that would lead her to raise or even consider the issue expressed by (13). This is very different in the case of (14): in this case, it is immediate to reconstruct the sort of motivation that may lead a speaker to consider the relevant issue. We suggest that the different cognitive plausibility of the two issues at stake underlies the difference in the perceived felicity of the associated questions.

## 9.2.3 Dynamic partition semantics

While Hamblin (1973), Karttunen (1977), and Groenendijk and Stokhof (1984) all operate under a static view on meaning, there are also a number of proposals that aim to capture the meaning of questions in a *dynamic* framework. The first such proposals, developed by Jäger (1996), Hulstijn (1997), and Groenendijk (1999), essentially reformulate the partition theory of questions in the format of an update semantics (Veltman, 1996).^{16} This means that they construe the meaning of a sentence as its *context change potential*, i.e., a function that maps a context to a new context. Just like we do here, these theories do not model a context simply as a set of worlds—embodying the information established in the conversation so far—but provide a more refined notion of context, one that also embodies the issues that have been raised so far. More specifically, a context *C* is modeled as an *equivalence relation* over a set of worlds $s\subseteq \mathcal{W}$. Such an equivalence relation, which induces a partition on *s*, can be taken to encode both information and issues. On the one hand, the information established in *C* is encoded by the set of all worlds that are in the *domain* of *C*, i.e., all worlds in *s*. On the other hand, the issues present in *C* are encoded by the partition that *C* induces: two worlds *w* and *v* are connected by *C* and therefore included in the same partition cell just in case the distinction between *w* and *v* is not (yet) at stake in the conversation. In other words, *C* is conceived of as a relation encoding *indifference* (Hulstijn, 1997): if
(p.180)
*w* and *v* are connected by *C*, the discourse participants have not yet expressed an interest in information that would distinguish between *w* and *v*.

Both statements and questions can then be taken to have the potential to change the context in which they are uttered. A statement restricts the domain *s* to those worlds in which the sentence is true (strictly speaking, it removes all pairs of worlds 〈*w*, *v*〉 from *C* such that the sentence is false in at least one of the two worlds). Questions *disconnect* worlds, i.e., they remove a pair 〈*w*, *v*〉 from *C* just in case the true exhaustive answer to the question in *w* differs from the true exhaustive answer to the question in *v*.

Thus, the dynamic systems of Jäger (1996), Hulstijn (1997), and Groenendijk (1999) provide a notion of context and meaning that embodies both information and issues in an integrated way, in terms of an equivalence relation encoding indifference. However, just as classical partition semantics, these dynamic systems are too restrictive to allow for a satisfactory treatment of conditional questions, disjunctive questions, and mention-some *wh*-questions.^{17}

# 9.3 Inquisitive indifference semantics

A core assumption of dynamic partition semantics is that indifference should be encoded by means of an *equivalence relation* between possible worlds. Mascarenhas (2009) and Groenendijk (2009) suggest that the limitations of the framework may be overcome by re-examining this assumption. Of course, whether indifference should be encoded by means of an equivalence relation between possible worlds depends on how exactly one conceives of the notion of indifference. One natural perspective is that an agent is indifferent between two worlds *w* and *v* just in case *w* and *v* agree on the truth value of all propositions that the agent deems relevant. Under this perspective, a relation encoding the agent’s indifference should indeed be an equivalence relation—it should be reflexive because every world will clearly agree with itself on the truth value of all relevant propositions; it should be symmetric because if *w*
(p.181)
agrees with *v* then *v* must also agree with *w*; and it should be transitive because if *w* agrees with *v* and *v* with *u*, then *w* must agree with *u* as well.

However, this is not the only sensible way to conceive of the notion of indifference. Another natural perspective is that an agent is indifferent between two worlds *w* and *v* just in case, if she were to be given the information that the actual world is either *w* or *v*, the issues that she entertains would be resolved and she would not require further information determining precisely which of *w* and *v* is the actual world. Under this perspective, indifference relations should still be reflexive and symmetric, but they do not necessarily have to be transitive. To see this, suppose that our agent entertains just one issue, namely the one expressed by the mention-some question in (19):

(19)

Now consider the following three possible worlds:

•

*w*_{1}: Italian newspapers are only sold at Central Station,•

*w*_{2}: Italian newspapers are only sold at the airport,•

*w*_{3}: Italian newspapers are sold in both places.

If the agent were to be told that the actual world is either *w*_{1} or *w*_{3}, her issue would be resolved: she could go get her newspaper at Central Station and would not require further information determining which of *w*_{1} and *w*_{3} is the actual world. The same holds if the agent were to be told that the actual world is either *w*_{2} or *w*_{3}: in this case she could get her newspaper at the airport. However, if she were told that the actual world is either *w*_{1} or *w*_{2}, then she would need further information in order to decide where to go: if the actual world is *w*_{1} she has to go to Central Station, but if it is *w*_{2} she has to go to the airport. Thus, the agent is indifferent between *w*_{1} and *w*_{3} as well as between *w*_{3} and *w*_{2}, but not between *w*_{1} and *w*_{3}. This means that her indifference relation is not transitive.

In view of such considerations, Groenendijk (2009) and Mascarenhas (2009) dropped the transitivity constraint on indifference relations. In the resulting framework, which they referred to as *inquisitive semantics*, the alternatives associated with a question are maximal sets of worlds such that each pair in the set stands in the indifference relation induced by the question. Since indifference relations are no longer required to be transitive, the alternatives associated with a question may overlap. For instance, in the above example, if we take our logical space to consist
(p.182)
of *w*_{1}, *w*_{2} and *w*_{3}, the alternatives associated with the question in (19) are {*w*_{1}, *w*_{3}} and {*w*_{2}, *w*_{3}}. These alternatives overlap, since they both contain *w*_{3}. Note that {*w*_{1}, *w*_{2}, *w*_{3}} is not an alternative associated with (19), because *w*_{1} and *w*_{2} do not stand in the indifference relation that (19) induces.

Allowing for issues with overlapping alternatives by dropping the transitivity requirement on indifference relations makes it possible to extend the scope of partition semantics. However, Ciardelli (2008) observed that the gain in generality resulting from this move is not yet sufficient. While conditional questions like (12a) and open disjunctive questions with two disjuncts like (12c) can be dealt with satisfactorily, disjunctive questions with three or more disjuncts remain problematic, and the same goes for mention-some *wh*-questions like (19).

To briefly illustrate what the problem is (see also Exercise 9.5), consider a context *c* consisting of three possible worlds, as above, but now let the availability of Italian newspapers in these worlds be as follows:

•

*w*_{1}: Italian newspapers are only sold at the Central Station and the zoo.•

*w*_{2}: Italian newspapers are only sold at the airport and the zoo.•

*w*_{3}: Italian newspapers are only sold at the Central Station and the airport.

The question in (19) is not resolved in this context, since *c* does not imply of any place that it sells Italian newspapers. However, for any pair of worlds *v*, *v′* from the context *c*, if we are given the information that the actual world is either *v* or *v′*, then more precise information distinguishing between the two worlds is not needed in order to resolve (19). Thus, the indifference relation expressed by (19) in the given context is the total relation—which amounts to a trivial issue. This shows that the approach of Groenendijk (2009) and Mascarenhas (2009) cannot detect the fact that (19) expresses a non-trivial issue in the given context.

This example shows that the resolution conditions of a question cannot in general be reconstructed from the indifference relation that it induces. Moreover, Ciardelli (2008) argued that it is impossible to overcome this problem without letting go of the most fundamental notion in the framework of Groenendijk and Mascarenhas, inherited from dynamic partition semantics, namely that of issues encoded by means
(p.183)
of indifference relations. This insight led to the inquisitive semantics framework presented in this book.^{18}

To distinguish the two stages in the development of inquisitive semantics, we refer to the framework proposed by Groenendijk (2009) and Mascarenhas (2009) as *inquisitive indifference semantics*.^{19} In terms of restrictiveness, inquisitive indifference semantics is situated in between partition semantics and the current inquisitive semantics framework, as indicated in Figure 9.1. This is a direct consequence of the main commonalities and differences between these frameworks. On the one hand, inquisitive indifference semantics is similar to dynamic partition semantics and differs from the present inquisitive semantics framework in that it encodes issues by means of indifference relations. On the other hand, it is similar to the present inquisitive semantics framework and different from dynamic partition semantics in that it ultimately characterizes issues in terms of what is needed to *resolve* them. Only, unlike in the present framework, this is not done directly, but *via* indifference relations. That is, an issue is encoded as an indifference relation, and whether this relation holds between two worlds *w* and *v* depends on whether the issue is resolved by the information that the actual world is either *w* or *v*. In the present framework, the connection between issues and resolution conditions is more direct, since indifference relations no longer play a role.

Summing up, while some important aspects of inquisitive indifference semantics persist in the present framework, its core notion—issues encoded by means of indifference relations—has been replaced by a more general one, and this generality is needed to suitably capture the full range of question types in natural languages. Thus, our framework naturally fits within the existing tradition of semantic theories of informative and inquisitive discourse, but it is more general and able to cover more empirical ground than its predecessors.

# (p.184) 9.4 Division of labor

So far, we have compared a number of approaches based on how they model question meanings—i.e., in terms of possible answers, true exhaustive answers, indifference relations, or resolution conditions. In this final section, we shift our attention to a different issue: What should the role of a compositional semantics be within a larger theory of question interpretation? That is, how should the labor be divided between the compositional semantics of questions and other components of the theory? We will first consider the received view on this issue, and then compare it to the one taken in inquisitive semantics.^{20}

## 9.4.1 The received view

The received view is one which, in fact, assigns a very minimal role to compositional semantics. In order to say more specifically what this role is, we have to look separately at matrix questions and embedded questions. In the case of matrix questions, it is assumed under this view that the issue raised in asking the question is not only determined by the compositionally derived semantic value of the question, but also in part by a specific *update rule* associated with question speech acts.

Similarly, in the case of embedded questions, it is assumed that the semantic contribution of the embedded clause is not only determined by the compositionally derived semantic value of the question, but also by an *answer operator* which is assumed to always accompany embedded questions.

Let us make this more concrete by briefly outlining a particular theory instantiating this view. We opt here for the theory presented in Heim (2016), which is very explicit about the interaction between the semantics of questions, update rules, and answer operators. But many other contemporary theories of questions assume a similar division of labor and would in principle serve our present purposes equally well.

First, let us exemplify the semantic values that Heim (2016) compositionally assigns to simple *wh*-questions and polar questions. The *wh*-question in (20a) receives the semantic value in (20b), a set containing the classical proposition |*Pd*| for every individual *d*.

(20)

(p.185)
On the other hand, the polar question in (21a) receives the semantic value in (21b), a singleton set containing just |*Pj*|.

(21)

Note that Heim operates in the alternative semantics framework of Hamblin (1973) and Karttunen (1977), where question meanings are arbitrary sets of classical propositions, i.e., they need not be downward closed, and they need not form partitions either. However, the semantic values that Heim derives, in particular for polar questions, do not coincide with those assumed by either Hamblin or Karttunen. That is, while Hamblin and Karttunen take a polar question like (21a) to have two ‘possible anwers’, |*Pj*| and |¬*Pj*|, the semantic value of (21a) in Heim’s system only contains the first of these two.

This should not be taken to reflect a disagreement as to what the ‘possible answers’ to polar questions like (21a) are. Rather, it reflects a different take on what the semantic value of a question is intended to represent. For Heim, the compositionally derived semantic value of a question is not intended to directly embody the set of ‘possible answers’ to that question, or any other type of answers for that matter. The semantic value of a question is just an abstract object that serves as input for the update rule associated with questions (in the case of matrix questions) or a suitable answer operator (in the case of embedded questions). We now turn to these ingredients of the theory.

In line with much earlier work, Heim assumes that in making a statement or asking a question, a speaker proposes to update the conversational context in a particular way. What the proposed update is, is determined by the update rules associated with statements and questions, respectively. To spell out what Heim takes these rules to be, we first need to briefly review how she models conversational contexts. In this, Heim essentially follows the work on dynamic partition semantics discussed above (Jäger, 1996; Hulstijn, 1997; Groenendijk, 1999). That is, she models a conversational context *C* as a set of mutually disjoint classical propositions. The union of these classical propositions, $\bigcup C$, is a set of worlds, embodying the information that has been commonly established among the conversational participants in *C*. On the other hand, the elements of *C* together form a partition over $\bigcup C$, embodying the exhaustive answers to the current question under discussion.

(p.186) With this notion of contexts in place, Heim defines the following update rules for statements and questions, respectively.

Definition 9.1 (Heim’s update rule for statements)

In making a statement, i.e., in uttering a declarative sentence

φ, a speaker proposes to replace the current contextCby a new context which is constructed by collecting all non-empty intersections of elements ofCwith |φ|, the classical proposition expressed byφ. That is,Cis to be replaced by:

• $\{p\cap |\phi |\mid p\in C\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}p\cap |\phi |\ne \varnothing \}$

Definition 9.2 (Heim’s update rule for questions)

In asking a question, i.e., in uttering an interrogative sentence

φ, a speaker proposes to replace the current contextCby a new context which is constructed by re-partitioningCinto cells consisting of worlds that agree on the truth of every element of [φ]. That is,Cis to be replaced by:

• $\left\{\phantom{\rule{0.3em}{0ex}}\{v\mid \forall p\in [\phi ]:w\in p\iff v\in p\}\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}w\in \bigcup C\right\}$

To illustrate these update rules, suppose that our initial context is one in which no information has been established and no questions have been asked yet, i.e., *C* = {*W*}. Now suppose a speaker utters the declarative sentence *Mary is going to the party*, translated as *Pm*. Then, according to the update rule for statements in Definition 9.1, the speaker proposes to replace *C* with the following context:

(22)

If the proposal is accepted by the other conversational participants, *C′* becomes the new context, which means that the information that Mary is going to the party becomes common ground, and no further information is requested.

Now let us return to our initial context *C* and suppose that the speaker instead utters the polar interrogative in (21a), *Is John going to the party?*. This time, according to the update rule for questions in Definition 9.2, the speaker proposes to replace *C* with the following context:

(23)

(p.187)
Thus, no new information becomes common ground, since $\bigcup {C}^{\prime}$ still covers the set of all possible worlds *W*. However, the context is now partitioned into two cells, |*Pj*| and |¬*Pj*|, which means that the participants become publicly committed to resolving the question whether John is going to the party or not.^{21}

Now let us move from matrix questions to embedded ones. Heim mainly focuses on questions embedded under *know*. For sentences in which *know* takes a declarative complement, she assumes the standard analysis. For instance, (24a) is assigned the truth-conditions in (24b):^{22}

(24)

Here, *σ*_{a}(*w*) is Ann’s information state in *w*, modeled as a set of possible worlds, and |*Pj*| is the classical proposition expressed by *Pj*, also a set of possible worlds.

What if *know* takes a question as its complement, whose semantic value is not a set of possible worlds but rather a set of classical propositions? Clearly, the standard analysis of *know*, exemplified in (4), cannot immediately be applied in this case. To overcome this obstacle, Heim assumes, as many other authors have done as well, that the semantic value of an embedded question, before it combines with that of the verb, is first transformed by a so-called *answer operator*, ANS. This answer operator takes as its input a possible world *w* and a question meaning *Q*, i.e., a set of classical propositions, and delivers as its output a single classical proposition ANS(*w*, *Q*). More specifically, Heim defines the answer operator in such a way that ANS(*w*, *Q*) is always the true exhaustive answer to *Q* in *w*.

Definition 9.3 (Heim’s answer operator)

For any possible world

wand any question meaningQ:

• ANS(

w,Q) := {v∣∀p∈Q:v∈p⇔w∈p}

Thus, a sentence like (25a) receives the truth-conditions in (25b), which is a satisfactory result. (p.188)

(25)

This completes our illustration of the received view on the division of labor between compositional semantics and other components of an overall theory of question interpretation. Note that, as indicated at the outset, the role assigned to compositional semantics on this view is very minimal. In the case of matrix questions a decisive role is played by the update rule, and in the case of embedded questions such a role is fulfilled by the answer operator. In both cases, the semantic value that is produced by the compositional semantics only serves as ‘raw material’ for these operators.

This, in our view, is a substantial weakness of the approach: it leaves the compositional semantics of questions highly *unconstrained*. That is, which values are produced by the compositional semantics does not matter all that much, as long as one can formulate an update rule and an answer operator which, when given these values as input, yield the desired output.

It is important to note in this regard that both the update rule and the answer operator are assumed to be specific to questions. Thus, they can be tailor-made for the purpose of transforming the compositionally derived semantic values to yield the desired output. They do not serve a broader purpose in the overall theory of interpretation, and are therefore not independently constrained. This diminishes the explanatory value of the approach.

It would be preferable to have a theory that does without any question-specific update rule or answer operator, i.e., one in which there is just one simple update rule that applies uniformly to statements and questions, and one in which the embedding operator—if at all present—is not specific to questions, but applies uniformly to declarative and interrogative embedded clauses. This is precisely the kind of approach that comes naturally with inquisitive semantics.

## 9.4.2 The inquisitive perspective

Let us first consider matrix questions and then embedded ones. We have already seen in Section 2.5.3 that inquisitive semantics comes with a natural notion of context update, which is simply defined in terms of (p.189) set intersection and applies uniformly to statements and questions. This allows for a unification and simplification of Heim’s update rules for statements and questions.

Definition 9.4 (Inquisitive update rule for statements and questions)

In uttering a sentence

φ, be it a declarative or an interrogative, a speaker proposes to replace the contextCby a new context which is constructed by intersectingCwith [φ]. That is,Cis to be replaced by $C\cap [\phi ]$.

For comparison, let us apply this rule to the examples considered above. As before, let the initial context *C* be one in which no information is available and no issues have been raised yet. In inquisitive semantics, this context is represented as {*W*}^{↓}. Now suppose a speaker utters the declarative sentence *Mary is going to the party*, translated as *Pm*. Then, according to our general update rule, the speaker proposes to replace *C* with the following context:

(26)

If the proposal is accepted by the other conversational participants, *C′* becomes the new context, which means that the information that Mary is going to the party becomes common ground, and no further information is requested.

Now let us again return to our initial context *C* and suppose that the speaker instead utters the polar interrogative in (21a), *Is John going to the party?* According to our general update rule, the speaker proposes to replace *C* with the following context:

(27)

In this case, the speaker does not provide any information herself, but she does raise an issue, requesting information from other participants in order to establish a common ground that is contained either in |*Pj*| or in |¬*Pj*|.

Thus, the results are essentially the same as in Heim’s system, but they are obtained by using a uniform update rule (indeed, the standard intersective update rule) rather than two separate update rules for statements and questions.

(p.190) It should be emphasized that what we have gained is not just simplicity. More importantly, the fact that the update rule determining the contextual effect of questions is not question-specific but rather plays a more general role in the overall theory means that it is constrained by considerations that are independent of questions altogether. This means that the approach leaves less room for ad-hoc stipulations, and is therefore more explanatory.

Now let us turn to embedded questions. We have seen in Section 8.2.2 that in inquisitive epistemic logic the standard analysis of *know* is generalized in such a way that it can deal uniformly with both declarative and interrogative complements. The support conditions for *K*_{a}*φ* are repeated in (28) and the truth-conditions which can be derived from this in (29).

(28)

(29)

Applying this analysis to the examples considered above yields the following results:^{23}

(30)

(31)

Thus, again, the same results are obtained as in Heim’s account, but this time without an answer operator. Just as in the case of matrix questions, this is not just a gain in simplicity, but also in explanatory force—assuming an answer operator that is specific to embedded questions and does not serve a more general purpose in the overall theory makes room for ad-hoc customization and leaves the compositional semantics of questions highly unconstrained. On the other hand, doing without such an answer operator leads to a theory in which the compositional semantics of questions has to deliver semantic values which can be fed immediately to the embedding verbs, without any transformation. This, together with the fact that the same semantic values should also serve
(p.191)
as input to the general intersective update rule discussed above in case the question occurs in matrix form, results in a much more constrained theory, and thus one with a greater explanatory value.^{24}

# 9.5 Exercises

## Exercise 9.1 Inquisitive semantics and alternative semantics

1. What is the technical difference between question meanings in alternative semantics and in inquisitive semantics?

2. Conceptually, what is the reason for this difference?

3. What are the repercussions of this difference for the analysis of questions in natural languages?

## Exercise 9.2 Conjunction in alternative semantics

Assume that in alternative semantics a polar question ?*p* is associated with the set of possible answers $[?p]=\{|p|,\overline{|p|}\}$.

1. Suppose conjunction is analysed in terms of intersection:

$$\begin{array}{rll}[Q\wedge {Q}^{\prime}]=[Q]\cap [{Q}^{\prime}]& & \end{array}$$What set of possible answers does this analysis yield for (32)?

(32)

2. Now suppose conjunction is analysed in terms of

*point-wise*intersection:$$\begin{array}{rll}[Q\wedge {Q}^{\prime}]=\{a\cap {a}^{\prime}\mid a\in Q\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{a}^{\prime}\in {Q}^{\prime}\}& & \end{array}$$What set of possible answers does this analysis deliver for (32)?

3. Can this point-wise conjunction operation be characterized as a meet operator with respect to some relation of entailment?

*Hint:*a meet operation must validate certain principles: in particular, it must be commutative (*a*∧*b*=*b*∧*a*), associative (*a*∧ (*b*∧*c*) = (*a*∧*b*) ∧*c*), and idempotent (*a*∧*a*=*a*). Does the above definition of conjunction validate these principles?

## (p.192) Exercise 9.3 Inquisitive semantics and partition semantics

Let *I* be an issue, *w* a world, and *a* a classical proposition. We say that *a* is a *true complete answer to I at w* if (i) *w* ∈ *a* and (ii) for every information state *s* with *w* ∈ *s*: $s\in I\phantom{\rule{2.77695pt}{0ex}}\iff \phantom{\rule{2.77695pt}{0ex}}s\subseteq a$.

1. Show that if a true complete anwer to

*I*at*w*exists, then it is unique.2. Let us say that an issue

*I*is a*partition issue*if it is induced by a partition, i.e., if there is a partition*ρ*such that*I*=*I*_{ρ}, where ${I}_{\rho}:=\{s\subseteq t\mid t\in \rho \}$. Show that*I*is a partition issue iff a true complete answer to*I*exists at each possible world.3. Show that if an issue

*I*has two overlapping alternatives, or its alternatives do not cover the whole logical space, then*I*is not a partition issue.

## Exercise 9.4 Approximate value questions

Consider the following question:

(33)

1. Describe the issue expressed by this question.

2. Using the characterization of partition issues given in the previous exercise, show that this issue is not induced by a partition of the logical space.

## Exercise 9.5 Inquisitive semantics versus indifference semantics

Determine the interpretation of a disjunction with three disjuncts, *p* ∨ *q* ∨ *r*, in the semantics of Groenendijk (2009) and Mascarenhas (2009). How does this differ from the proposition assigned to *p* ∨ *q* ∨ *r* in InqB? How does this difference arise?

## Notes:

(^{1})
One prominent approach that we will not discuss here is the *functional* approach (sometimes also called the *categorial* or the *structured meanings* approach), which has its roots in the work of Hull (1975), Tichý (1978), and Hausser and Zaefferer (1978), and has been further developed by Ginzburg and Sag (2000), Krifka (2001a), and Ginzburg (2005), among others. For overviews of the literature on questions, we refer to Groenendijk and Stokhof (1997), Ginzburg (2010), Krifka (2011), Cross and Roelofsen (2014), Dayal (2016), and Dekker et al. (2016).

(^{2})
It should be noted that there are significant differences between Hamblin’s and Karttunen’s approach concerning the *compositional derivation* of question meanings. while Karttunen sticks to the standard Montagovian architecture, Hamblin proposes a rather radical departure from it, adapting the semantic type of all lexical items and letting the operation that is standardly used to compose the meanings of two constituents, i.e., *function application*, operate in a pointwise fashion. This compositional architecture, however, faces a number of thorny problems (see, e.g., Shan, 2004; Novel and Romero, 2010; Charlow, 2014). In inquisitive semantics these problems can be overcome in a principled way. A detailed discussion of compositionality, however, is beyond the scope of this book; we refer to Ciardelli, Roelofsen, and Theiler (2017a).

(^{3})
For the knowledge test, readers concerned about Gettier cases should replace the term ‘information state’ by ‘knowledge state’, and assume that only information that properly qualifies as knowledge is reflected in *s*.

(^{4})
By the information state corresponding to a declarative sentence we mean the set of worlds where the sentence is true, in an information model containing no background information.

(^{5})
Recall from footnote 3 in Chapter 2 that some propositions in InqB do not contain any alternatives. According to the characterization of minimal resolving answers just given, questions expressing such propositions do not have any minimal resolving answers. See Ciardelli (2010), Ciardelli et al. (2013b), and Roelofsen (2013a) for further discussion of such cases.

(^{6})
It is important to note that the Hamblin/Karttunen notion of a possible answer does not in general coincide with the notion of a minimal resolving answer. For instance, consider the question (i) under its usual, mention-all reading.

((i))

((ii))

Assuming that it is known which individuals make up the intended domain, the classical proposition in (iia) qualifies as a minimal resolving answer to (i). The one in (iib) does not, since it provides no information on which individuals passed the exam besides Alice. For Hamblin and Karttunen, the situation is reversed: the proposition in (iib) qualifies as a possible answer to (i), but not the one in (iia). Thus, the Hamblin/Karttunen notion of possible answers cannot be explicated in terms of minimal resolving pieces of information.

(^{7})
Recall from Section 5.6.2 that the issue expressed by questions in natural language is sometimes not completely determined by linguistic conventions; various contextual factors may play a role as well (the intended domain of quantification, the intended method of identification, the intended level of granularity, and the general decision problem that the speaker aims to resolve in asking the question). The challenges that are involved in modeling this context-sensitivity are orthogonal to the challenge of suitably representing the issue expressed by a question under fixed assumptions about the relevant contextual parameters. Inquisitive semantics, just like the work of Hamblin (1973) and Karttunen (1977), addresses the latter challenge, but remains neutral with respect to the former.

(^{8})
See Roelofsen (2013a); Ciardelli and Roelofsen (2017a); Ciardelli et al. (2017a) for more elaborate discussion of this point, and a critical assessment of some concrete notions of entailment and conjunction that may be considered in alternative semantics.

(^{9})
It is not the formal notion of meaning as such that stands in the way of a suitable notion of entailment, but really the conception of these meanings in terms of possible answers. For instance, if we construe the meaning of a sentence as a set of classical propositions, as in alternative semantics, but think of these propositions as those that the sentence *draws attention to*, rather than as possible answers, then it is quite straightforward to define a suitable notion of entailment, which compares two sentences/meanings in terms of their attentional strength (Roelofsen, 2013b).

(^{10})
Katzir and Singh (2013)’s proposal is relativized to a context of utterance *c*. Since context-dependency plays no role in our discussion, we omit reference to contexts for ease of exposition.

(^{11})
It should be noted that there are apparent counterexamples to Hurford’s constraint, which may seem to undermine the argument that we are making here. For instance:

((i))

At first blush, it seems that the second disjunct entails the first, and yet the sentence is felicitous. However, as argued in detail by Chierchia et al. (2009), in such cases the weaker disjunct receives an exhaustive interpretation—here, that Bill solved *only* two of the problems—which in effect makes it logically independent from the other disjunct. In fact, Hurford’s constraint allows us to explain why the only reading of (i) is one in which the first disjunct receives an exhaustive reading. So, far from undermining the existence of Hurford’s constraint, cases like (i) provide further evidence for it. For a more detailed exposition of the argument that we are presenting here, taking cases like (i) into account, we refer to Ciardelli and Roelofsen (2017a).

(^{12})
For a more detailed discussion of the relations between inquisitive semantics and partition semantics, see Ciardelli et al. (2015, §5) and Ciardelli (2017b).

(^{13})
Krifka (2001b) endorses Szabolcsi’s claim, though he offers a different explanation, based on the assumption that questions do not express sets of propositions or partitions, but rather *speech acts*, which Krifka models as operations on commitment states. Speech act disjunction does not exist according to Krifka, because it “would lead to disjunctive sets of commitments, which are difficult to keep track of” (Krifka, 2001b, p. 16).

(^{14})
Haida and Repp (2013) also challenge Szabolcsi’s empirical claim, although they maintain a weaker version of it: questions can only be disjoined in downward entailing or non-veridical contexts. Our example (17) presents a challenge for this weaker claim as well.

(^{15})
We are grateful to Donka Farkas, Anikó Liptak, and Anna Szabolcsi for discussion of this datapoint.

(^{16})
See the book *Questions in dynamic semantics* (Aloni et al., 2007) for several papers elaborating on these early proposals.

(^{17})
At least not without further amendments. Isaacs and Rawlins (2008) develop a dynamic partition semantics that allows for hypothetical updates of the context of evaluation. This framework allows for a natural analysis of conditional questions. However, open disjunctive questions and mention-some *wh*-questions remain beyond its reach.

(^{18})
A bit more historical detail: Groenendijk’s 2009 paper was written and started circulating in 2007, but appeared in print only in 2009. Mascarenhas’s 2009 master thesis was also largely written in 2007, but only presented in its final form in 2009. Ciardelli’s 2008 term paper was written in the fall of 2008, for a course taught by Groenendijk. The arguments presented in the term paper were further elaborated in Ciardelli (2009) and Ciardelli and Roelofsen (2011).

(^{19})
In previous work (e.g., Ciardelli, 2009; Ciardelli and Roelofsen, 2011), inquisitive indifference semantics has also been referred to as ‘inquisitive *pair* semantics’, since it can be characterized in terms of a notion of support with regard to pairs of worlds (see Groenendijk, 2009). Here, we instead use the term ‘inquisitive *indifference* semantics’ because it refers more transparently to the framework’s central concept.

(^{21})
The reader is invited to verify that the partition induced by a *wh*-question like (23) according to Heim’s update rule for questions is precisely the one that is associated with it in partition semantics.

(^{22})
We simplify here somewhat, leaving factivity presuppositions out of consideration. These are orthogonal to the main issues that we will be concerned with in this section.

(^{23})
The derivation in (31b) assumes that *σ*_{a} is factive, i.e., that for any *w*, *w* ∈ *σ*_{a}(*w*). For discussion of this constraint, see page 152.

(^{24})
A detailed inquisitive account of declarative and interrogative embedded clauses and the verbs that take such clauses as their complement is given in Theiler et al. (2016b). This account, unlike the one presented here, assumes that embedded clauses always involve a so-called *embedding operator*. This operator, however, is not specific to embedded questions. It applies uniformly to both declarative and interrogative embedded clauses. Thus, it does not make the overall theory less constrained.