Inquisitive Modal Logic: A Preview

Abstract

Throughout this book, we have emphasized different ways in which questions are relevant in logic. We saw that questions can be seen as names for types of information, and that by generalizing logic to questions we can capture logical relations holding between information types. We also saw that in the inquisitive setting, a more general account of certain logical operators emerges, which boils down to the classical one in the special case of statements, but which also covers the role of these operators in questions. Furthermore, we saw that questions may be used in inferences as placeholders for arbitrary information of a certain type, and that by reasoning with questions we can build formal proofs of the existence of certain logical dependencies (and of certain logical relations of answerhood and presupposition).

Throughout this book, we have emphasized different ways in which questions are relevant in logic. We saw that questions can be seen as names for types of information, and that by generalizing logic to questions we can capture logical relations holding between information types. We also saw that in the inquisitive setting, a more general account of certain logical operators emerges, which boils down to the classical one in the special case of statements, but which also covers the role of these operators in questions. Furthermore, we saw that questions may be used in inferences as placeholders for arbitrary information of a certain type, and that by reasoning with questions we can build formal proofs of the existence of certain logical dependencies (and of certain logical relations of answerhood and presupposition). However, there is another important role for questions in logic that we have not yet touched upon. This role becomes apparent when we equip inquisitive logic with modal operators that capture question-directed modal notions. This takes us into the realm of inquisitive modal logic.

Since inquisitive modal logic is the topic of a freshly started research project, significant developments in this area can be expected in the next few years. For this reason, we leave a comprehensive exposition of the topic for a future occasion. In this final chapter, we will however give a preview of this sub-field of inquisitive logic. First, in Sect. 8.1 we will explain why the prospect of adding modalities to inquisitive logic is especially promising. In Sects. 8.2 and 8.3 we will sketch how different kinds of modalities have so far been added to inquisitive logic, illustrating the significance of these modalities in the setting of one particular interpretation, and discussing one important aspect that distinguishes inquisitive modal logic from the inquisitive propositional and predicate logics discussed in this book. Finally, in Sect. 8.4 we mention some directions for future work.

8.1 Issue-Directed Modal Notions

Modal logic is an incredibly versatile sub-field of logic, which is used to formally analyze a range of notions of great interdisciplinary importance. In its various interpretations, it is used to capture notions such as knowledge and belief, permission and obligation, different varieties of necessity and possibility, provability in a theory, truth in the past or in the future, strategic ability, and much more. What do these notions have in common? They can all be analyzed as properties of propositions: it is propositions that can be necessary or possible, known or believed, true in the past or in the future, etcetera. Not surprisingly, in the sort of English sentences which are formalized in standard modal logic, the argument of a modal operator is typically given by a ‘that’ clause, whose content is a (standard) proposition. This is illustrated by the following examples:

Now consider the following sentences:

These sentences express instances of important modal notions. Sentence (2-a) is an example of a supervenience claim: given the electoral system, there can be no difference in winner without an underlying difference in the number of votes. Supervenience is a modal notion that plays a key role in all areas of analytic philosophy, at least as important as that of possibility and necessity. Sentences (2-b) and (2-c) ascribe a certain ‘inquisitive attitude’ to Brown: they characterize her as being in a state that bears a certain relation to a question content, just like the belief ascription in (1-b) characterizes her as standing in a certain relation to a proposition. Inquisitive attitudes have recently come under attention in philosophy of mind; various authors (Friedman [1], Carruthers [2]) have emphasized that such attitudes are just as important to the analysis of inquiry and agency as the much more widely studied propositional attitudes. Finally, sentence (2-c) is a control claim, stating that a certain aspect of the world—in this case the outcome of the election—is under Brown’s control. To be able to express and reason about what aspects of a situation each agent controls is important for the analysis of action and strategic reasoning in a multi-agent setting.

What these three examples have in common is that they ascribe a modal property, not to a proposition, but to a question, namely, the embedded interrogative ‘who wins the election’. Or rather, more precisely, they ascribe a modal property to the content expressed of this question. Let us refer to the content expressed by a question as an issue. We can then say that the notions illustrated by the examples in (2) are issue-directed: it is issues that are the relata of a supervenience claim, the objects of wondering and caring, and the sort of things that an agent may or may not have control over.

In standard modal logic, the language does not contain formulas that stand for questions. This is no accident, since standard modal logic builds on truth-conditional semantics, which as we discussed is not suitable to interpret questions. As a consequence, issue-directed modal notions have so far remained outside of the scope of modal logic. This is a significant limitation: we saw three examples of important issue-directed modal notions, and they are not isolated cases: on the contrary, once we look for them, interesting issue-directed notions can be identified in all areas of application of modal logic.

This limitation can be overcome by building a new framework for modal logic based on inquisitive logic. As we saw, in inquisitive logic we have not only formulas that stand for statements, but also formulas that stand for questions. The semantics allows us to model the content of a question as a set of information states—those in which the question is supported. In this setting, we can naturally equip our language with modal operators O that can apply to a question $\mu$ to yield a statement $O\mu$, the truth conditions of which are defined in terms of the content of $\mu$. In this way, a broad range of new, interesting modal notions can be formally analyzed and brought within the purview of modal logic. As a result, the domain of application of modal logic can be extended substantially.

In the next two sections, we will make the idea more concrete by looking at two particular kinds of modal operators that can be added to an inquisitive logic and which have received attention in the inquisitive logic literature (see [3,4,5,6,7,8,9,10,11]).

8.2 Generalizing Kripke Modalities to Questions

Modalities in Kripke semantics. In standard Kripke semantics, modalities are analyzed as quantifiers over a set of accessible worlds. Formally, this works by extending the language with a new unary operator $\square$ (a dual is defined by letting $\Diamond \varphi$ abbreviate $\lnot \square \lnot \varphi$). Models for the language are obtained by equipping a set W of possible worlds with an accessibility relation $R:W\times W$. The semantics, which is given in terms of truth conditions relative to a world, interprets $\square$ by means of the following clause, where $R[w]=\{v\in W\mid wRv\}$:

$$M,w\models \square \varphi \iff \forall v\in R[w]: M,v\models \varphi $$

Conceptually, the relation R can be given many different interpretations. For instance, R[w] could be viewed as the set of worlds that are possible according to what a certain agent knows or believes at w; it could be the set of worlds that conform to a certain normative code, or the set of worlds which are compatible with certain background facts about the world (e.g., the laws of physics), or the set of worlds which are possible future instants relative to w, etcetera. Each of these interpretations gives rise to a corresponding reading for modal formulas. For instance, $\square p$ could be read as “the agent knows/believes p”, “it is obligatory that p”, “necessarily p”, “it will always be the case that p”, and so on.

Kripke modalities in inquisitive logic. Let us see how this treatment of modalities can be extended to the inquisitive setting. Syntactically, we may just extend the language of inquisitive (propositional or predicate) logic with a new unary operator $\square$. Models are defined simply by extending an information model M with an accessibility relation R on the set of possible worlds. Note that the resulting models coincide with standard Kripke models (in the predicate logic case, Kripke models with a constant domain).

The semantics is obtained by extending the definition of support with the following clause¹:

• $M,s\models \square \varphi \iff \forall w\in s: M,R[w]\models \varphi .$

By specializing this clause to singleton states, we get the following truth conditions for $\square \varphi$:

• $M,w\models \square \varphi \iff M,R[w]\models \varphi .$

It is clear from these clauses that for any $\varphi$—regardless of whether $\varphi$ is a statement or a question—$\square \varphi$ is truth-conditional: it supported at a state s iff it is true at each world $w\in s$. As a consequence , the semantics of $\square \varphi$ is fully determined at the level of truth conditions. In words, these truth conditions say that $\square \varphi$ is true at a world w iff $\varphi$ is supported by the set R[w] of successors of w.

In case the argument of our modality is a statement $\alpha$, our semantics boils down to Kripke semantics. Indeed, if $\alpha$ is truth-conditional, we have:

$$\begin{aligned} M,w\models \square \alpha\iff & {} M,R[w]\models \alpha \\\iff & {} \forall v\in R[w]: M,v\models \alpha . \end{aligned}$$

If we restrict the language to classical formulas not containing inquisitive operators ( or, in the predicate logic case, ), we thus obtain a language that can be fully identified with the language of standard modal logic. All formulas in this fragment are truth-conditional, and the truth conditions for them are the same as in standard modal logic. This means that our inquisitive modal logics (in the plural, since different logics arise from different classes of frames, as usual) are conservative extensions of the corresponding classical systems.

Illustration in the epistemic setting. To appreciate how inquisitive semantics allows us to extend the operator $\square$ to questions, it is helpful to have in mind the epistemic interpretation of $\square$ as formalizing the verb ‘know’. In this interpretation, the information state R[w] models the epistemic state of the agent at world w—the set of worlds compatible with what the agent knows at w.

Consider the knowledge ascriptions in (3). In inquisitive modal logic, they can be formalized straightforwardly by applying the knowledge modality to the translations of the complements ‘that Alice passed the test’ (Pa), ‘whether Alice passed the test’ (?Pa) and ‘who passed the test’ ($\forall x?Px$).

Let us see what our semantics predicts for these formulas. First, we saw that all of them are truth-conditional, which is in line with the fact that the sentences in (3) are statements. We also saw that for the ‘standard’ knowledge ascription $\square Pa$, the truth conditions are the same as given by standard epistemic logic:

$$M,w\models \square Pa\iff \forall v\in R[w]: M,v\models Pa.$$

In words, $\square Pa$ is true in case Pa is true in all worlds compatible with the agent’s knowledge.

Now let us consider the formula $\square ?Pa$, where the argument is the polar question ?Pa. Using the support conditions for polar questions, which are familiar by now, we have:

$$\begin{aligned} M,w\models \square {?Pa}\iff & {} M,R[w]\models {?Pa}\\\iff & {} \forall v,v'\in R[w]: (M,v\models Pa\iff M,v'\models Pa). \end{aligned}$$

Thus, $\square ?Pa$ is true just in case the truth value of Pa is settled in the agent’s epistemic state—i.e., if the agent has no uncertainty concerning this truth value. This is intuitively the correct prediction for a knowing-whether ascription such as (3-b). We also have:

$$\begin{aligned} M,w\models \square ?Pa\iff & {} M,R[w]\models {?Pa}\\\iff & {} M,R[w]\models Pa\text { or }M,R[w]\models \lnot Pa\\\iff & {} M,w\models \square Pa\text { or }M,w\models \square \lnot Pa\\\iff & {} M,w\models \square Pa\vee \square \lnot Pa. \end{aligned}$$

Thus, $\square ? Pa$ has the same truth conditions as $\square Pa\vee \square \lnot Pa$, and since the two formulas are both truth-conditional, they are equivalent: $\square ? Pa\equiv \square Pa\vee \square \lnot Pa$. This is an intuitive result: knowing whether Alice passed the test amounts to knowing either that Alice passed the test, or that she did not pass the test.

Finally, consider the formula $\square \forall x?Px$, where the argument is the mention-all question$\forall x?Px$ asking for the extension of P. Using the support conditions for mention-all questions (cf. Example 5.2.5 on Sect. 5.2), we have:

$$\begin{aligned} M,w\models \square {\forall x?Px}\iff & {} M,R[w]\models \forall x{?Px}\\\iff & {} \forall v,v'\in R[w]: P_v=P_{v'}. \end{aligned}$$

Thus, $\square {\forall x?Px}$ is true at a world w in case the extension of P is settled in the agent’s epistemic state—i.e., if the agent has no uncertainty about this extension. This is the intuitively correct prediction for (3-c): to know who passed the test is to know the extension of the predicate ‘having passed the test’.

As these examples illustrate, our generalized semantics for $\square$ allows us to give a neat uniform account of knowledge ascriptions involving statements (‘knowing that’) and questions (‘knowing whether, who, what, ...’). Note that in this case, the point is not that knowledge ascriptions such as (3-b) and (3-c) cannot be captured in standard modal logic: they can. We already saw that $\square ?Pa$ is equivalent to the standard modal formula $\square Pa\vee \square \lnot Pa$, and for (3-b), we have:

$$\square \forall x?Px\;\equiv \;\forall x(\square Px\vee \square \lnot Px).$$

The point is, rather, that in inquisitive modal logic the semantics of knowledge attributions can be derived compositionally in a principled way from a general semantics for ‘know’ and the semantics of the embedded complements. The fact that, say, knowing whether p amounts to knowing p or knowing $\lnot p$ does not have to be stipulated, but is derived in the logic, which is a welcome result. Moreover, such an account sheds light on how verbs like ‘know’ (but also ‘remember’, ‘tell’, and many others) work in natural language (for the linguistic relevance of such an account, see Ciardelli and Roelofsen [12], Theiler et al. [13, 14]).

Non-epistemic interpretations. We illustrated the semantics for $\square$ in the epistemic setting, but there are many other natural interpretations. Just to give a hint, let us consider the pair of formulas $\square p$ and $\square {?p}$. In a legal setting, if $\square p$ says that the law mandates that p, then $\square {?p}$ says that the law mandates whether p. In the setting of provability logic, if $\square p$ says that the theory proves p, then $\square {?p}$ says that the theory decides p. In the setting of temporal logic, if $\square p$ says that p will henceforth always be the case, then $\square {?p}$ says that p is henceforth immutable. And yet other natural interpretations can easily be given.

Notes on the logic. The logic of the generalized modality $\square$ turns out to be very simple. The distributivity axiom of standard modal logic is generally valid, including when $\varphi$ and $\psi$ are questions:

$$\square (\varphi \rightarrow \psi )\rightarrow (\square \varphi \rightarrow \square \psi ).$$

The necessitation principle also holds: if $\varphi$ is a validity, then so is $\square \varphi$. From this, additional facts follow, such as the monotonicity of $\square$ and the commutation of $\square$ with $\wedge$. In the predicate logic case, the constant domain setup ensures that the Barcan and coverse Barcan formulas also hold, i.e., $\square$ commutes with the universal quantifier²:

$$\forall x\square \varphi \equiv \square \forall x\varphi .$$

A distinctive feature of $\square$ in the inquisitive setting is the validity of the following pseuso-commutation principles, which say that an inquisitive operator under $\square$ is equivalent to the corresponding classical operator above $\square$.

These principles allow us to push a $\square$ modality through an inquisitive operator. In the propositional case, this can used to show that any formula of the form $\square \varphi$ is equivalent to a formula of standard modal logic see Sect. 6 of [15]. This is not the case in the first-order setting, however: there are modal formulas of the form $\square \varphi$, where $\varphi$ is a question, which, while being statements, are not equivalent to any formula of standard first-order modal logic.³ Thus, by extending $\square$ to questions we obtain a logic which is more expressive than standard modal logic, even in restriction to statements.

In the propositional setting, Ciardelli [15, Sect. 6] has described a strategy for turning complete axiomatizations for standard (normal) modal logics into complete axiomatizations of the corresponding inquisitive modal logics. This strategy applies to all the most familiar modal logics, yielding completeness results for their inquisitive extensions.

8.3 Properly Inquisitive Modalities

Basic setup. In the literature on inquisitive modal logic, a different modal operator has also been studied. This modality is standardly denoted $\boxplus$, and as in the case of $\square$, it can be added to the language of inquisitive propositional or predicate logic. As we will see, this is an example of a modality that allows us to capture issue-directed modal notions that cannot be captured in standard modal logic.

In order to interpret this modality, we equip an information model with a relation $\mathcal {R}:W\times \wp (W)$ between worlds and information states. Such a relation allows us to associate to each world w a set of information states $\mathcal {R}[w]=\{s\subseteq W\mid w\mathcal {R}s\}$.⁴$^,$⁵ Let us call the resulting model an inquisitive modal model.

The semantics is obtained by extending the definition of support with the following clause for $\boxplus$:

• $M,s\models \boxplus \varphi \iff \forall w\in s,\forall t\in \mathcal {R}[w]:M,t\models \varphi$.

This clause makes the formula $\boxplus \varphi$ truth-conditional. We can thus study its semantics by looking at its truth conditions, which are as follows:

• $M,w\models \boxplus \varphi \iff \forall t\in \mathcal {R}[w]:M,t\models \varphi$.

Thus, $\boxplus \varphi$ is a statement which is true at a world w just in case $\varphi$ is supported at all $\mathcal {R}$-successors of w. Note that the relation $\mathcal {R}$ also induces a Kripke-style accessibility relation $R\subseteq W\times W$, given by:

$$R[w]:=\bigcup \mathcal {R}[w].$$

This means that in the context of an inquisitive modal model we can also interpret a modality $\square$, which uses the induced relation R in the way discussed in the previous section.

When applied to a truth-conditional formula, the two modalities $\boxplus$ and $\square$ coincide with each other and with the universal modality of standard modal logic. That is, when $\alpha$ is truth-conditional we have:

$$\begin{aligned} M,w\models \boxplus \alpha\iff & {} M,w\models \square \alpha \\\iff & {} \forall v\in R[w]: M,v\models \alpha . \end{aligned}$$

Thus, the classical fragment of our modal logic is just standard modal logic, with $\boxplus$ collapsing onto $\square$. However, things become interesting as soon as we consider formulas obtained by applying $\boxplus$ to a question. To appreciate the results, it is helpful to have a concrete interpretation of inquisitive modal models in mind.

Illustration in the inquisitive-epistemic setting. In the inquisitive epistemic logic proposed by Ciardelli and Roelofsen [3], an inquisitive modal model is given the following intuitive interpretation: given a world w, we have $w\mathcal {R}s$ just in case all the issues the agent is interested in are settled in the state s—that is, if s is an information state where the agent’s curiosity is satisfied. The set of states $\mathcal {R}[w]$ thus captures the inquisitive state of the agent at world w, encoding the issues that the agent is interested in. The information state $R[w]=\bigcup \mathcal {R}[w]$ is viewed as reflecting the agent’s knowledge at w.

To make this more concrete, consider the three situations depicted in Fig. 8.1, where as usual, $w_{pq}$ stands for a world where p and q are both true, $w_{p\overline{q}}$ for a world where p is true and q false, etcetera. In Fig. 8.1a–c, the maximal elements of the agent’s inquisitive state $\mathcal {R}[w]$ are drawn in solid lines; the corresponding information state $R[w]=\bigcup \mathcal {R}[w]$ is drawn by the dashed line. Here, w stands for an arbitrary world in the model (note that the different sub-figures depict different models, since the accessibility relation $\mathcal {R}$ is different).

• In Fig. 8.1a, the epistemic state of the agent is $R_1[w]=\{w_{pq},w_{p\overline{q}}\}$. This means that the agent knows that p and does not know whether q. The agent’s inquisitive state is $\mathcal {R}_1[w]=\{\{w_{pq},w_{p\overline{q}}\}\}^{\downarrow }$. This means that the issues of the agent are already settled by the agent’s knowledge state (and, therefore, also by any stronger body of information). Thus, Fig. 8.1a represents a situation where the agent knows that p and has no further open issues.

• In Fig. 8.1b, the epistemic state of the agent is the entire set of worlds $R_2[w]=\{w_{pq},w_{p\overline{q}},w_{\overline{p} q},w_{\overline{pq}}\}$. This means that the agent has no information. The agent’s inquisitive state is $\mathcal {R}_2[w]=\{\{w_{pq},w_{p\overline{q}}\},\{w_{\overline{p}q},w_{\overline{pq}}\}\}^{\downarrow }$. This means that the issues of the agent can be settled either by reaching a state at least as strong as $\{w_{pq},w_{p\overline{q}}\}$—i.e., by establishing that p—or by reaching a state at least as strong as $\{w_{\overline{p}q},w_{\overline{pq}}\}$—i.e., by establishing that $\lnot p$. In other words, the agent’s issues are settled just in case the question ?p is resolved. Thus, Fig. 8.1b represents a situation where the agent has no knowledge and is interested (only) in the issue of whether p.

• In Fig. 8.1c, the situation is parallel to the one in the previous case, but with the roles of p and q swapped: the agent has no knowledge and is interested (only) in the issue of whether q.

Fig. 8.1

Three inquisitive states for an agent. The maximal elements of the inquisitive state are depicted in solid lines, the corresponding epistemic states in dashed lines. The last figure shows the alternatives for the question ?p

With this particular intuitive interpretation of inquisitive modal models in mind, let us now examine the different significance of the modal claims $\square \mu$ and $\boxplus \mu$ when $\mu$ is a question. The truth conditions for $\square \mu$ are:

• $M,w\models \square \mu \iff M,R[w]\models \mu$

Thus, $\square \mu$ is true if what the agent knows suffices to resolve the question $\mu$. As we saw in the previous section, this corresponds to the intuitive truth conditions of the statement ‘the agent knows $\mu$’. Now let us consider $\boxplus \mu$. We have:

• $M,w\models \boxplus \mu \iff \forall t\in \mathcal {R}[w]: M,t\models \mu$

Thus, $\boxplus \mu$ is true just in case any information that settles the agent’s issues also settles $\mu$—in other words, if settling $\mu$ is necessary in order to satisfy the agent’s curiosity. This could be the case in a trivial way if the agent’s current information settles $\mu$, i.e., if $\square \mu$ is true. But the truth-conditions of $\boxplus \mu$ are more lenient: it could be the case that $\mu$ is not settled by the agent’s current information, but it is settled by all “target” information states where the agent’s issues are settled. This situation is described by the formula $\lnot \square \mu \wedge \boxplus \mu$, which [3] propose to adopt as a formalization of the statement ‘the agent wonders about $\mu$’. Let us illustrate this with the pictures in Fig. 8.1.

• In Fig. 8.1a, the agent’s epistemic state (dashed) supports the question ?p. Thus, we have $w\models \square {?p}$ (and then, by persistency, also $w\models \boxplus {?p}$). Thus, this is classified as a situation where the agent knows whether p.

• In Fig. 8.1b, the agent’s state epistemic state (dashed) does not support ?p. However, each element of the agent’s inquisitive state (the two solid blocks and their subsets) supports ?p. Thus, in this state we have $w\models \lnot \square {?p}\wedge \boxplus {?p}$. So, this is classified as a situation where the agent wonders whether p.

• In Fig. 8.1c, the agent’s epistemic state again does not support ?p. In this case, however, some elements of the agent’s inquisitive state, for instance $\{w_{pq},w_{\overline{p}q}\}$, do not support ?p either. In this situation, we have $w\models \lnot \square {?p}\wedge \lnot {\boxplus {?p}}$. So this is classified as a situation where the agent neither knows whether p, nor wonders about it.

This illustrates how in inquisitive modal logic we can express not only facts about the knowledge agents have, but also facts about the issues they entertain. As we are now going to see, in order to express these facts, questions are crucial.

Modal statements about questions. Throughout the book, when setting up a system of inquisitive logic, we started out with a classical logic of statements and we added questions to it. While the addition of questions resulted in a more expressive language, this gain in expressive power did not concern statements: as witnessed by Corollary 3.4.5 and Proposition 5.3.7, any truth-conditional formula in inquisitive propositional or predicate logic is equivalent to a classical formula that does not contain any question operator—and thus equivalent to a formula of classical propositional or predicate logic.

In inquisitive modal logic, the situation is different. Consider again the formula $\boxplus {?p}$. Like any modal formula, this is a statement, i.e., truth-conditional. However, one can prove (cf. Prop. 7.1.18 in Ciardelli [15]) that $\boxplus {?p}$ is not equivalent to any -free formula. Thus, in inquisitive modal logic, the presence of questions has repercussions also on the range of statements that the language can express: by embedding questions under modal operators, we can express modal statements that are not expressible without referring to questions.

This is a significant difference: it means that in inquisitive modal logic—unlike in the systems considered in the previous chapters—questions are not merely added on top of a pre-existing logic of statements. Rather, statements and questions are crucially intertwined in the way illustrated in Fig. 8.2: questions are built up from statements by means of and ; at the same time, by embedding questions under $\square$ and $\boxplus$ we can form new statements.

Fig. 8.2

In inquisitive modal logic, statements and questions are intertwined in an essential way

This brings out a further role for questions in logic, in addition to the ones discussed in detail in this book: questions give us names for issues; by defining modal operators that can apply to questions we can then capture modal facts about issues—the kind of issue-directed modal notions mentioned in Sect. 8.1. We have illustrated this potential in this section by describing the proposed analysis of wondering in inquisitive modal logic, but similar ideas can in principle be deployed for many other issue-directed notions. This territory, however, is almost entirely uncharted, and remains to be explored in future research.

Notes on the logic. Like the modality $\square$, the inquisitive modality $\boxplus$ also has very familiar features. In this case as well, the distributivity axiom

$$\boxplus (\varphi \rightarrow \psi )\rightarrow (\boxplus \varphi \rightarrow \boxplus \psi )$$

is valid for any formulas $\varphi$ and $\psi$, and so is the necessitation principle: if $\varphi$ is valid then so is $\boxplus \varphi$. As usual, this implies that $\boxplus$ is monotonic and commutes with $\wedge$. In the predicate logic case, $\boxplus$ also commutes with $\forall$ due to the constant domain setup of the semantics. The main difference with $\square$ is that $\boxplus$ does not validate the pseudo-distributivity over inquisitive operators:

As shown by Ciardelli [15, Chap. 7], distributivity and necessitation are in fact all we need to completely axiomatize the logic of $\boxplus$ in the propositional case, which shows that $\boxplus$ is an extremely natural generalization of the standard universal modality. This completeness result is also extended to a range of modal logics obtained by imposing some salient conditions on the relation $\mathcal {R}$.

8.4 Looking Ahead

The foregoing discussion has, hopefully, given the reader an idea of why combining questions with modalities is especially interesting, and of how extending modal logic into the inquisitive territory has the potential to bring new interesting modal notions within the purview of logical analysis. Although inquisitive modal logic has received some attention in recent years (see the list of references in Appendix A), much more work is needed to bring out this potential.

First, the role of questions in modal logic has to be demonstrated by showing that a range of interesting modal notions can be analyzed in this setting. In addition to the analysis of issue-directed attitudes such as wondering illustrated above, the range of natural applications include the analysis of supervenience and strategic control. In the case of supervenience, the idea is, at a first pass, that property P supervenes on property Q in case, in the relevant domain of possibilities, the extension of Q is determined by the extension of P, which can be formalized in inquisitive modal logic by $\square ({\forall x?Px}\rightarrow {\forall x?Qx})$ (Ciardelli [17]). Things are somewhat more complicated, however: for reasons that we cannot discuss in detail here, a proper analysis of different kinds of supervenience in fact requires the tools of the logic ${{\textsf {InqBQ}}}^+$ developed in Sect. 7.5, where information states are modeled as sets of world-assignment pairs. In the case of strategic control, the idea is that, within an appropriate interpretation of inquisitive modal models, a formula such as $\lnot \square {?p}\wedge \boxplus {?p}$ says that the truth value of p is not settled before the agent’s action, but becomes settled as soon as the agent has acted; this captures the fact that whether p comes about is determined by the agent’s choice at a certain point in time. The possibility of this interpretation of inquisitive modalities is mentioned by Ciardelli [15, Sect. 7.5], but its integration in the context of logics of actions like stit logic (Belnap et al. [18]), coalition logic (Pauly [19]) and alternating-time temporal logic (Alur et al. [20]) remains to be developed and investigated.

Moreover, modal logic has an extremely rich mathematical theory. Once we extend modal logic to questions, such a theory needs to be reconstructed in the generalized setting. Ciardelli and Otto [8], Meißner and Otto [9] have recently made a first step towards a model theory of inquisitive modal logic, characterizing the expressive power of the logic in terms of a suitable notion of bisimulation, defining translations to first-order predicate logic, and proving analogues of the classical van Benthem theorem, which characterizes modal logic as the bisimulation invariant fragment of first-order predicate logic. Much more remains to be done, however. One topic that remains entirely to be studied in the inquisitive setting is frame definability, where it is natural to look for analogues of Sahlqvist theory and the Goldblatt-Thomason theorem. Other areas to be explored are the proof theory of inquisitive modal logic (where, for instance, tableaux systems might be fruitfully developed), the range of modal operators definable in the inquisitive setting, and the properties of first-order inquisitive modal logic, which has so far not been systematically investigated.