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Inquisitive Semantics$

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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Propositional attitudes

Propositional attitudes

Chapter:
(p.143) 8 Propositional attitudes
Source:
Inquisitive Semantics
Author(s):

Ivano Ciardelli

Jeroen Groenendijk

Floris Roelofsen

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198814788.003.0008

Abstract and Keywords

In Chapter 8 it is shown that inquisitive semantics gives rise to a new view on propositional attitudes, especially those that are relevant for information exchange. Namely, besides the familiar informationdirected attitudes like knowing and believing, it allows us tomodel issuedirected attitudes like wondering and being curious as well. This also leads to a semantic treatment of the verbs that are used to express such attitudes. Among other things, this treatment explains the selectional restrictions of verbs like wonder, i.e. why they only take interrogative complements, while verbs like know take both declarative and interrogative complements.

Keywords:   propositional attitudes, attitude verbs, information-directed, issue-directed, selectional restrictions

In the previous chapters we have seen that inquisitive semantics provides a new notion of semantic content, which does not just embody informative content but also inquisitive content, as well as a new notion of conversational contexts, which does not only capture the information that has been established in the conversation so far but also the issues that have been raised. In this chapter we will show that the framework also gives rise to a new view on propositional attitudes, especially those that are relevant for information exchange. Namely, besides the familiar information-directed attitudes like knowing and believing it also allows us to model issue-directed attitudes like wondering and being curious.

A perspicuous and widely adopted formal treatment of information-directed attitudes is provided by epistemic logic (EL), sometimes also called the logic of knowledge and belief, which has its roots in the work of Hintikka (1962) and has been further developed by many authors in subsequent work (see, e.g., Fagin et al., 1995; van Ditmarsch et al., 2007; van Benthem, 2011). In this framework, the information state of an agent is modeled as a set of possible worlds, namely those worlds that are compatible with the information available to the agent. As we have seen, this notion of information states also plays an important role in inquisitive semantics. However, while the information-directed attitudes of an agent can be captured in terms of her information state, this clearly does not hold for issue-directed attitudes. In order to capture the attitude of wondering, we need a description of the agent’s inquisitive state, i.e., a representation of the issues that she entertains.

To this end, we will define an inquisitive epistemic logic (IEL, Ciardelli and Roelofsen (2015)), which brings together ideas from standard EL and InqB. This logic enriches InqB with two modal operators: K, which is used to talk about the agents’ knowledge, and E, which is used to talk about the issues that the agent entertains.

One purpose of IEL is to serve as a formal framework to describe and reason about information- and issue-directed attitudes as such. (p.144) However, this is not the only purpose. Of equal importance, it also provides a basic semantic treatment of verbs in natural languages that are used to report such attitudes. In English, such verbs include know and wonder, and many other languages have verbs that fulfil precisely the same purpose. While the semantics of know and its cross-linguistic kin has been considered extensively, its treatment in IEL differs from most previous accounts in that it deals completely uniformly with cases where know takes a declarative complement, as in (1a), and cases where it takes an interrogative complement, as in (1b–d).

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Propositional attitudes

As for wonder, IEL does not only capture its interpretation when taking an interrogative complement, as in (2a) below, but it also provides an explanation of the fact that the verb cannot take a declarative complement, illustrated in (2b).

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Propositional attitudes

1 We will proceed as follows. Section 8.1 describes the standard approach to propositional attitudes, focusing on the analysis of know in epistemic logic. Section 8.2 presents the IEL framework, and shows how inquisitive semantics allows us to obtain a more general account of know and an account of wonder. Finally, Section 8.3 broadens the scope of the discussion, looking at the general view of modal operators that emerges from IEL and sketching some directions for future work.

8.1 Propositional attitudes: the standard account

Information-directed attitudes like know, believe, and remember are usually analysed as relations between agents and classical propositions, and referred to as propositional attitudes. The traditional analysis of such attitudes goes back to Hintikka (1969), and it lies at the heart of the framework of epistemic logic.

(p.145) Focusing on the case of knowledge, the standard approach can be described succinctly as follows. For each agent a we consider a map σ‎a which assigns to each possible world w a set σ‎a(w) of possible worlds—those worlds which are compatible with what the agent knows at w. The set of worlds σ‎a(w) is referred to as the agent’s epistemic state in the world w.2

In the logical language, one can then add a modal operator Ka that allows us to talk about what the agent a knows. A formula Kaφ‎ is true at a world w if it follows from what a knows at w that φ‎ is true. More formally, Kaφ‎ is assigned the following truth-conditions, where |φ‎| denotes the set of possible worlds where φ‎ is true.3

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Propositional attitudes

This logical analysis is not only used in contexts where we want to model and reason about information, which is often the case in logic, economics, artificial intelligence and computer science; it is also standardly assumed in linguistics to be at the core of the workings of the verb know and its cross-linguistic kin.4

Other information-directed attitudes like believe and remember can be analyzed in much the same way: in this case, the set σ‎a(w) will consist of those worlds that are compatible with what the agent (p.146) believes/remembers, and a corresponding modality can be added to the logical language, which will be interpreted by means of a clause analogous to (3).

Notice that, from a formal point of view, Ka operates by comparing two sets of worlds—two propositions in the classical sense: the epistemic state of the agent, and the proposition expressed by the complement. We will see in Section 8.3 that, in the inquisitive setting, modalities have essentially the same behavior, but they will compare two propositions in the inquisitive sense: both the state of an agent and the proposition expressed by the complement will be modeled as sets of information states, encoding both information and issues.

Before turning to the inquisitive take on modality, however, we will discuss some of the limitations of the standard view on propositional attitudes and attitude verbs which are most relevant for our purposes.

Limitation 1: know + interrogatives. Some verbs expressing information-directed attitudes, like know and remember, can combine not only with declarative complement clauses, but also with interrogative ones. Focusing on the case of know, in English we find not only sentences like (4), which can be analysed directly by means of clause (3), but also sentences like (5a–c).

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Propositional attitudes

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Propositional attitudes

In order to account for the semantics of (5a–c) while maintaining that know primarily operates on a classical proposition, two types of approaches have been pursued. Groenendijk and Stokhof (1984) proposed a uniform approach: in their theory, complement clauses, whether declarative or interrogative, have the same type of denotation—a classical proposition; in the case of a declarative clause that α‎, the denotation is the set of worlds where α‎ is true, |α‎|; in the case of an interrogative clause μ‎, the denotation is the true complete answer to μ‎ at the world of evaluation. In either case, the attitude verb can then apply directly to the denotation of the embedded clause according to the clause in (3).

By contrast, Karttunen (1977) proposed an approach based on type-shifting: on this approach, an interrogative clause denotes a set Q of propositions—the set of true answers to the interrogative. When know (p.147) combines with such a clause, we proceed in two steps: first, from the denotation Q of the embedded interrogative we derive the classical proposition Q, which represents the complete answer to Q at the world of evaluation as given by Karttunen’s theory; second, an analysis of know in line with (3) is applied to the resulting classical proposition.5

Both these approaches rest on the assumption that sentences like (5a–b) always express relations between an individual and a specific classical proposition—the complete answer to the embedded question. But this is problematic. To see why, consider a mention-some question like (6).

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Propositional attitudes

This question can be completely resolved by providing only one of multiple places where Italian newspapers are sold. Now consider:

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Propositional attitudes

There is no single classical proposition that Alice needs to know in order for (7) to be true; rather, (7) may be true by virtue of Alice knowing one among various classical propositions: that one can buy an Italian newspaper at Central Station, that one can buy an Italian newspaper at the airport, etc. Thus, the approaches of Groenendijk and Stokhof (1984) and Karttunen (1977)—which would interpret sentence (7) as a claim that Alice knows a certain classical proposition—cannot assign the right truth-conditions in this case.6

In the next section, we will develop an inquisitive counterpart of the epistemic logic account of knowledge. On this account, the attitude of knowing relates an agent to a proposition in the inquisitive semantics sense. Since in inquisitive semantics both declaratives and interrogatives express propositions, we can analyse (4) and (5) in a uniform way, without assuming any type-shifting and without relying on the existence of a complete true answer to the embedded question.

(p.148) Limitation 2: issue-directed attitudes

Besides information-directed attitudes like know and believe, there are also issue-directed attitudes like wonder and be curious. Such issue-directed attitudes play a key role in cognition and inquiry (see Friedman, 2013). However, these attitudes cannot be construed at attitudes towards classical propositions: if I wonder who is coming for dinner, the object of my wondering is not a specific classical proposition, but rather the issue as such. For this reason, issue-directed attitudes fall squarely beyond the scope of the traditional approach to propositional attitudes.

These attitudes are also associated with corresponding verbs/ constructions in English. Interestingly, these verbs can only take an interrogative complement. For instance, (8a) is grammatical, but (8b) is not.

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Propositional attitudes

In the literature, a verb like wonder has mostly been treated as an un-analysed relation between individuals and question meanings (e.g., see Groenendijk and Stokhof, 1984). However, without some analysis of this relation, we lack an account of the entailments licensed by sentences involving wonder. We cannot predict, for instance, that the conclusion in (9c) follows from the premises in (9a) and (9b), but not from either premise alone.

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Propositional attitudes

As we will see, in the inquisitive setting issue-directed attitudes can be analysed as relations between agents and inquisitive propositions. The crucial difference between information-directed attitudes and issue-directed attitudes is that, in order to analyse the latter, it is not sufficient to equip an agent with an information state. Rather, we need to be able to represent an agent’s inquisitive state—encoding the issues that the agent is interested in.7

(p.149) This inquisitive analysis of wondering as an attitude also suggests an analysis of the corresponding verb in natural language. As we will see, this analysis accounts for the validity of entailments such as the one in (9), and provides an explanation for the fact that wonder does not embed declarative complements.

8.2 Inquisitive epistemic logic

In this section, we illustrate the inquisitive approach to propositional attitudes by presenting the framework of inquisitive epistemic logic (IEL, Ciardelli and Roelofsen, 2015), which is designed to model not only the knowledge that certain agents have, but also the issues that they are interested in. This framework also provides an inquisitive analysis of know and wonder that addresses the limitations pointed out in the previous section.

8.2.1 Inquisitive epistemic models

In standard epistemic logic, every agent a is equipped with a map σ‎a, which gives for every possible world w a description of the epistemic state of the agent at w, modeled as an information state σ‎a(w). In inquisitive epistemic logic, we model not only the information that an agent has, but also the issues that she entertains. This is done by means of a map Σ‎a which gives, for every possible world w, an issue Σ‎a(w) over the information state σ‎a(w), called the inquisitive state of agent a at w. Intuitively, an information state sσa(w) is in Σ‎a(w) if and only if all the issues that the agent entertains at w are resolved in s. In other words, the states s ∈Σ‎a(w) are those that contain enough information to satisfy the agent’s curiosity. We may view them as the states that the agent would like to reach through inquiry, but one should not read too much into this characterization: in particular, reaching a state in Σ‎a(w) need not be desirable for the agent in an absolute sense (the agent’s issues may well be resolved in ways that the agent finds quite unpleasant).

The constraint that Σ‎a(w) be an issue over the information state σ‎a(w), i.e., that Σa(w)=σa(w), can be viewed as resulting from two requirements. In one direction, we model an agent’s inquisitive state by specifying which enhancements of the agent’s epistemic state contain enough information to resolve the agent’s issues. This means that every s ∈Σ‎a(w) must be a subset of σ‎a(w), which implies Σa(w)σa(w). For the converse, recall from Section 2.3 that the information state (p.150) Σa(w) captures the information assumed by the issue Σ‎a(w), i.e., the information needed to guarantee that Σ‎a(w) can be truthfully resolved. For example, if Σ‎a(w) is the issue of what Alice’s dog is called, then Σa(w) is the information that Alice has a dog. By requiring that σa(w)Σa(w), we capture the idea that having the relevant information is a prerequisite for entertaining the issue.8

Now, since σa(w)=Σa(w), the agent’s epistemic state, σ‎a(w) can always be retrieved from her inquisitive state, Σ‎a(w). Thus, in effect, Σ‎a(w) encodes both the knowledge and the issues of agent a in world w. This means that the map Σ‎a suffices as a specification of the state of the agent at each world, and we do not have to list σ‎a explicitly as an independent component of our models. This leads to the following definition of inquisitive epistemic models.

Definition 8.1 (Inquisitive epistemic models)

A first-order inquisitive epistemic model for a set of agents A is a quadruple M=W,D,I,ΣA, where:

  • W,D,I is a first-order information model, in the sense of Definition 4.1.

  • ΣA={Σa|aA} is a set of state maps Σ‎a, one for each agent aA, each of which assigns to any world w an issue Σ‎a(w).

    We refer to Σ‎a(w) as the inquisitive state of a at w. Moreover, we let σa(w):=Σa(w), and we refer to σ‎a(w) as the epistemic state of a at w.

Just like in standard epistemic logic, this general characterization of inquisitive epistemic models may be supplemented with certain constraints on the agents’ information states and inquisitive states. For instance, the following conditions may (but need not) be imposed on an inquisitive epistemic model:

  • Factivity: for any wW, wσ‎a(w)

  • Introspection: for any w,vW, if vσ‎a(w), then Σ‎a(v) =Σ‎a(w)

The factivity condition, which is exactly as in standard epistemic logic, requires that the agent’s knowledge be truthful. The introspection condition requires that the agent knows exactly what her state is, with regard (p.151) to both knowledge and issues: according to this condition, only worlds where the state of the agent is the same as in w can be compatible with what the agent knows at w; that means that an agent can never be uncertain about what her own inquisitive state is.

These conditions are intended here just as an illustration: the choice of the particular conditions to be imposed on the state maps Σ‎a will depend on the particular intended application of the framework, and in any case, it is orthogonal to the main novelties introduced by IEL.

8.2.2 Knowledge

To obtain an inquisitive account of knowledge, we extend the first-order language of InqB by means of modal operators Ka, where a is the index for an agent, which can be applied without restrictions to any formula in the language.

The semantics of InqB needs to be extended with an inductive clause that specifies how a formula of the form Kaφ‎ is to be interpreted. In Chapter 4, we saw that in the inquisitive setting, a semantics can be specified recursively in two ways: one can directly define the proposition [φ‎] expressed by a formula φ‎, or one can define a relation of support sφ‎ between information states and formulas, and then define the proposition expressed by φ‎ as the set of states that support φ‎: [φ‎] = {ssφ‎}. In the present context, the latter presentation will be more convenient. We will augment the recursive support clauses given by Fact 4.8 with the following inductive clause for Kaφ‎:

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Propositional attitudes

Notice that this clause ensures that the interpretation of Kaφ‎ is persistent: if sKaφ‎ and ts, then tKaφ‎. Moreover, Kaφ‎ is vacuously supported by the empty information state. This guarantees that the set [Kaφ‎] of supporting states is indeed a proposition in the inquisitive sense—a non-empty and downward closed set of information states—and thus that the system we are defining fits within the general inquisitive semantics framework defined in Chapter 2.

To understand the clause, it is useful to look at the truth-conditions to which it gives rise. Recall that we say that a formula φ‎ is true at a world w in case w ∈info(φ‎). Also, recall from Fact 2.14 that truth relative to a world w always amounts to support at the singleton state {w}. By specializing the support clause in (10) to a singleton state {w}, we obtain the following truth-conditions for a formula Kaφ‎. (p.152)

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Given these truth conditions, it becomes clear that Kaφ‎ is supported by a state s just in case it is true at any world in s. This means, by Fact 2.19 on page 27, that modal formulas are always non-inquisitive.

Fact 8.2

For any φ‎, Kaφ‎ is non-inquisitive.

This also means that, in order to understand the semantics of Kaφ‎, we just need to understand at which worlds Kaφ‎ is true. The set |Kaφ‎| of these worlds will be the unique alternative in the proposition expressed by Kaφ‎.

According to clause (11), the truth-conditions of Kaφ‎ are very simple: Kaφ‎ is true at w just in case φ‎ is supported by the epistemic state of a at w. To see what this predicts for some particular cases, consider again the examples in (4) and (5). We assume that these sentences can be translated to our logical language by applying the operator Ka to the translations of the embedded clauses, which we assume to be identical to the translation of the corresponding main clause. Given the translations of these main clauses suggested in Chapter 5, this leads to the following translations:

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The truth-conditions for these modal formulas are as follows:

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For sentence (12a), translated as KaCb, we obtain the same predictions that standard epistemic logic would deliver: KaCb is true if Alice’s knowledge implies that Bob is coming. This holds whenever the complement of Ka is non-inquisitive.

Fact 8.3

If φ‎ is non-inquisitive, then wKaφσa(w)|φ|

Recall from Chapter 6 that we assume that declarative clauses always involve a projection operator ‘!’ which makes them non-inquisitive. (p.153) Thus, we obtain generally that, for natural language sentences in which know embeds a declarative clause, our account coincides with the one given by standard epistemic logic: a sentence Alice knows that S is predicted to be true in case it follows from Alice’s knowledge that S is true.

Now, however, the same clause for K can be applied directly to analyze sentences (12b)–(12d), in which know embeds an interrogative clause. The predictions are the expected ones: (12b) is true in case Alice’s knowledge implies that Bob is coming, or it implies that Bob is not coming; (12c) is true in case Alice’s knowledge implies that Bob is coming, or it implies that Charlie is coming; finally, (12d) is true in case for each individual d in the domain, Alice’s knowledge implies either that d is coming, or that d is not coming; in other words, (12d) is true if Alice’s knowledge determines exactly what is the set of individuals who are coming.

Notice that this treatment of interrogatives embedded under know, unlike the ones of Groenendijk and Stokhof (1984) and Karttunen (1977), has no problems dealing with knowledge of mention-some questions. To see this, consider again the mention-some question in example (6), repeated below as (14).

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Propositional attitudes

If Ix stands for ‘x is a place where one can buy an Italian newspaper,’ then (14) is naturally translated as ∃x.Ix: it expresses an issue which is resolved in a state s just in case s implies of some d that d is a place where one can buy an Italian newspaper (see Chapter 5).

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Propositional attitudes

As a consequence, the statement in (7), repeated here as (16), will be translated as Ka(∃xIx).

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Propositional attitudes

This yields the following truth-conditions: (16) is predicted to be true in case for some d, it follows from Alice’s knowledge that d is a place where one can buy an Italian newspaper.

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Thus, we correctly capture that (16) may be true by virtue of Alice knowing different things: in one case, it may be true because she knows (p.154) that one can buy an Italian newspaper at Central Station; in another case, it might be true because she knows that one can buy an Italian newspaper at the airport. The problem that we pointed out for the analyses of Groenendijk and Stokhof (1984) and Karttunen (1977) is avoided, because we do not try to reduce (16) to a claim that there is a specific piece of information that Alice knows.

As a last example illustrating the generality of the inquisitive account of knowledge, consider (18).

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Propositional attitudes

In this case, the embedded clause is a conditional question Pb?Wb, and so (18) as a whole is translated as Ka(Pb?Wb). This predicts the following truth-conditions for (18):

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That is, (18) is predicted to be true if Alice’s knowledge, restricted to those worlds where Bob puts down his Ace, settles the question whether Bill will win or not. This is the desired prediction, and it is another case where the inquisitive account goes beyond what could be predicted by existing theories of question embedding.9

This illustrates how, in inquisitive epistemic logic, the standard analysis of knowledge generalizes smoothly to the case in which the prejacent has non-trivial inquisitive content. This makes it possible to obtain a general account of the verb know in combination with both declarative and interrogative clauses—an account that does not require any type-shifting to take place and which extends the empirical coverage of existing accounts—dealing smoothly, among other things, with mention-some questions and conditional questions.

(p.155) 8.2.3 Wondering

To provide an analysis of the attitude of wondering, the language of inquisitive epistemic logic is equipped with a second kind of modal operator. Besides the modality Ka, for each agent a we also have a modality Ea which allows us to talk about the issues that the agent entertains (whence the notation Ea).10 A modal formula Eaφ‎ is interpreted by means of the following support clause.

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As in the case of Kaφ‎, it is easy to see that the support conditions for Eaφ‎ are persistent and vacuously satisfied by the empty information state, which ensures that [Eaφ‎] is a proposition in the inquisitive semantics sense. By specializing the support condition to the case of a singleton state {w}, we obtain the following truth-conditions for Eaφ‎.

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Propositional attitudes

Given these truth conditions, it is clear that Eaφ‎ is supported by a state s just in case it is true at any world in s. Thus, just like Kaφ‎, also Eaφ‎ is always non-inquisitive, regardless of whether φ‎ is inquisitive or not.

Fact 8.4

For any φ‎, Eaφ‎ is non-inquisitive.

This means that, in order to understand the semantics of Eaφ‎, we just need to understand at which worlds this formula is true. The proposition expressed by Eaφ‎ will then have a unique alternative, namely, the set |Eaφ‎| of all worlds where the sentence is true.

According to (22), Eaφ‎ is true at a world w in case φ‎ is supported by all elements s ∈Σ‎a(w) of the agent’s inquisitive state. Now, recall that the states t ∈Σ‎a(w) are precisely those states where the issues that a entertains at w are resolved. Thus, Eaφ‎ is true in case, if the issues that a entertains were resolved, φ‎ would be supported.

Notice that there is a trivial way in which Eaφ‎ may be true: φ‎ might already be supported by the current epistemic state of the agent, σ‎a(w). In this case, since all elements of Σ‎a(w) are enhancements of σ‎a(w) and support is persistent, all these elements will support φ‎ as well, and Eaφ‎ will be true at w. Now, this trivial case holds precisely when Kaφ‎ is true. This allows us to define a new formula Waφ‎ which says that Eaφ‎ holds non-trivially. (p.156)

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Propositional attitudes

The formula Waφ‎ is true at w in case (i) the current epistemic state of the agent does not support φ‎, but (ii) if the agent’s issues were resolved, φ‎ would come to be supported. We may read this less formally as: the agent epistemic state does not support φ‎, but the agent strives to reach a state where φ‎ is resolved.

The operator Wa gives us a reasonable analysis of the issue-directed attitude of wondering, and of the corresponding verb in natural language. To see what this analysis predicts, consider (24). By analysing the verb wonder as Wa and the embedded interrogative in the usual way, we obtain the translation Wa?Cb.

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Propositional attitudes

Let us consider the truth-conditions that are predicted: Wa?Cb is true at a world w in case:

  • Alice’s epistemic state σ‎a(w) does not support ?Cb, that is, σ‎a(w) contains both Cb-worlds and ¬Cb-worlds;

  • all the states s ∈Σ‎a(w) support ?Cb, that is, they consist either exclusively of Cb-worlds, or exclusively of ¬Cb-worlds.

Condition (i) means that Alice’s current knowledge does not determine whether Bob is coming; condition (ii) means that resolving Alice’s issues is bound to lead to a state that determines whether Bob is coming; in other words, Alice’s issues are not resolved unless it is established whether Bob is coming or not. This sounds like a reasonable analysis of the truth-conditions of (23).

As another example, consider (24), translated as Wa(∀x?Cx).

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This sentence is true in case Alice’s current knowledge does not determine exactly which individuals are coming, but in order to resolve Alice’s issues it would be necessary to establish exactly which individuals are coming.

This analysis of wonder provides us with an account of the validities of inferences such as (9), repeated below as (25), as the reader is asked to show in Exercise 8.5.

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(p.157) Moreover, this account of wonder also suggests an explanation of the ungrammaticality of sentences like (26), in which wonder embeds a declarative clause.

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Propositional attitudes

According to our analysis, this sentence would be translated as WaCb. When is this formula true? By definition, WaCb amounts to the conjunction ¬KaCbEaCb. Now let us consider what each conjunct requires:

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Propositional attitudes

These two conditions are contradictory: (27b) implies Σa(w)|Cb|, that is, σa(w)|Cb|, which contradicts (27a). Hence, WaCb is a contradiction.11

Notice that, in deriving this result, we have not used anything specific about the embedded clause other than the fact that it expresses a non-inquisitive proposition—a fact which is common to all declarative complements. This means that combining wonder with a declarative complement systematically results in a contradiction. This can be taken to explain the ungrammaticality of this sort of construction (for the connections between systematic contradictions and ungrammaticality, see Gajewski, 2002; Chierchia, 2013; Abrusán, 2014).12,13

To conclude this brief exposition of IEL, let us illustrate the workings of the modal operators of IEL with an example. Figures 8.1(a)–8.1(c) represent the inquisitive states of three agents at a given world w. In each case, the solid blocks represent the maximal elements of the agent’s inquisitive state, while the dashed area—corresponding to the union of these blocks—represents the agent’s knowledge. Formally, the agents’ (p.158) inquisitive states at the given world are as follows, where S denotes the downward closure of the set of states S:

  • Σ‎a(w) = {{11}, {10}}

  • Σ‎b(w) = {{11, 10}, {01, 00}}

  • Σ‎c(w) = {{11, 10, 01}, {00}}

As a consequence, the agents’ epistemic states are as follows:

  • σ‎a(w) = {11, 10}

  • σ‎b(w) = σ‎c(w) = {11, 10, 01, 00}

We take the name of each world to reflect the truth value of two atomic sentences p and q: at world 11 both are true, at world 10 only p is true, and so on. The alternatives for the polar question ?p are depicted in Figure 8.1(d).

Propositional attitudes

Figure 8.1 The inquisitive states of three agents and the alternatives for ?p.

Our three agents each stand in a different relation to the question ?p. Alice’s epistemic state implies that p is true and thereby supports the question: thus, Ka?p is true, i.e., Alice knows whether p. From this it follows that Wa?p is false (because the condition ¬Ka?p fails), i.e., Alice does not wonder whether p.

Bob’s epistemic state is trivial, and it does not support the question ?p. Thus, Kb?p is false, i.e., Bob does not know whether p. On the other hand, the information states where Bob’s issues are resolved—the elements of Σ‎b(w)—are precisely the information states in which ?p is supported. This means that Eb?p is true. Together with ¬Kb?p, this implies that Wb?p is true, i.e., Bob wonders whether p.

Charlie’s epistemic state is trivial as well, which means that, like Bob, Charlie does not know whether p. However, some elements of his inquisitive state (e.g., the information state {11, 10, 01}) fail to support the question ?p. This means that Ec?p is false, and therefore, Wc?p is false as well: thus, unlike Bob, Charlie does not wonder whether p.

(p.159) Interestingly, both Alice and Charlie do not wonder whether p, but for different reasons: in Alice’s case, it is because she has already resolved the question, while in Charlie’s case, it is because he is not interested in resolving it.

8.3 Beyond know and wonder

We have focused our attention in this chapter on some specific modal notions, in a particular logical setting. However, we think that the general approach illustrated by IEL is applicable beyond this restricted setting as well, giving rise to a richer view on the linguistic notion of modality in general. We end this chapter with some programmatic remarks on the potential benefits of such an enriched perspective.

In linguistics, modal expressions are standardly viewed as sentential operators that relate the proposition expressed by their argument (their prejacent) to a proposition encoding a set of relevant background assumptions (the modal base). Some modal expressions indicate that the prejacent is consistent with the modal base (possibility modals), while others indicate that the prejacent is entailed by the modal base (necessity modals). The nature of the modal base depends on the particular flavor of the modal expression. For instance, epistemic modals relate their prejacent to a relevant body of information, while deontic modals relate their prejacent to a modal base determined by a relevant set of rules. Finally, modal expressions differ in their grammatical category. Among the most widely investigated kinds of modal expressions are attitude verbs like know, believe, want, and hope, and auxiliary verbs like might, may, must, and should.

Sophisticated theories have been developed to capture the core mechanisms that underlie the linguistic behavior of all these different types of modal expressions in a unified way (see in particular Kratzer 2012 for a collection of influential articles, and Kaufmann and Kaufmann 2015 for a recent survey). However, while the domain that is covered by these theories is indeed impressively broad, the approach taken in inquisitive epistemic logic suggests a substantial further generalization, both of the linguistic notion of modal expressions as such, and of the theories that deal with them.

Namely, rather than construing modal expressions as relating two classical propositions, we may construe them as relating two propositions in the inquisitive sense. For instance, both modalities Ka and Ea (p.160) of IEL can be seen as expressing relations between an agent’s inquisitive state Σ‎a(w) and the inquisitive proposition [φ‎] expressed by the prejacent. This becomes evident once their clauses are re-stated as follows:

  • wKaφΣa(w)[φ]

  • wEaφΣa(w)[φ]

This shift in perspective broadens our linguistic view on modality in three ways. First, as exemplified in a very concrete way in IEL, the class of modal expressions becomes richer, now also including ones that take inquisitive constructions as their argument. Thus, it becomes possible to pursue a unified account of propositional attitude verbs like know, believe, want, and hope on the one hand, and issue-directed attitude verbs like wonder, be curious, and care on the other. Second, a more fine-grained notion of modal bases becomes available: we can now interpret modal expressions not only in the context of a certain body of information, but also in the context of a relevant background issue. And third, while on the standard account there are only two salient relations between the prejacent and the modal base, i.e., inclusion (entailment) and overlap (consistency), in the inquisitive setting there are many more, due to the fact that inquisitive propositions carry more structure than simple sets of worlds. This allows for a refinement of the standard dichotomy between possibility and necessity modals.

While these remarks are admittedly very programmatic and clearly stand in need of concrete substantiation, the research programme that they suggest seems an exciting one to pursue. The treatment of know and wonder developed in IEL just constitutes the first step in this direction.

8.4 Pointers to further work

For a more detailed presentation of inquisitive epistemic logic, the reader is referred to Ciardelli and Roelofsen (2015) and Ciardelli (2016d). Besides the knowledge and issues of individual agents, these references are also concerned with collective notions of knowledge and issues—in particular, with the common knowledge and common issues which are publicly shared among the group. The modal logic arising from IEL has been investigated and axiomatized in Ciardelli (2014, 2016d). Various extensions and refinements of IEL have been explored in the recent literature as well. Ciardelli and Roelofsen (2015) and van Gessel (2016) equip the IEL framework with a dynamics that models the way in which a multi-agent scenario evolves when a statement is made or a question is asked, generalizing the analysis of public and (p.161) private announcements in dynamic epistemic logic (van Ditmarsch et al., 2007). The logic of public announcements in the inquisitive setting is axiomatized in Ciardelli (2017a). Ciardelli and Roelofsen (2014) develop a refinement of IEL that does not only deal with ‘hard knowledge’ but also with defeasible beliefs, which may be revised or retracted. This inquisitive belief revision framework can be used to model not just linguistic information exchange, but also other information-related processes such as rational inquiry, where the interplay between issues and beliefs has been argued to play a crucial role (see, e.g., Olsson and Westlund, 2006). In Theiler et al. (2016a,b) the linguistic treatment of know suggested in IEL is refined in order to capture, among other things, the observation that the truth of a knowledge attribution requires not only that the agent have enough knowledge to resolve the embedded question, but also that she not believe any pieces of information that would falsely resolve the question (Spector, 2005; George, 2013; Cremers and Chemla, 2016). In Theiler et al. (2017), IEL is extended with a treatment of believe, which accounts for the fact that this epistemic verb, unlike know, is neg-raising (e.g., Alice doesn’t believe that Bill did it typically leads to the inference that Alice believes that Bill didn’t do it) and for the fact that it does not take interrogative complements (e.g., Alice believes whether Bill did it is ungrammatical). Finally, Roelofsen and Uegaki (2016) and Cremers et al. (2017a) discuss possible refinements of the IEL treatment of wonder and believe in order to obtain a more comprehensive account of the ignorance inferences that these verbs trigger. For instance, when wonder embeds an alternative question, it does not just imply that the issue expressed by the question is not yet resolved in the subject’s current information state, but also that all the alternatives that make up the issue are still compatible with the subject’s information state (e.g., Alice wonders whether Bob, Charlie, or Daniel did it does not only imply that Alice does not know yet who the culprit is, but also that she cannot yet rule out any of Bob, Charlie, and Daniel as a potential culprit).

8.5 Exercises

Exercise 8.1 Truth conditions of knowledge and wonder attributions

Consider again the situation described in Figure 8.1. For each question μ‎ in the list below, determine for which agents x the formulas Kxμ‎ and Wxμ‎ are true.

  • ?q

  • ?(pq)

  • (p.162) ?!(pq)

  • ¬q?p

Exercise 8.2 Reasoning about knowledge and wondering

Consider the inference in (9), repeated here as (28):

(28)

Propositional attitudes

  1. 1. Translate the premises and the conclusion of the inference in the language of IEL. Use a predicate C which stands for ‘is the culprit’, and assume that this predicate is satisfied by exactly one individual at each world in the model. Recall that embedded declarative clauses are translated as involving a projection operator ‘!’.

  2. 2. Prove that if (28a) and (28b) are true at a world, so is (28c).

  3. 3. Prove that (28a) and (28b) entail (28c) in the sense of inquisitive semantics.

Exercise 8.3 Inquisitive epistemic logic

For {Ka,Ea,Wa}, determine whether the following logical principles are valid or invalid. Provide proofs or counterexamples.

  • Normality: (φψ)(φψ)

  • Distribution laws:

    • (φψ)φψ

    • (φψ)φψ

    • (φψ)!(φψ)

  • Monotonicity: if φ‎ψ‎, then φψ

  • Necessitation: if ⊧φ‎, then φ

Exercise 8.4 Ignorance

Consider a new modal operator Na in IEL, where Naφ‎ is informally read as ‘a is completely ignorant with regard to φ‎’. Define a suitable semantic interpretation of Naφ‎, which ensures that whenever φ‎ is non-inquisitive, Naφ‎ is equivalent to ¬Ka ?φ‎.

Notes:

(1) Following the standard linguistic notation, we indicate the ungrammaticality of a sentence by marking it with a ‘*’.

(2) Epistemic logic is most commonly presented in terms of a binary relation RaW×W, where wRav holds if v is compatible with what a knows at w. This presentation is equivalent to the one in terms of functions from worlds to information states: given a relation Ra, we can define a corresponding map σRa by letting σRa(w):={vWwRav}; conversely, given a map σ‎a : W (W), we can define a binary relation Rσa by letting wRσavvσa(w). It is immediate to see that the correspondence is one-to-one.

(3) A well-known problem with this analysis of knowledge is the problem of logical omniscience: an agent is always taken to know all the consequences of the things that she knows (for discussion of this problem and for some solutions see, among others Eberle, 1974; Hintikka, 1975, 1979; Fagin and Halpern, 1987; Fagin et al., 1995; Stalnaker, 1991; Artemov and Kuznets, 2006). Our proposal will be an extension of epistemic logic, and it will inherit the logical omniscience problem. On the other hand, the epistemic logic analysis of knowledge and belief has proven very fruitful in many domains, from epistemology and logic to economics, linguistics, and computer science, and thus provides a natural starting point for our inquisitive extension.

(4) One aspect of the semantics of the verb know which is not captured by the epistemic logic analysis is the fact that a knowledge attribution presupposes the truth of the complement. This problem becomes especially apparent when know appears under negation or in a conditional antecedent: in English, the sentence Alice doesn’t know that Bob is coming still implies that Bob is coming, but this is not captured by the rendition of this sentence in epistemic logic, ¬KjCb, which is true in case Bill is not coming, or he is and Alice doesn’t know this. To do justice to these observations, one should supplement the epistemic logic analysis with the stipulation that Kaφ‎ presupposes that φ‎ is true, and with clauses that determine how presuppositions are projected from sub-sentential constituents (in the style of Karttunen, 1973, 1974).

(5) In addition to this, Karttunen (1977) and Groenendijk and Stokhof (1984) also differ in their notion of complete answer to a wh-question. In the subsequent literature, the two notions have come to be referred to, respectively, as the weakly exhaustive and the strongly exhaustive answer to the wh-question. This difference is orthogonal to our main concerns in this chapter.

(6) To deal with these cases, Groenendijk and Stokhof (1984) propose to treat (6) as having a higher semantic type and to type-shift the entry for the embedding verb so that it can apply to objects of this type. We will not discuss this option in detail here. A problem with this lifting strategy is pointed out in footnote 20 of Ciardelli (2017b).

(7) A particularly interesting inquisitive attitude is caring. Although the verb care does embed both declaratives and interrogative complements, the attitude of caring itself cannot in general be viewed as having a classical proposition as its object, but should rather be seen as oriented towards an issue. See Elliott et al. (2017) and Ciardelli and Roelofsen (2018) for discussion.

(8) In an enrichment of IEL which also models defeasible beliefs, these requirements may be weakened. For instance, Ciardelli and Roelofsen (2014) propose a framework in which we can distinguish between the agent’s prior issues and the agent’s current issues—the issues entertained given the agent’s current beliefs.

(9) Isaacs and Rawlins (2008), who are specifically concerned with conditional questions, do provide an analysis of such questions embedded under know. However, since their analysis of conditional questions is based on discourse dynamics, it requires a significant complication of the lexical entry for know. In addition, since their analysis of questions is based on binary relations, their account is less general (due to the limitations discussed in Section 9.3 below). For example, it cannot account for an embedded mention-some conditional question like the one in (i).

((i))

Propositional attitudes

Finally, essentially due to a technical problem pointed out in footnote 8 of Sano and Hara (2014), Isaacs and Rawlins wrongly predict that knowing a conditional question implies knowing whether the antecedent is true.

(10) Although we read Ea as ‘entertain’—for lack of a better term—we do not propose to regard the modality Ea as an analysis of the verb entertain (or any other English verb).

(11) Notice that if a sentence φ‎ is false at all worlds, this ensures that φ‎ is a contradiction not only in the truth-conditional sense, but also in the inquisitive sense, i.e., that [φ‎] = {}. For by definition, the set of worlds where φ‎ is true is info(φ)=[φ]. But the only way that [φ] can be empty is if [φ‎] = {}, i.e., if φ‎ is a contradiction in the inquisitive sense.

(12) The explanation given here for the infelicity of wonder that does not need the strong connection between information and issues that we are assuming by letting σa(w)=Σa(w): it suffices to assume that σa(w)Σa(w), i.e., that the agent knows (or indeed believes) the information presupposed by the issues that she entertains. For discussion of this point, see Ciardelli and Roelofsen (2018).

(13) This explanation can be adapted naturally to other verbs such as be curious and investigate, which can be analysed along the same lines as wonder. An explanation of the fact that verbs like believe and think do not take interrogative complements can also be given in inquisitive semantics, see Theiler et al. (2017).