## Ivano Ciardelli, Jeroen Groenendijk, and Floris Roelofsen

Print publication date: 2018

Print ISBN-13: 9780198814788

Published to Oxford Scholarship Online: December 2018

DOI: 10.1093/oso/9780198814788.001.0001

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# Basic notions

Chapter:
(p.13) 2 Basic notions
Source:
Inquisitive Semantics
Publisher:
Oxford University Press
DOI:10.1093/oso/9780198814788.003.0002

# Abstract and Keywords

Chapter 2 specifies how issues, information states, propositions, and conversational contexts are formally defined in inquisitive semantics. It also defines a number of fundamental relations that may hold between such entities. For instance, it specifies what it means for an information state to resolve an issue or to support a proposition, what it means for a context to be updated with a proposition, and when one proposition entails another.

In the previous chapter we have argued that a formal notion of issues is of crucial importance for the analysis of linguistic information exchange. The present chapter specifies how issues are formally defined in inquisitive semantics. It also defines three other basic notions—information states, propositions, and conversational contexts—and a number of fundamental relations that may hold between such entities. In particular, as depicted in Figure 2.1, we will specify what it means for an information state to resolve an issue or to support a proposition, what it means for a context to be updated with a proposition, when one context is an extension of another, when one proposition entails another, when one information state is an enhancement of another, and when one issue is a refinement of another.

Before turning to the inquisitive setting, however, we first briefly review how these notions—with the exception of issues—are standardly defined.

# 2.1 The standard picture

The simplest way to construe information states, propositions, and conversational contexts is as sets of possible worlds (see, e.g., Hintikka, 1962; Stalnaker, 1978). A set of possible worlds can be thought of as representing a certain body of information, namely the information that the actual world corresponds to one of the worlds in the set. Such a body of information may be seen as the information available to a certain conversational participant; in that case it can be taken to represent the information state of that participant. On the other hand, a body of information may also be seen as the information conveyed by a certain sentence; in that case it can be taken to constitute the semantic content of that sentence, the proposition that it expresses. And finally, a body of information could be seen as the information that has so far been commonly established by all the participants in a conversation; in (p.14) that case it embodies the common ground of the conversation, which constitutes a minimal representation of the conversational context.1 Thus, depending on the perspective one takes, one and the same type of formal object—a set of possible worlds—can be used to model all three basic notions.

Figure 2.1 Basic notions in inquisitive semantics.

Let us now turn to the notions of enhancement (between information states), entailment (between propositions), and extension (between contexts). One information state s is an enhancement of another information state s′ just in case all the information available in s′ is also available in s, i.e., if every candidate for the actual world that is ruled out by s′ is also ruled out by s. This holds just in case $s⊆s′$. Similarly, one proposition p entails another proposition p′ if and only if p contains at least as much information as p′ does, i.e., if $p⊆p′$, and one context c is an extension of another context c′ if and only if all the information that is commonly established in c′ is also commonly established in c, i.e., if $c⊆c′$. Thus, enhancement, entailment, and extension again formally all amount to the same relation, i.e., set inclusion, though in each case we take a somewhat different perspective on what this formal relation encodes, mirroring the different perspectives on sets of possible worlds (p.15) when viewed as information states, propositions, and conversational contexts.

Now let us turn to the notion of support, which relates information states to propositions. An information state s is standardly taken to support a proposition p just in case the information embodied by p is already available in s, i.e., if every candidate for the actual world that is ruled out by p is ruled out by s as well. This holds just in case $s⊆p$. So support, just like entailment, enhancement, and extension, formally amounts to set inclusion.

Finally, let us consider the notion of update. The result of updating a context c with a proposition p is a new context c[p] which, besides the information already present in c, also contains the information embodied by p. That is, a candidate for the actual world is ruled out by c[p] if it was already ruled out by the information established in the old context c, or if it is ruled out by the new information embodied by p. Formally, this means that update amounts to set intersection: $c[p]=c∩p$.

What we have just reviewed is the simplest possible way to define information states, propositions, conversational contexts, and the relations that may hold between them in possible world semantics. Various more fine-grained versions of these basic notions have been proposed in the literature. Our goal here, however, is to construct the direct counterparts of these basic notions, together with a new notion of issues, in the inquisitive setting. Once these elementary notions are in place, one could set out to adapt the various refinements that have been proposed in the standard setting to the inquisitive setting as well. This will not be our direct concern in this book, but we will point to other work where such refinements have been pursued.

We are now ready to start building up the inquisitive semantics framework, starting with the notion of information states.

# 2.2 Information states

Information states are modeled in inquisitive semantics just as they are in the standard setting, namely as sets of possible worlds—those worlds that are compatible with the information available in the state. There is no need to change the notion of information states since—unlike in the case of propositions and conversational contexts, as we will see in Sections 2.4 and 2.5—this notion is just supposed to capture a body of information, and not anything issue-related.

(p.16) Even though we straightforwardly adopt the standard notion of information states, we will define, discuss, and exemplify the notion somewhat more explicitly here than in the brief review in 2.1, in preparation of what is to come next. We use $W$ to denote the entire logical space, i.e., the set of all possible worlds.

Definition 2.1 (Information states)

An information state s is a set of possible worlds, i.e., $s⊆W$.

We will often refer to information states simply as states. Figure 2.2 depicts some examples of information states in a logical space consisting of just four possible worlds: w1, w2, w3, w4. Intuitively, an information state can be thought of as locating the actual world within a certain region of the logical space. For instance, the state in Figure 2.2(d) contains the information that the actual world is located in the upper left corner of the logical space, while the state in Figure 2.2(c) contains the information that the actual world is located in the upper half of the logical space.

Figure 2.2 Information states.

If s and t are two information states and $t⊆s$, then t contains at least as much information as s; it locates the actual world with at least as much precision. In this case, we call t an enhancement of s.

Definition 2.2 (Enhancements)

A state t is called an enhancement of s just in case $t⊆s$.

Note that we do not require that t is strictly contained in s, i.e., that it contains strictly more information than s. If $t⊂s$, then we say that t is a proper enhancement of s.

The four information states depicted in Figure 2.2 are arranged from left to right according to the enhancement order. The state in Figure 2.2(b) is an enhancement of the state in Figure 2.2(a), and so on. The state consisting of all possible worlds, $W$, depicted in Figure 2.1(a), is the least informed of all information states: any possible world is still taken to be a candidate for the actual world, which means that we have (p.17) no clue at all what the actual world is like. This state is therefore referred to as the ignorant state. Every other state is an enhancement of it.

At the other far end of the enhancement order is the empty state, . This is an enhancement of any other state. It is a state in which all possible worlds have been discarded as candidates for the actual world, i.e., the available information has become inconsistent. It is therefore referred to as the inconsistent state.

# 2.3 Issues

We now turn to the notion of issues, in a sense the most central notion in inquisitive semantics. How should issues be represented formally? Our proposal is to characterize issues in terms of what information it takes to resolve them. That is, an issue is identified with a set of information states: those information states that contain enough information to resolve the issue.

We assume that every issue can be resolved in at least one way, which means that issues are identified with non-empty sets of information states. Moreover, if a certain state s contains enough information to resolve an issue I, then this must also hold for every enhancement $t⊆s$. This means that issues are always downward closed: if I contains a state s, then it contains every $t⊆s$ as well. Thus, issues are defined as non-empty, downward closed sets of information states.2

Definition 2.3 (Issues)

An issue is a non-empty, downward closed set of information states.

Definition 2.4 (Resolving an issue)

We say that an information state s resolves an issue $I$ just in case $s∈I$. If s resolves I, we will sometimes also say that I is settled in s.

Figure 2.2 displays some issues. In order to keep the figures neat, only the maximal elements of these issues are depicted. Since issues are downward closed, we know that all enhancements of these maximal elements are also included in the issues at hand. The issue depicted in subfigure (a) can only be settled consistently by specifying precisely which world is the actual one. The issue depicted in subfigure (b) can (p.18) be settled either by establishing that the actual world is an element of the set {w1, w2}, or by establishing that it is an element of {w3, w4}. The issue depicted in subfigure (c) can be settled either by establishing that the actual world is an element of {w1, w3, w4}, or by establishing that it is an element of {w2, w3, w4}. Finally, the issue in subfigure (d) is trivially settled: it does not require any information as to what the actual world may be.

Given an issue I, the information state $s:=⋃I$ (the union of all the elements in I) contains exactly the information that is necessary and sufficient to guarantee that I can be truthfully resolved, i.e., to guarantee that there is an information state that resolves the issue and contains the actual world. For, if the actual world is located in s, this means that it belongs to some tI: in this case t is an information state that resolves I and contains the actual world, which means that I can indeed be truthfully resolved. On the other hand, if the actual world lies outside of s, then it does not belong to any information state that resolves I, which means that I cannot be resolved truthfully.

We think of the information state $⋃I$ as capturing the information assumed by the issue I, and we say that I is an issue over $⋃I$.

Definition 2.5 (Issues over a state)

Let I be an issue and s an information state. Then we say that I is an issue over s if and only if $⋃I=s$.

The issues depicted in Figure 2.3 are all issues over the information state W = {w1, w2, w3, w4}. Notice that an issue $I$ over a state s may contain s itself. In this case resolving $I$ does not require any information beyond the information that is already available in s. If so, we call $I$ a trivial issue over s. Downward closure implies that for any state s there is precisely one trivial issue over s, namely the issue consisting of all enhancements of s, i.e., the powerset of s, which we denote as (s). On the other hand, if (p.19) $I$ is an issue over s and $s∉I$, then in order to settle $I$ further information is required, that is, a proper enhancement of s must be established. In this case we call $I$ a proper issue over s. The issue in Figure 2.3(d) is the trivial issue over W; all the other issues in Figure 2.3 are proper issues over W.

Figure 2.3 Issues.

Two issues over a state s can be compared in terms of what it takes for them to be settled: one issue $I$ is at least as inquisitive as another issue $J$ just in case any state that settles $I$ also settles $J$. In this case we also say that I is a refinement of J. Since an issue is identified with the set of states that settle it, the refinement order on issues just amounts to inclusion.

Definition 2.6 (Issue refinement)

Let $I,J$ be two issues over a state s. Then $I$ is at least as inquisitive as $J$ if and only if $I⊆J$. In this case we say that $I$ is a refinement of J.

Among the issues over a state s there is always a least and a most inquisitive one. The least inquisitive issue over s is the trivial issue (s), whose resolution, as we saw, requires no information beyond the information already available in s. The most inquisitive issue over s is ${{w}|w∈s}∪{∅}$, which can only be settled consistently by providing a complete description of what the actual world is like. Among the issues in Figure 2.3, the issue in subfigure (a) is the most inquisitive issue over W, and thus a refinement of all other issues; the issue in subfigure (d) is the least inquisitive issue over W, and all other issues are refinements of it. As for the issues in subfigures (b) and (c), neither is a refinement of the other.

Suppose that a given information state s resolves an issue I, and there is no weaker information state $t⊃s$ that also resolves I. Then s contains just enough information to resolve I, it does not contain any superfluous information. Such information states are precisely the maximal elements of I, since information states consisting of more worlds contain less information. We will refer to these maximal elements as the alternatives in I.

Definition 2.7 (Alternatives in an issue)

The maximal elements of an issue I are called the alternatives in I.

If an issue I over a state s is trivial, i.e., if sI, then s is the unique maximal element of I, i.e., the unique alternative in I. On the other hand, if I contains two or more alternatives, then it must be a non-trivial (p.20) issue. If an issue contains only finitely many information states, which will be the case in all the examples that we will consider here, then the connection between containing multiple alternatives and being a non-trivial issue also holds in the other direction.3

Fact 2.8 (Multiple alternatives and proper issues)

An issue containing finitely many elements is non-trivial if and only if it contains at least two alternatives.

This fact makes it very easy to see whether an issue is inquisitive, given a visual representation of it. For instance, the issue in Figure 2.3(d) is not inquisitive because it contains a single alternative, while all the other issues in Figure 2.3are inquisitive because they contain multiple alternatives.

Note that, as exemplified in Figure 2.3(c), two alternatives in an issue I may very well overlap, they do not have to be mutually exclusive. However, since alternatives are defined as maximal elements, one alternative can never be fully contained in another.4

# 2.4 Propositions

Traditionally, the semantic content of a sentence, the proposition that it expresses, is intended to capture the information that a speaker conveys in asserting the sentence (as per the conventions of the language; additional information may be conveyed through pragmatic implicatures). In inquisitive semantics, propositions are not just intended to capture the information that is conveyed in uttering a sentence, but also the issue that may be raised in doing so. In short, propositions are intended to embody both informative and inquisitive content.

How should such more versatile propositions be modeled formally? The most straightforward option would be to construe a proposition (p.21) P as a pair 〈infoP, issueP〉, where infoP is a classical proposition, i.e., a set of possible worlds, embodying the informative content of P, and issueP an issue, embodying the inquisitive content of P. We can then think of a speaker who utters a sentence expressing the proposition P as (i) providing the information represented by infoP, and (ii) raising the issue represented by issueP. By the latter we mean that the speaker proposes to enhance the current common ground of the conversation in such a way that it comes to settle issueP; for short, we will say that the speaker ‘steers the common ground towards a state in issueP’.

This notion of propositions is a natural starting point, but note that it does not impose any constraints on how the two components of a proposition should be related to each other. There are two constraints that we think should be enforced. First, all states that resolve issueP should be enhancements of infoP. It would not make sense for a speaker to steer the common ground towards a state where the information embodied by infoP itself is not commonly established. Formally, this means we should have that $⋃issueP⊆infoP$.

Second, the information that a speaker conveys should ensure that the issue she raises can be resolved truthfully. This means that we should have that $infoP⊆⋃issueP$. To see this, suppose that $infoP⫅̸⋃issueP$. Then there is a world w ∈infoP which is not contained in any state that resolves issueP. According to infoP, w may well be the actual world. Now, suppose it is the actual world. Then there is no way of resolving issueP without discarding the actual world. So, in this case infoP does not ensure that issueP can be resolved truthfully.

Putting the two constraints together, we get that issueP should be an issue over infoP: $⋃issueP=infoP$. Given this, our formal notion of propositions can be simplified considerably. After all, since infoP can always be retrieved from issueP, it can just as well be left out of the representation of P. Thus, a proposition P can simply be represented as a non-empty, downward closed set of information states. The informative content of P is then represented by the union of all these states, $⋃P$, while the issue embodied by P is the one which is resolved in a state s just in case sP.

Definition 2.9 (Propositions)

• A proposition P is a non-empty, downward closed set of information states.

• The set of all propositions will be denoted by $Q$.

(p.22)

Definition 2.10 (Informative content)

For any proposition P: $info(P):=⋃P$

Definition 2.11 (The issue embodied by a proposition)

The issue embodied by a proposition P is the one that is resolved in a state s just in case sP.

## 2.4.1 Truth and support

We say that a proposition P is true in a world w just in case w is compatible with the informative content of P, i.e., w ∈ info(P).

Definition 2.12 (Truth)

A proposition P is true in a world w just in case w ∈ info(P).

We say that an information state s supports a proposition P just in case it implies the informative content of P, i.e., $s⊆info(P)$, and it resolves the issue embodied by P, i.e., sP. But note that if sP, then it must also be the case that $s⊆info(P)$. So support just amounts to membership.

Definition 2.13 (Support)

An information state s supports a proposition P if and only if sP.

From the fact that propositions are downward closed it follows that truth and support are closely connected.

Fact 2.14 (Truth and support)

A proposition P is true in a world w if and only if P is supported by {w}.

The notion of support will become very useful later on. Notice that the relation between propositions and support is exactly the same as that between issues and resolution: a proposition consists of all states that support it; an issue consists of all states that resolve it. Moreover, the relation between propositions and support in inquisitive semantics is also parallel to the relation between classical propositions and truth: a classical proposition is the set of all worlds in which it is true. In the present setting, truth does not relate directly to propositions in this way, but rather to the informative content of a proposition: the informative content of a proposition is the set of all worlds in which the proposition is true. Evidently, the fact that the connection between truth and propositions is more direct in the classical setting is an immediate consequence of the fact that classical propositions exclusively encode informative content. (p.23)

## 2.4.2 Informative and inquisitive propositions

We will say that a proposition P is informative just in case its informative content is non-trivial, i.e., $info(P)≠W$. On the other hand, we will say that P is inquisitive just in case establishing its informative content is not sufficient to settle the issue that it raises, i.e., info(P)∉P.

Definition 2.15 (Informative and inquisitive propositions)

• A proposition P is informative iff $info(P)≠W$.

• A proposition P is inquisitive iff info(P) ∉P.

Just as we did in the case of issues, we refer to the maximal elements of a proposition as the alternatives in that proposition. These are states that support the proposition and cannot be weakened in any way without losing support. That is, they contain just enough information to support P.

Definition 2.16 (Alternatives in a proposition)

• The maximal elements of a proposition P are called the alternatives in P.

• The set of alternatives in P is denoted as alt(P).

When discussing issues, we noted that there is a close connection between containing multiple alternatives and being non-trivial. For propositions, there is a parallel connection between containing multiple alternatives and being inquisitive. Namely, if P contains two or more alternatives, then it cannot contain info(P) and therefore must be inquisitive. On the other hand, if a proposition P is non-inquisitive, i.e., if info(P) ∈ P, then it always contains a unique alternative, namely info(P). If a proposition contains only finitely many information states, which is the case in all the examples that we will consider, then the connection between multiple alternatives and inquisitiveness is even stronger. Namely, a proposition with finitely many elements is inquisitive if and only if it contains multiple alternatives.5

Fact 2.17 (Inquisitiveness and alternatives)

A proposition containing finitely many elements is inquisitive if and only if it contains multiple alternatives.

(p.24) Figure 2.4 depicts a number of propositions. In each case, we only depict the alternatives that the proposition contains. The proposition depicted in Figure 2.4(a) contains just one alternative and is therefore not inquisitive, but it is informative, since its informative content does not cover the entire logical space. The proposition depicted in Figure 2.4(b) contains two alternatives and is therefore inquisitive; on the other hand, it is not informative, because its informative content, i.e., the union of the two alternatives, covers the entire logical space. The proposition depicted in Figure 2.4(c) is both informative and inquisitive, since it contains two alternatives and the union of these two alternatives does not cover the entire logical space. Finally, the proposition depicted in Figure 2.4(d) contains a single alternative, which covers the entire logical space; it is therefore neither informative nor inquisitive.

Figure 2.4 Propositions.

We will refer to a proposition that is both informative and inquisitive as a hybrid proposition, and to one that is neither informative nor inquisitive as a tautology.

Propositions can be thought of as inhabiting a two-dimensional space, as depicted in Figure 2.5. The horizontal axis is inhabited by non-inquisitive propositions, the vertical axis by non-informative propositions, the ‘zero-point’ of the space by tautologies, and the rest of the space by hybrids.

Figure 2.5 Propositions in a two-dimensional space.

(p.25) Spelling out what it means to be informative and/or inquisitive we obtain the following direct characterization of propositions that are non-informative, non-inquisitive, or tautological.

Fact 2.18

• P is non-inquisitive iff info(P) ∈ P.

• P is non-informative iff $info(P)=W$.

• P is a tautology iff $W∈P$.

It will be insightful (and useful for later) to consider a number of alternative characterizations of non-inquisitive propositions as well.

Fact 2.19 (Alternative characterizations of non-inquisitive propositions)

The following are equivalent for any proposition P:

1. 1. P is non-inquisitive;

2. 2. P = (info(P));

3. 3. P has a greatest element;6

4. 4. P is supported by a state s just in case P is true in all worlds in s.

Proof. We will prove the chain of implications $(1)⇒(2)⇒(3)⇒(4)⇒(1)$.

• $(1)⇒(2).$ By definition, $info(P)=⋃P$, so for every tP we have $t⊆info(P)$, which means that t (info(P)). This shows that $P⊆℘(info(P))$, regardless of whether P is non-inquisitive. Now suppose P is non-inquisitive, i.e., suppose info(P) ∈ P: by downward closure, every substate of info(P) must be in P as well, so $℘(info(P))⊆P$. Putting the two inclusions together, we obtain P = (info(P)).

• $(2)⇒(3).$ If P = (info(P)), clearly info(P) is the greatest element in P.

• $(3)⇒(4).$ Suppose s supports P, i.e., sP. By downward closure, {w}∈ P for all ws. By Fact 2.14, P is true at each ws.

Conversely, suppose P is true at each ws. By Fact 2.14, this means that {w}∈ P for all ws. Suppose now that P has a greatest element, and call this element $smax$. Then for all ws, {w} must be included (p.26) in $smax$, and so also $s⊆smax$. Since $smax∈P$, it follows by downward closure that sP, that is, P is supported at s.

• $(4)⇒(1)$. Suppose (4) holds. Take any $w∈info(P)=⋃P$: then wt for some tP. Since P is supported by t, by (4) P is true at w. This shows that P is true at all w ∈info(P): by (4) it follows that P is supported by info(P), i.e., that info(P) ∈ P. $□$

The characterization of non-inquisitive propositions given in (3) makes it particularly easy to say whether a proposition is non-inquisitive given a visualization of it—we just have to check whether it has a greatest element. We already established in Fact 2.17 that a proposition containing finitely many elements is inquisitive if and only if it contains at least two alternatives, and thus non-inquisitive if and only if it contains just one alternative, i.e., one maximal element. The present characterization in terms of greatest elements is more general since it applies to infinite propositions as well.

The characterization of non-inquisitive propositions in (4) brings out the fact that such propositions are fully characterized by their truth-conditional content. This is not the case for inquisitive propositions. For instance, the proposition depicted in Figure 2.4(b) is true at all possible worlds—it has tautological truth-conditions—yet it does not coincide with the tautological proposition in Figure 2.4(d).

Finally, one particular consequence of the characterization of non-inquisitive propositions given in (2) is that there is only one proposition that counts as a tautology, namely (W). After all, tautologies are not only non-inquisitive but also non-informative. So if P is a tautology, then we must have that info(P) = W. But then, according to the characterization in (2), it must be the case that P = (W).

## 2.4.3 Entailment

Propositions can be ordered both in terms of their informative content and in terms of their inquisitive content. A proposition P is at least as informative as another proposition Q if and only if the informative content of P determines with at least as much precision what the actual world is like as the informative content of Q, i.e., $info(P)⊆info(Q)$.

Definition 2.20 (Informative order on propositions)

For any $P,Q∈Q$:

• PinfoQ iff $info(P)⊆info(Q)$

(p.27) Similarly, we say that P is at least as inquisitive as Q just in case any state that settles the issue embodied by P also settles the issue embodied by Q, i.e., if and only if $P⊆Q$.

Definition 2.21 (Inquisitive order on propositions)

For any $P,Q∈Q$:

• PinqQ iff $P⊆Q$

Combining these two orders, we say that P entails Q just in case P is both at least as informative and at least as inquisitive as Q. But note that if $P⊆Q$, then it must also automatically hold that $info(P)⊆info(Q)$. So entailment simply amounts to inclusion.

Definition 2.22 (Entailment)

For any $P,Q∈Q$:

• PQ iff $P⊆Q$

Entailment between two propositions can also be characterized as preservation of support, just like classical entailment can be characterized as preservation of truth: one proposition entails another just in case any state that supports the former also supports the latter.

Fact 2.23 (Entailment in terms of support)

For any $P,Q∈Q$:

• PQ iff any state that supports P also supports Q

Entailment forms a partial order on the set of all propositions, i.e., it is a reflexive, transitive, and anti-symmetric relation. The tautology, (W), is entailed by any other proposition, i.e., it is the weakest element of the partial order. On the other hand, the partial order also has a strongest element, namely {}, which entails all other propositions. We refer to this proposition as the contradictory proposition. We will denote the tautological and the contradictory proposition as ⊤ and ⊥, respectively.

• $⊤:=℘(W)$

• ⊥:={}

Fact 2.25 (Partial order)

• ⊧ forms a partial order on $Q$

• For every $P∈Q$: ⊥⊧P and P⊧⊤

## (p.28) 2.4.4 Some linguistic examples

The notion of propositions as non-empty, downward closed sets of information states allows us to capture the informative and inquisitive content of a wide range of declarative and interrogative sentences in natural languages in a uniform and transparent way. We provide a brief illustration here; more elaborate linguistic analyses will be presented in Chapters 5–8. Imagine a context in which we are dealing with a two-digits code, where each digit can be either 1 or 0. Consider the following sentences in English.

(1)

Among these sentences, the first three are declaratives, while the remaining five are interrogatives. These sentences can all be analysed uniformly in terms of the notion of propositions developed in this section. The propositions that they express are shown in Figure 2.6, where possible worlds are identified with the corresponding codes and where, as before, only the maximal elements in each proposition—the alternatives—are displayed.

Figure 2.6 Propositions expressed by (1a)-(1h), exemplifying several types of declarative and interrogative sentences.

The propositions expressed by the declaratives in (1a)–(1c) each contain a single alternative, which does not cover the entire logical (p.29) space. Thus, these propositions are informative but not inquisitive. This captures the fact that, in uttering one of (1a)–(1c), a speaker provides some information but does not raise any issue.

While the semantic contents of (1a)–(1c) could have been captured just as well by means of the standard notion of a proposition as a set of possible worlds, the enriched notion of propositions allows us to analyse also the interrogatives in (1d)–(1h). These are naturally taken to express the propositions depicted in Figure 2.6(d)–(h). In each case, the relevant proposition is inquisitive, since it contains multiple alternatives, and it is not informative, since these alternatives jointly cover the entire logical space. This captures the fact that, in uttering one of (1d)–(1h), a speaker raises an issue and does not provide any information.

Let us consider in some more detail the issues expressed by these questions. The polar question (1d) raises an issue which is resolved by an information state s just in case the information in s implies that the code is 11 ($s⊆{11})$ or it implies that the code is not 11 ($s⊆{10,01,00}$). The wh-question (1e) raises an issue which is resolved by an information state s just in case the information available in s determines exactly what the first digit of the code is, i.e., it implies that the first digit is 1 ($s⊆{11,10}$) or that the first digit is 0 ($s⊆{01,00}$). The situation is analogous for the question (1f). The wh-question (1g) raises an issue which is resolved by an information state s just in case the information available in s determines exactly what the code is, that is, in case $s⊆{11}$ or $s⊆{10}$ or $s⊆{01}$ or $s⊆{00}$.

Finally, the conditional wh-question (1h) raises an issue which is resolved by an information state s just in case the information available in s restricted to those worlds where the first digit is 1 determines exactly what the second digit is. This condition amounts to $s∩{11,10}⊆{11}$ or $s∩{11,10}⊆{01}$, and it is easy to see that this holds if and only if $s⊆{00,01,11}$ or $s⊆{00,01,10}$. Thus, the issue expressed by (1h) is one that can be settled either by establishing that if the first digit is 1, the second is also 1 ($s⊆{00,01,11}$) or by establishing that if the first digit is 1, the second is 0 ($s⊆{00,01,10}$).

The uniform perspective on the semantics of (1a)–(1h) afforded by the inquisitive notion of a proposition also gives rise to a uniform perspective on the entailment relations that hold between these sentences. As far as the declarative sentences (1a)–(1c) are concerned, we predict the same entailments that truth-conditional semantics predicts: (1a) entails (1b), which in turn entails (1c). This is as it should be: in the absence (p.30) of (proper) inquisitive content, entailment still amounts to comparing informative strength, just like in the classical case.

Now, however, the same notion of entailment can also be used to compare the questions in (1d)–(1h): in this case, informative content is trivial, and entailment will compare inquisitive strength; an entailment between a pair of questions will hold if the issue expressed by the first is at least as demanding as the issue expressed by the second. Thus, for instance, (1g) entails any of the other questions: if one establishes what the code is, one also thereby establishes whether the code is 11, what the first digit is, and so on. More generally, the issue expressed by (1g) is the strongest possible issue over the ignorant state.

The analysis also captures that (1f) entails (1h): if one establishes what the second digit is, one also thereby establishes what the second digit is if the first digit is 1. Another prediction is that the questions in (1d) and (1e) are incomparable in terms of inquisitive strength: on the one hand, the information that the code is not 11 resolves (1d) but not (1e); on the other hand, the information that the first digit is 1 resolves (1e) but not (1d); thus, neither of these questions entails the other.

Finally, notice that entailment can also be used to compare declaratives with interrogatives. A declarative entails an interrogative just in case the information provided by the former suffices to resolve the issue raised by the latter. Thus, (1a), which completely specifies the code, entails all the questions in (1d)–(1h); (1b), which only specifies the second digit, entails only (1f) and (1h); and (1c), which gives conditional information about the second digit, entails only (1h).

Conversely none of the above questions entails any of the given declaratives. This is because, in terms of informative content, (1d)–(1h) are trivial, while (1a)–(1c) are not.

# 2.5 Contexts

In Section 1.1.1 we reviewed a number of reasons why conversational contexts should be modeled in a way that does not only take account of the information that has been established in the conversation so far, but also of the issues that have been brought up, often referred to as questions under discussion. This can be done using the notion of issues introduced above.7 The most straightforward way of doing so would (p.31) be to model a context C as a pair 〈infoC, issuesC〉, where infoC is an information state representing the information that the conversational participants have commonly established so far, and issuesC a set of issues which have been raised in the conversation so far and which the conversational participants would like to see commonly resolved. That is, while infoC represents the current common ground, the issues in issuesC determine what kind of common ground the conversational participants would like to establish, namely one in which every issue in issuesC is settled. The initial context would then be $〈W,∅〉$, consisting of the trivial information state, which does not rule out any world, and the empty set of issues. As the conversation progresses, worlds would be removed from infoC and issues would be added to issuesC.

This way of modeling conversational contexts is a good starting point, but just as in the case of propositions, it is natural to impose certain constraints on how the informative and the inquisitive component of a context are related to each other. First, every issue I ∈issuesC should be one that is settled only in information states that enhance the current common ground, infoC. Formally, this means that for every I ∈issuesC we should have that $⋃I⊆infoC$.8

Second, for every issue I ∈issuesC, the information available in the current common ground should ensure that I can be resolved truthfully, i.e., without discarding the actual world. This means that for every I ∈issuesC we should have that $infoC⊆⋃I$. To see this, we can follow the same line of reasoning that we followed when considering propositions above. Thus, for every I ∈issuesC, it should hold on the one hand that $⋃I⊆infoC$ and on the other hand that $infoC⊆⋃I$. Putting the two together, we get that every I ∈issuesC should be an issue over infoC: $⋃I=infoC$.

If we impose this constraint, the considered notion of conversational contexts is in principle a suitable notion. However, for our current purposes, it can be simplified. We will do this in two steps. First, rather than thinking of a context C as a pair 〈infoC, issuesC〉 where issuesC is a set of issues over infoC, we may just as well think of it (p.32) as a pair 〈infoC, issueC〉 where issueC is a single issue over infoC. This simplification is justified by the fact that any set of issues Ω‎ over a state s can be merged into a single issue over s:

$Display mathematics$

which is settled precisely by those enhancements $t⊆s$ that settle all issues in Ω‎. Notice that if Ω‎≠ the issue $IΩ$ amounts to the intersection $⋂Ω$ of all issues in Ω‎, whereas if Ω‎ = , $IΩ$ amounts to the trivial issue (s) over s.9,10

So we can think of a context C as a pair 〈infoC, issueC〉, where infoC is an information state and issueC a single issue over infoC. We can then take the initial context to be the pair $〈W,℘(W)〉$, consisting of the trivial information state, which does not rule out any world, and the trivial issue over this state, which is settled even if no information is present yet.

But this representation can be simplified further. After all, since issueC is an issue over infoC, we always have that $infoC=⋃issueC$. That is, infoC can always be retrieved from issueC. But then infoC can just as well be left out of the representation of C. Thus, a context C can simply be represented as an issue, i.e., a non-empty, downward closed set of information states. The information commonly established in C is then embodied by $⋃C$.

Definition 2.26 (Contexts)

• A context C is a non-empty, downward closed set of information states.

• The set of all contexts will be denoted by $C$.

Definition 2.27 (The information available in a context)

• For any context C: $info(C):=⋃C$

(p.33) We have moved from the commonplace notion of a context as a set of possible worlds—representing the information established so far—to a richer notion of contexts as non-empty, downward closed sets of information states—representing both the information established and the issues raised so far. We will now identify some special properties that contexts may have (section 2.5.1), some relations that may hold between them (section 2.5.2), and some operations that can be performed on them (section 2.5.3).

## 2.5.1 Informed and inquisitive contexts

First of all, we say that a context C is informed just in case some non-trivial information has been established in it, i.e., $info(C)≠W$. Otherwise we say that the context is uninformed.

Definition 2.28 (Informed and uninformed contexts)

• A context c is informed iff $info(C)≠W$.

• A context c is uninformed iff $info(C)=W$.

Similarly, we say that a context C is inquisitive just in case the information that has been established so far does not yet settle the issues that have been raised, i.e., info(C)∉C. On the other hand, if all issues are settled we say that C is indifferent.

Definition 2.29 (Inquisitive and indifferent contexts)

• A context C is inquisitive iff info(C)∉C.

• A context C is indifferent iff info(C) ∈ C.

There are two special contexts: the initial and the absurd context. The initial context, C, is the only context that is both uninformed and indifferent. The absurd context, C, is one in which the established information is inconsistent and therefore rules out all possible worlds.

Definition 2.30 (The initial and the absurd context)

• $C⊤:=℘(W)$

• C:={}

Some example contexts are depicted in Figure 2.7, whereas before it is assumed that W = {w1, w2, w3, w4}. Only the maximal states in each context are depicted. Since contexts are downward closed, we know that all enhancements of these maximal states are also part of the context at hand. The context in (a) is the initial context, (W), which is neither (p.34) informed nor inquisitive. The one in (b) is still not informed, but it is inquisitive. In order to resolve the issue that is present in this context, it either needs to be established that the actual world is one of {w1, w2} or that it is one of {w3, w4}. The context in (c) is both informed and inquisitive. In this context it is common ground that the actual world is one of {w1, w2, w3}, i.e., w4 has been ruled out as a candidate for the actual world, but in order to resolve the issue that has been raised, more precise information is needed—namely, it either needs to be established that the actual world is one of {w1, w2} or that it is one of {w1, w3}. Finally, the context in (d) is informed, but not inquisitive. It is common ground in this context that the actual world is one among {w1, w2}, and no issues have been raised whose resolution would require more precise information.

Figure 2.7 Some contexts.

## 2.5.2 Context extension

Two contexts can be compared in terms of the information that has been established or in terms of the issues that have been raised. One context C′ is at least as informed as another context C if and only if $info(C′)⊆info(C)$.

Definition 2.31 (Informative order on contexts)

For any contexts C, C′:

• C′infoC iff $info(C′)⊆info(C)$

Similarly, we say that C′ is at least as inquisitive as C if and only if every state that settles all the issues that have been raised in C′ also settles all the issues that have been raised in C, i.e., if and only if $C′⊆C$.

Definition 2.32 (Inquisitive order on contexts)

For any contexts C, C′:

• C′inqC iff $C′⊆C$

(p.35) Combining these two orders, we say that C′ is an extension of C just in case C′ is both at least as informed and at least as inquisitive as C. But note that if $C′⊆C$, then it must also be the case that $info(C′)⊆info(C)$. So context extension simply amounts to inclusion.

Definition 2.33 (Extending contexts)

For any contexts C, C′:

• C′ is an extension of C, C′C, iff $C′⊆C$

The extension relation forms a partial order on $C$, and C and C constitute the extremal elements of this partial order: C is an extension of every context, and every context is in turn an extension of C.

Fact 2.34 (Partial order)

• ≥ forms a partial order on $C$

• For every $C∈C$: CC and CC

In Figure 2.7, the contexts in (b), (c), and (d) are all extensions of the trivial context in (a). Moreover, (d) is also an extension of (b) and (c), but neither (b) nor (c) is an extension of the other.

## 2.5.3 Updating contexts

Recall that in the standard setting, where both contexts and propositions are construed as sets of possible worlds, the result of updating a context c with a proposition p is a new context c[p] which, besides the information already present in c, also contains the information embodied by p. That is, a candidate for the actual world is ruled out by c[p] if it was already ruled out by the information established in the old context c, or if it is ruled out by the new information embodied by p. Thus, formally, update amounts to set intersection in the standard setting: $c[p]=c∩p$.

In inquisitive semantics, we want the result of updating a context C with a proposition P to be a new context C[P] which incorporates both the informative content of P and the issue that it embodies. Thus, on the one hand, a candidate for the actual world must be ruled out by the information established in C[P] if it was either already ruled out by the information established in the old context C, or if it is ruled out by the informative content of P. Formally, this means that we must have that $info(C[P])=info(C)∩info(P)$. On the other hand, a state must resolve the issues present in C[P] if and only if it resolves the (p.36) issues already present in C and also the issue embodied by P. Formally, this means that we must have that $C[P]=C∩P$. Now, note that if the latter condition is satisfied, then the former condition is automatically satisfied as well. This means that, just as in the standard setting, update can simply be defined as set intersection.

Definition 2.35 (Updating contexts)

For any $C∈C$ and any $P∈Q$:

• $C[P]:=C∩P$

Some examples of context update are given in Figure 2.8. In the first case, the initial context is informed but not inquisitive. More specifically, in this context it is commonly established that the actual world is one among {w1, w2, w3}, and no issues have been raised that require more precise information. This context is updated with a proposition which is informative—embodying the information that the actual world is one among {w1, w2, w4}—but not inquisitive. The result of the update, obtained by intersection, is a new context in which it is established that the actual world is among {w1, w2}, and where there are still no issues (p.37) that require more precise information. Note that in this case, where neither the initial context nor the proposition involved in the update are inquisitive, our framework reproduces exactly the same result that is obtained in the standard setting. This holds in full generality.

Figure 2.8 Some update examples.

Fact 2.36 (Update without inquisitiveness yields standard results)

For any non-inquisitive context C and any non-inquisitive proposition P, C[P] is a non-inquisitive context as well, and its unique maximal element is the intersection of the unique maximal element of C and that of P.

The second example in Figure 2.8 is one where the initial context is the same as in the first example, but now the proposition with which it is updated is inquisitive, embodying the issue of whether the actual world is among {w1, w2} or among {w3, w4}. The context resulting from the update is one in which this issue is present, together with the information that was already available beforehand. That is, after the update it is still established that the actual world is among {w1, w2, w3}, as in the initial context, but now there is also an issue as to whether it is w3 or among {w1, w2}. Note that in order to obtain this result simply by means of intersection, it is important that both contexts and propositions are downward closed. Made fully explicit, the initial context is represented as the following set of information states:

$Display mathematics$

The proposition considered is:

$Display mathematics$

Applying intersection to these two sets yields the new context:

$Display mathematics$

whose two maximal elements, {w1, w2} and {w3}, are the ones that are depicted. This result would not be obtained if we discharged downward closure and identified contexts and propositions exclusively with their maximal elements. In that case, the initial context would be represented as {{w1, w2, w3}}, the proposition at hand as {{w1, w2}, {w3, w4}}, and applying intersection to these two sets would yield the empty set, clearly not the desired result.

The third example in Figure 2.8 is one where the initial context is already inquisitive. The issue that is present is whether the actual world is among {w1, w3} or among {w2, w4}. The proposition with which this (p.38) context is updated is the same as in the previous example, embodying the issue whether the actual world is among {w1, w2} or among {w3, w4}. The update results in a context in which these two issues have been merged. In order to resolve the issue that is present in this new context it is necessary to determine exactly which of w1, w2, w3, and w4 is the actual world. That is, it is necessary to resolve the issue that was already present in the initial context, and also the issue that was embodied by the proposition involved in the update.

Thus, while our update procedure yields standard results in the case of non-inquisitive contexts and propositions, it also smoothly generalizes to cases involving inquisitive contexts and/or propositions.

# 2.6 Summary and pointers to possible refinements

We have now introduced all the notions that we set out to introduce (recall the diagram in Figure 2.1 at the beginning of the chapter). We adopted the standard notion of information states as sets of possible worlds. In terms of this familiar notion, we defined a new notion of issues. We represent an issue as a non-empty, downward closed set of information states, namely those information states that contain enough information to resolve the issue. With this crucial notion in place, we turned to propositions and contexts. We moved from the standard notion of a proposition as a set of possible worlds, which just allows us to capture the information that a sentence conveys, to a more fine-grained notion, which also allows us to capture the issue that a sentence raises. Similarly, we replaced the standard minimal notion of contexts, which just captures the information that has been commonly established in the conversation so far, by a richer notion that also allows us to capture the issues that have been brought up. Formally, both propositions and contexts are not modeled as sets of possible worlds in our framework, but rather, just like issues, as non-empty, downward closed sets of information states.

Turning to the relations that may hold between the various kinds of objects, we have seen that entailment between propositions, enhancement of information states, and extension of contexts all amount to set inclusion, just as in the standard setting, and the same is true for the new notion of issue refinement. Support, a relation between information states and propositions, is no longer defined as inclusion, but rather as membership. This is a consequence of the fact that an information state (p.39) no longer necessarily supports a proposition if it implies the informative content of that proposition; rather, it should also contain enough information to resolve the issue embodied by the proposition. Finally, context update still amounts to set intersection. However, since the operation no longer applies to sets of worlds but rather to sets of information states, we have seen that it can deal in a uniform way with cases involving purely informative propositions and indifferent contexts, and with cases involving inquisitive propositions and/or contexts.

We briefly illustrated how the informative and inquisitive content of various types of sentences in English can be captured using the proposed notion of propositions. There are also several aspects of meaning that are beyond the scope of the basic inquisitive semantics framework that we are presenting here. However, the framework is set up in such a way that it allows for several natural refinements. We briefly mention four such refinements, with references to other work for further detail.

First, instead of the static view on meaning that we have assumed here, one may also adopt a dynamic view on meaning (see, e.g., Kamp, 1981; Heim, 1982; Groenendijk and Stokhof, 1991; Veltman, 1996). Under this view, the meaning of a sentence is conceived of as its context change potential, modeled formally as a function F that maps any context C to a new context F(C), which would result from uttering the given sentence in C. This new context F(C) need not necessarily be obtained by intersecting C with the proposition P expressed by the sentence. In fact, on a dynamic view, there is no need to associate sentences with propositions at all. This allows for greater flexibility, which has led to important advances in the treatment of various linguistic phenomena, including anaphora and presuppositions. While we have taken a static perspective here, the formal notions that we have introduced, in particular the notion of context, also form a suitable starting point for a dynamic inquisitive semantics (see Ciardelli et al., 2012, 2013a, for initial work in this direction, though much remains to be done).

Second, rather than starting out with the commonplace notion of information states as sets of possible worlds, which imposes a very specific (Boolean) structure on the space of information states, we may also work with other notions of information states, giving rise to information spaces with different structures. This strategy can be used, in particular, to implement inquisitive semantics in a context where the underlying view of information is non-classical (Punčochář, 2017; Ciardelli et al., 2017b).

(p.40) Third, in order to model more than just informative and inquisitive content we may further enrich our notion of propositions and/or contexts, either by explicitly encoding additional dimensions of meaning (see, e.g., Roelofsen and Farkas, 2015; AnderBois, 2016b), or by weakening the downward closure constraint that we have placed on contexts and propositions here (Ciardelli et al., 2014; Punčochář, 2015; Groenendijk and Roelofsen, 2015). Such amendments lead to richer notions of meaning, and further broaden the range of linguistic phenomena that can be captured in the framework. However, these refinements also involve certain complications that do not arise in the basic framework presented here.

Finally, in addition to the basic, incremental notion of context update discussed in this chapter—where contexts are always enhanced by adding more information or more issues—one might consider more complex and realistic models of conversation, allowing conversational participants to reject a given proposal or resist it in less drastic ways (Bledin and Rawlins, 2016), as well as to retract previous claims and challenge some of the information in the common ground.

# 2.7 Exercises

## Exercise 2.1 Contexts

1. 1. Give a representation of the following contexts:

1. (a) it is established that Bill is going to the party, and there is an issue as to whether Mary is going as well;

2. (b) it is established that if Bill goes to the party, then Mary will go as well, and there is an issue as to whether Bill is going;

3. (c) it is established that only one of Bill and Mary is going to the party, and there is an issue as to which of them is going.

2. 2. For each of these contexts, determine whether it is an extension of the others.

## Exercise 2.2 Propositions

1. 1. Give a representation of the propositions encoding the following information and issues:

1. (a) Information: that Bill is only going to the party if Mary is going.

Issue: whether Mary is going. (p.41)

2. (b) Information: none.

Issue: which among Bill and Mary are going to the party (only Bill, only Mary, both, or neither).

3. (c) Information: that only Mary is going to the party, not Bill.

Issue: none.

2. 2. For each of the above propositions, determine whether it entails the others.

## Exercise 2.3 Update

1. 1. Determine the result of updating each of the contexts in Exercise 2.1 with each of the propositions in Exercise 2.2.

2. 2. Prove Fact 2.36, showing that the notion of update as intersection yields standard results when applied to non-inquisitive contexts and propositions.

## Exercise 2.4 Informational and inquisitive triviality

Let P and P′ be two non-inquisitive propositions, and Q and Q′ two non-informative propositions.

1. 1. Is $P∩P′$ guaranteed to be non-inquisitive? If so, give a proof; if not, give a counterexample.

2. 2. Is $Q∩Q′$ guaranteed to be non-informative? If so, give a proof; if not, give a counterexample.

3. 3. Is $P∩Q$ either guaranteed to be non-inquisitive or to be non-informative? If so, give a proof; if not, give a counterexample. (p.42)

## Notes:

(1) Sometimes a distinction is made between the common ground of a conversation and the context set (Stalnaker, 1978). The common ground is then construed as the set of pieces of information that are publicly shared among the conversational participants, and the context set as the set of possible worlds that are compatible with all these pieces of information. For our purposes, it will not be necessary to make this distinction, so we simply construe the common ground as the set of possible worlds that are compatible with the commonly established body of information.

(2) Notice that this means that the inconsistent information state, , is an element of every issue. Thus, it is assumed that every issue is resolved in the inconsistent information state. This limit case may be regarded as a generalization of the ex falso quodlibet principle to issues.

(3) To see that this does not generally hold for issues containing infinitely many states, consider an issue I that contains an infinite chain of states, $s1⊂s2⊂s3⊂…$, without any maximal element. Such an issue is non-trivial, since $⋃I∉I$, but it does not contain any alternatives. So, if an issue contains at least two alternatives, then it is always non-trivial, but the reverse implication only holds if we restrict ourselves to finite cases (Ciardelli, 2009).

(4) Our use of the term alternatives here is closely related to its use in the framework of alternative semantics (cf., Hamblin, 1973; Kratzer and Shimoyama, 2002; Simons, 2005; Alonso-Ovalle, 2006; Aloni, 2007, among others). One difference, however, is that in alternative semantics one alternative may very well be fully contained in another. We will discuss the commonalities and differences between inquisitive semantics and alternative semantics in more depth in Section 4.8 and Section 9.1.

(5) See footnote 3 for an example showing that an inquisitive issue with infinitely many states does not necessarily contain multiple alternatives; it may not contain any alternatives at all. A parallel example can easily be constructed for propositions.

(6) By a greatest element we mean a state $smax∈P$ such that for every sP, $s⊆smax$. Notice that if P has a greatest element $smax$, then $smax$ is the unique maximal element in P. Conversely, if P has a unique maximal element, then in the finite case (but not in the general case) this is guaranteed to be the greatest element in P.

(7) It can also be done—and indeed has been done—using different formal notions from the literature on questions (e.g., Hamblin, 1973; Groenendijk and Stokhof, 1984; Groenendijk, 2009; Mascarenhas, 2009). Chapter 9 provides a detailed comparison between the current notion of issues and these previous notions.

(8) In a conversation, it is of course possible to raise issues which call into question some of the propositions which were part of the common ground. However, in order to countenance such an issue, the common ground first needs to be weakened so as to make the relevant issue again open for debate. We refer to Ciardelli and Roelofsen (2014) for some discussion of how the process of dropping a certain belief could be modeled in the inquisitive setting.

(9) Recall from footnote 1 that we are implicitly already assuming a similar simplification concerning the informative component of a context: we do not keep track of all the separate pieces of information that have been established in the conversation so far, but rather of the set of worlds that are compatible with all these pieces of information—formally, this is again the intersection of all the separately established pieces of information. For certain purposes it is necessary to keep track of all the separate pieces of information and/or issues that have been established/raised in a conversation (see, e.g., Roberts, 1996; Farkas and Bruce, 2010; Farkas and Roelofsen, 2017). For our current purposes, however, this would only add unnecessary complexity.

(10) Notice that $IΩ$ is guaranteed to be an issue in the sense of Definition 2.3. In particular, it is guaranteed to be non-empty, since it always contains the inconsistent information state.