Free Choice in Modal Inquisitive Logic

This paper investigates inquisitive extensions of normal modal logic with an existential modal operator taken as primitive. The semantics of the existential modality is generalized to apply to questions, as well as statements. When the generalized existential modality is applied to a question, the result is a statement that roughly expresses that each way of resolving the question is consistent with the available information. I study the resulting logic both from a semantic and from a proof-theoretic point of view. I argue that it can be used for reasoning about a general notion of ignorance, and for reasoning about choice-offering permissions and obligations. The main technical results are sound and complete axiomatizations, both for the class of all Kripke frames, and for any class of frames corresponding to a canonical normal modal logic.

sentence is supported by an information state if the information it encodes implies that Jane takes the bus to work. The latter sentence is supported by an information state if the information it encodes settles the question of how Jane travels to work.
Modal inquisitive logic has been studied in several works. 1 Some approaches extend existing modal logics to the inquisitive semantics setting. Some examples include inquisitive extensions of: normal modal logic [10], logics of strict conditionals [11], two-dimensional modal logic [36], and Propositional Dynamic Logic [28]. These extensions typically generalize the standard possible world semantics to one based on information states. Other works introduce essentially new models and modalities, which do not have any counterpart in standard possible worlds semantics, e.g. inquisitive epistemic logic [16]. However, all of the mentioned approaches focus on universal modal operators. An equally important role in modal logic is played by existential modal operators. Because of the way negation works in the inquisitive logic setting, universal and existential modal operators are not interdefinable in the usual way.
In this paper, I consider inquisitive extensions of normal modal logic with an existential modality as primitive modal operator. The existential modality, which I denote by , is generalized to apply not only to statements, as in standard modal logic, but also to questions. This extension was first suggested by Ciardelli [10, p. 248], and a restricted version is considered in [27]; however, it has not yet been systematically explored in full generality. I study the resulting logic both from a semantic and from a proof-theoretic point of view. The main technical results are sound and strongly complete axiomatizations, both for the class of all Kripke frames, and for any class of frames corresponding to a canonical normal modal logic.
In propositional inquisitive semantics, every formula is associated with a set of alternatives, which are defined as the least informative information states supporting the formula. Typically, formulas formed using inquisitive disjunction have multiple alternatives, roughly corresponding to the classical propositions expressed by the disjuncts. A formula of the form expresses that each alternative for is true in some accessible world. When applied to a statement, behaves just like an ordinary existential modal operator in standard modal logic. When applied to a question, on the other hand, the result is a statement roughly expressing that each way of resolving the question is possible, or on an epistemic reading, that each way of resolving the question is consistent with the available information. For example, the formula expresses the polar (yes/no) question whether . Provided is contingent in the relevant model, the formula expresses that both and are consistent with the available information.
The generalized existential modality has interesting applications in epistemic logic and in deontic logic. In an epistemic setting, can be used to formulate a notion of question-directed ignorance, according to which ignorance with respect to a question means that the available information is too weak to even partially resolve the question. This question-directed notion of ignorance can be seen as a generalization of the concept of being ignorant about the truth-value of a statement [34,35]. On a deontic reading, can be used to express a form of permission. Accounting for free-choice inferences is a longstanding puzzle in deontic logic and in the theory of deontic modals (see e.g. [1,3,5,21,23,32]). Intuitively, from the sentence 'Jane may take an apple or a pear' one can infer that Jane may take an apple and that Jane may take a pear; in other words, Jane is free to choose which fruit to take. In general, the free-choice principle states that from a permitted disjunction, one can infer the permission of each disjunct. The logic of the generalized existential modality validates a conditional version of the free-choice principle. From one can infer and , provided that no alternative for is strictly included in an alternative for , and vice versa. This conditional version of the free-choice principle turns out to be key to understanding the logic of the generalized existential modality. Using an additional modal operator, the conditional free-choice principle can be expressed as a validity, which plays a key role in the axiomatization of the logic.
The paper is structured as follows. Section 2 contains some preliminaries on inquisitive propositional logic and the modal extension from [10]. This section also contains some additional results that will be used later in the paper. Then, in Section 3, I introduce the inquisitive extension of normal modal logic with a generalized existential modality as primitive modal operator, as well as an extension with an additional modal operator. Section 4 discusses applications to epistemic and deontic logic. In Section 5, I consider some expressivity results for the logics. In Section 6 I introduce an axiom system and prove soundness and strong completeness. Section 7 concludes the paper.

Preliminaries
In this section, I briefly present the inquisitive propositional logic InqB and the inquisitive extension of normal modal logic InqBK , and state some results that will be used in this paper. For detailed expositions, see [10,12,14,15].

Inquisitive Propositional Logic
Let Prop be a countable set of atomic formulas. The language InqB of inquisitive propositional logic InqB is defined by the following grammar, where ranges over Prop: . is the constant for contradiction and the familiar connectives (implication) and (conjunction) have their usual intended readings. Negation, classical disjunction and equivalence are defined in the standard way: , and . The connective is the inquisitive disjunction, which is used to formulate questions. For example, the formula expresses the question whether or ; the formula expresses the polar question whether ; the formula expresses the conditional question whether , if , and so on.
An information model is a pair consisting of a non-empty set of possible worlds and a valuation function assigning a subset of to every atomic formula in Prop. Given an information model , a set is called an information state, or simply a state. Support of a formula at a state in a model , written , is recursively defined as follows [10, pp. 47, 50]: iff iff iff for all implies iff and iff or .
The support set of a formula in a model is defined as the set . The following proposition states two fundamental properties of the support relation (see Proposition 2.2.3. in [10, p. 50]).

Proposition 2.1 For all models , all states
in , and all formulas : 1. If and then (Persistency of support); 2.
Persistency of support is motivated by the idea that if some information state supports a sentence, then any information state that provides at least as much information -i.e. any state -also supports the sentence. Taken together, persistency of support and the empty state property makes the support set of any formula in any model closed under subsets. In general, a set of information states closed under subsets is called an inquisitive proposition; sometimes, I will also refer to the support set of a formula as the inquisitive proposition expressed by .
Truth of a formula relative to a world in a model can be recovered from the support conditions by defining truth at (written ) as support at . The truth conditions defined in this way boil down to the standard truth conditions for classical logic, with identified with classical disjunction. The truth set of a formula in a model is defined as the set . With the notion of truth in place, the following support conditions for the defined operators and can be derived: iff for all iff for all or .
Persistency guarantees that if a formula is supported at a state , then it is true in all worlds . When the converse holds as well, the formula is said to be truth-conditional. That is, a formula is truth-conditional if for all models and all states of , if and only if for all . In general, truth-conditional formulas are also called statements and formulas that are not truthconditional are called questions.
Given the support conditions, it is easily verified that the only way to form a question is by using inquisitive disjunction. Say that a formula is classical if it does not contain any occurrences of inquisitive disjunction .

Proposition 2.2 All classical formulas are truth-conditional.
While all classical formulas are truth-conditional, not all truth-conditional formulas are classical. For example, any negated formula is truth-conditional, but need not be classical.
Each formula is associated with a set of classical formulas called the resolutions of [10, p. 56].

Normal Modal Inquisitive Logic
Ciardelli [10, Chapter 6] defines the modal inquisitive logic InqBK , which is an inquisitive extension of the minimal normal modal logic K. The language is the extension of the language InqB with formulas of the form . On an epistemic reading, is used to talk about the knowledge of an implicit agent. If is a statement, then expresses that the implicit agent knows that is true. If is a question, on the other hand, expresses that the implicit agent's information state settles . For example, expresses that the agent's information state settles whether . The language is interpreted on Kripke models , where is a non-empty set of possible worlds, is an accessibility relation on and is a valuation function for atomic formulas. Note that Kripke models can be seen as information models enriched with an accessibility relation. The support conditions for the connectives of InqB are defined as before. Where is a Kripke model, a state of and is the set of worlds accessible from the world , the support conditions for modal formulas are defined as follows: iff for all .
It is easily verified that formulas of the form are truth-conditional. If is a truth-conditional formula, then has completely standard truth conditions: iff .
Define the classical fragment of , denoted , as the set of -free formulas. It is easily verified that all classical formulas are truth-conditional. Note also that is effectively the language of standard modal logic, which can be given a straightforward Kripke semantics. Truth of a formula of in a world in a model , written , is defined as usual (in particular, iff for all ). Note that the symbol ' ' is used for the support relation in the semantics based on information states, whereas the symbol ' ' is used for the truth relation in standard Kripke semantics. The following lemma shows that for classical formulas, the support-based semantics and standard Kripke semantics coincide. As in InqB, every formula in can be associated with a set of resolutions (see [10, pp. 206-209] As in InqB, the resolutions of formulas of are classical formulas. The resolutions of a formula are used to express its normal form (see Proposition 6.3.13. in [10, p. 209]). 2 The definition of resolutions for and the subsequent proofs involving the notion rely on a complexity ordering of formulas of on which the resolutions of a formula are less complex than the modal formula . For details, see [10, p. 208]. See also Section 5.2. 3 To guarantee uniqueness, I assume that there is a fixed enumeration of the formulas of , and that the disjuncts in 1 occur in accordance with that enumeration.

Proposition 2.8 (Normal form for
) For any formula , is equivalent to .

Alternatives
In propositional inquisitive semantics, sentences are associated with sets of alternatives. Intuitively, the alternatives of a sentence are those information states that contain just enough information to support the sentence. Formally, if is a set of information states, then ALT , called the alternative reduct of , is defined as the set of information states in that are maximal with respect to the subset relation: ALT for all if then .
The set ALT of alternatives for a formula in a model is defined as the alternative reduct of the support set of , i.e. ALT ALT . If is truthconditional, then the truth set of is the only alternative for , i.e. ALT . If is a question, on the other hand, then is associated with multiple alternatives. Figure 1 illustrates the alternatives for some formulas.
In the case of InqB and InqBK , the inquisitive proposition expressed by a formula is completely characterized by the alternatives for , in the sense that is supported by if and only if for some alternative for . This property does not hold in general; for example, it fails for certain formulas of first-order inquisitive logic [10, p. 106-107]. If the inquisitive proposition is normal, then is represented, or generated, by its set of alternatives ALT . In general, a set of information states is a generator for if the following property holds [12, p. 328]: for some .
A generator for is said to be minimal if there is no such that is a generator for , and is said to be finite if it contains finitely many information states.

A Generalized Existential Modality
In InqBK , the universal modality is taken as the sole primitive modal operator. An existential modality can be defined in the standard way as . Whereas the support conditions for formulas of the form depend on the support conditions for , the support conditions for only depend on the truth conditions for : iff for all .
The reason for this is that negated formulas are truth-conditional, and the only alternative for a truth-conditional formula is the truth set of that formula. Thus, for example, is equivalent to , which merely states that there is a world accessible from the world of evaluation. In general, if is a question then merely states that the presupposition of is true in some accessible world [10, p. 248]. Hence, the defined existential modality in InqBK is not very interesting from an inquisitive point of view.
In this section, I consider the basic modal inquisitive logic InqBK featuring a generalized existential modal operator . This modal operator was first suggested by Ciardelli [10, pp. 247-249]. 5 InqBK can be seen as an inquisitive extension of the minimal normal modal logic K with an existential modality as the sole primitive modal operator. A formula of the form is supported at a state if and only if each alternative for is consistent with the information state , for each . If is truth-conditional, then the truth conditions for coincide with the standard truth conditions for existential modal operators. If is a question, on the other hand, expresses that all ways of resolving are consistent with the available information. For example, is true at if and only if all alternatives for are consistent with . Assuming that the model contains both worlds where is true and worlds where is false, this amounts to and .
A key feature of InqBK is the following conditional free-choice principle: From one can infer and , provided that no alternative for is strictly included in an alternative for , and vice versa.
I also consider the logic InqBK , which is the extension of InqBK with the binary modal operator . Formulas of the form express that no alternative for is strictly included in an alternative for , and vice versa. The operator makes it possible to express the conditional free-choice principle as a validity.

Language and Semantics
In this section, I present the language and semantics of the logic InqBK . . The intended reading of the connectives , and are as before, and the abbreviations , and introduced previously are used. On an epistemic interpretation, if is a question, expresses that the information available to an implicit agent is consistent with any of the answers to . On a deontic interpretation, expresses a notion of permission.
Formulas of the language are interpreted in terms of support relative to an information state in a Kripke model. The support clauses for atomic formulas, , and the connectives , and are defined as before. Support conditions for modal formulas are defined as follows, where ALT is defined as in Section 2.3, i.e. as the alternative reduct of the support set of : iff for all for all ALT .
It is easily verified that formulas of the form are truth-conditional. It is also easily verified that the counterparts to the properties stated in Proposition 2.1, i.e. persistency of support and the empty state property, hold for the extended semantics. This guarantees that the support sets of formulas of are indeed inquisitive propositions. Formulating the support conditions for in terms of the alternatives for assumes that the inquisitive proposition expressed by is normal in the sense defined in Section 2.3. This is indeed the case for all formulas of , which I will show below. If expressed a non-normal inquisitive proposition, would be 'vacuously' supported at any state in the model since the set of alternatives for would be empty. 6 6 One may want to formulate the support conditions for in a more general way to allow for cases where normality fails. Here is one suggestion, where no reference to the alternatives for is made (cf. [

Definition 3.2 (Declarative fragment
) The declarative fragment of is defined by the following grammar, where ranges over Prop, and ranges over : . I refer to formulas of the declarative fragment as declarative formulas, or sometimes simply as declaratives. Declarative formulas can be equivalently characterized as those formulas where can only occur within the scope of a modal operator. Note that any classical formula is also a declarative formula. I use symbols for declarative formulas and symbols for arbitrary formulas. Since atomic formulas, and modal formulas are truth-conditional, and the operations and preserve truth-conditionality, the following result holds.

Proposition 3.3 Any declarative formula is truth-conditional.
As before, each formula of is associated with a set of resolutions. . Any resolution is a declarative formula. In contrast to the resolutions of formulas of InqB and , resolutions of formulas of need not be classical. As will be shown in Section 5, there are truth-conditional formulas of that are not equivalent to any classical formula. As before, resolutions are used to express normal forms of formulas (cf. Proposition 6.3.5. in [10, p. 207]).

Proposition 3.5 (Normal form for ) For any formula , is equivalent to .
The next result shows that any formula of expresses a normal inquisitive proposition. This implies that formulas of the form are never 'vacuously' true, This clause coincides with the support clause formulated in terms of the alternatives for whenever expresses a normal inquisitive proposition. since all formulas have non-empty sets of alternatives in all models. The proof is analogous to the proof of Proposition 2.11.

Proposition 3.6 (Normality for ) Let be a model and a state of . Then for any formula , if and only if there is
ALT such that .

Some Properties
When is a truth-conditional formula, the truth set is the only alternative for . Hence, when is truth-conditional, the truth conditions for coincide with the standard truth conditions for the existential modality: iff .
The semantics for the operator can be seen as a conservative generalization of the semantics for the existential modal operator in standard normal modal logic. This is stated in more formal terms in the lemma below. Let be the classical fragment of , consisting of all -free formulas. Let mean that is true in world in model according to standard Kripke semantics (in particular, iff there is such that ).

Lemma 3.7 For any classical formula , any model and any state of , if and only if for all .
Define . Since negated formulas are truth-conditional, formulas of the form behave classically: iff .
The truth conditions for crucially depend on the support conditions for , whereas the truth conditions for only depend on the truth conditions, and not on the support conditions, for . Thus, while is equivalent to by definition, is not equivalent to when is a question. Things become more interesting when the argument of is a question associated with several alternatives. For example, while is true in a world if is either true or false in some world in , i.e. if is non-empty, is true in if contains a world where is true and a world where is false (provided that is contingent in the model). Figure 2 illustrates how alternatives and accessibility relations can interact.
In order to get a first understanding of the logical behavior of when it is applied to questions, one may begin by noting the following validity, which is easily established using the lemma below.  implies .
This conditional crucially depends on and being logically independent in the model. To see this, suppose that entails but does not entail in the model , i.e. . In this case, the alternative set of contains , but not . Now, suppose that is not empty whereas is. Then is true in , whereas is false in . In the general case of two arbitrary formulas and , it may happen that some alternative for is properly contained in some alternative for , or vice versa. It cannot in general be assumed that ALT ALT ALT , so quantifying over the union of the alternatives for and the alternatives for cannot always be reduced to quantifying over the alternatives for . Say that two information states and are non-subordinate if neither of them is a proper subset of the other, i.e. if and . 7 Intuitively, if two information states are non-subordinate then neither of them provides more information than the other. This relation is readily lifted to sets of information states. Here are some basic properties of the non-subordination relation, all of which have straightforward proofs which I omit here.

Lemma 3.10 For any sets of information states
and : A key observation about the non-subordination relation is the following (cf. Proposition 3.2 in [27, p. 349]).  Using the above fact, the conditional free-choice principle stated below can be verified. In general, the free-choice principle says that from a disjunction embedded under a permission modal, one can infer that each disjunct is permitted; see e.g. [1,3,5,21,23,32,37]. If is taken to express permission, the conditional free-choice principle licenses such inferences when the sets of alternatives for the disjuncts are non-subordinate in the sense of Definition 3.9.

Proposition 3.12 (Conditional free-choice principle) Let be any model and any formulas. Then
Proof Let be a model. Suppose ALT # ALT . The formula is truth-conditional; hence, suppose for some in . Then for all ALT , . Since and are normal inquisitive propositions by Proposition 3.6, Proposition 3.11 implies that ALT ALT . Hence for all ALT , so . The proof that is analogous. It follows that , so since was chosen arbitrarily, .

Extension
In light Proposition 3.12, it is natural to augment the language with a binary modal operator capable of expressing the non-subordination relation.
The language is defined by the following grammar, where ranges over Prop: . Formulas of the form express that ALT and ALT are nonsubordinate in the sense of Definition 3.9. Formally, is interpreted by the following clause: iff ALT # ALT .
Since the support conditions for formulas of the form do not depend on the current state of evaluation at all, it is easily seen that such formulas are truthconditional. It can also be noted that persistency of support and the empty state property hold. Hence, all formulas of express inquisitive propositions. I use InqBK to refer to the logic featuring both the operator and the operator . The notions of validity in a model, validity, entailment and equivalence for InqBK are defined in ways analogous to before. I use InqBK to denote the entailment relation, omitting the subscript when it is clear from context.
Using the modality , the conditional free-choice principle can be stated as a validity: or equivalently .
The declarative fragment of is the fragment where only occurs within the scope of a modal operator. Definition 3.14 (Declarative fragment ) The declarative fragment of is defined by the following grammar, where ranges over Prop, and and range over : .

Proposition 3.15 Any declarative formula is truth-conditional.
As before, each formula of is associated with a set of resolutions, which can be used to define the normal form of the formula. The next result shows that the inquisitive propositions expressed by formulas of are normal, and the proof is analogous to that of Proposition 2.11. The result guarantees that the support clauses for formulas of the form , which are defined in terms of alternatives, work as intended.

Model-Filtered Resolutions
Any alternative ALT corresponds to a resolution in the sense that ALT . However, it is not necessarily the case that any resolution corresponds to an alternative for . For instance, is a resolution of , but if the model contains some world where is true, is not an alternative for . As noted by Ciardelli [10, p. 56, n. 8], one can easily obtain a perfect correspondence between the set of truth sets of the resolutions of and the set of alternatives for by filtering out any resolution of whose truth set is a proper subset of the truth set of another resolution of . Here is some intuition behind this definition. Fix a model and define the relation on the set of declarative formulas by setting if and only if . Clearly, is a preorder (i.e. a transitive and reflexive relation) on . Since anti-symmetry may fail, is not necessarily a partial order. Now, is the set of -maximal elements of . Since is finite, is non-empty.
The following result shows that the truth sets of the -filtered resolutions of indeed correspond one-to-one with the alternatives for relative to .

The Global Modality
Define . The only alternative for in a model is the whole domain of , i.e. for any , ALT . Define u .
The following support conditions can then be derived.  [20]. 8 If is truth-conditional, the support conditions for u coincide with the standard semantics for the global modality: Define u u . The truth conditions for u only depend on the truth conditions, and not on the support conditions, for : On the other hand, the truth conditions for u crucially depend on the support conditions for . This means that the equivalence between u and u holds by definition, whereas u and u are not equivalent when is a question. The following lemma highlights the logical behavior of u . The first two items show that u is a special case of the box modality in InqBK (see [10, pp. 205, 214]); the remaining three items make u a type of S5 modality. 9 Lemma 3.23 (Properties of u ) All formulas of the following forms are valid in InqBK : A useful feature of the global modality is that it makes it possible to express identity of inquisitive propositions at the level of the object language. Say that a formula of the form u is a global equivalence between and . Basically, a global equivalence between and expresses that and are supported at exactly the same states.  By the above lemma, the property of replacement of global equivalents is easily verified.

Proposition 3.27 (Replacement of global equivalents) Let
, and let be obtained from by replacing one or more occurrences of in by . Then u .
As a final remark of this section, I want to highlight that the inclusion of the global modality leads to failure of the disjunction property, which is otherwise normally enjoyed by inquisitive logics. This can be shown by a simple counterexample: u u , but u and u . The disjunction u u is supported if either is supported by the maximal information state (since is truth-conditional, this amounts to being true in all possible worlds), or is false in at least one possible world. One of the two cases holds in any model. Hence, the disjunction property fails because of the global nature of u . 10

Applications
In this section, I consider applications to reasoning about ignorance in epistemic logic, and to reasoning about choice-offering permissions and obligations in deontic logic.

Epistemic Logic and Ignorance
The epistemic notion of ignorance is usually assumed to concern statements. In the epistemic logic literature, focus is typically on the notion of being ignorant about the truth value of a statement (see e.g. [34,35]; see also [18] on the related notion of knowing whether). Being ignorant about the truth value of a statement is typically analyzed as meaning that one's information is compatible both with and with . However, ignorance can also be seen as a more general question-directed attitude, in the sense that one can be ignorant of (the answers to) a question [25,29]. For example, one can be ignorant of who the current president of the United States is, or of where to buy tickets to the metro in Paris. Question-directed ignorance is arguably interesting in its own right, but the notion is also involved in the semantics of certain predicates that take interrogative clauses as their complement, such as wonder or investigate [30].
A general question-directed notion of ignorance can be defined by combining the operator from Ciardelli's modal inquisitive logic [10, Chapter 6] (cf. Section 2.2) with the generalized existential modality in the following way: . Taking and to refer to the same accessibility relation, the following semantic clause can be derived from the support conditions for and : iff for all for all ALT and .
Given that the underlying accessibility relation represents the epistemic relation of an implicit agent, expresses that the agent's information state does not resolve , and that each alternative for is compatible with the agent's information state. Being ignorant about the truth value of a statement can now be expressed as ignorance directed at the polar question whether , i.e.
. In general, if is a question, means that the agent considers to be completely open, in the sense that her available information cannot even partially resolve . In other words, says that the agent is totally ignorant about the answer to the question . For an example of total ignorance, consider the formula The notion of total ignorance should be distinguished from the notion of not knowing the answer: not knowing the answer to a question is compatible with being able to partially resolve the question. For example, not knowing the answer to the question of what day of the week it is, is compatible with knowing that it is Saturday or Sunday.
The account sketched here is in line with recent work on the semantics of wonder, where total ignorance is an important component [30]. For example, the sentence 'Jane wonders whether Alice, Bob or Carol came to the party' implies that Jane is ignorant as to whether Alice came, as to whether Bob came, and as to whether Carol came; in other words, Jane is totally ignorant of the embedded question [30, p. 1008].

Deontic Logic and Free-Choice Inferences
A longstanding problem in deontic logic and natural language semantics is to account for choice-offering permissions and obligations (see e.g. [1,5,23,31,32,37]; see also [21] for a comprehensive overview). Choice-offering permissions and obligations are typically expressed by a disjunctive clause under the scope of a deontic operator. Consider the following sentences: (1) a. Jane may buy the green car or the blue car.
b. Jane may buy the green car and Jane may buy the blue car.
An intuitive interpretation of (1a) is that it conveys a choice between two individually permitted options: to buy the green car, and to buy the blue car. Hence, one would expect (1b) to be entailed by (1a). However, validating inferences of this type without running into undesirable consequences has proved to be difficult [21]. For another example, consider the following two sentences: (2) a. Jane is obliged to post the letter. b. Jane is obliged to post the letter or burn the letter.
An intuitive reading of (2b) is that it offers a choice between two options: Jane can fulfill her obligation either by posting the letter, or by burning it. Since (2a) does not offer Jane the choice to burn the letter, (2a) should not entail (2b). However, many standard accounts take obligation to be upwards monotonic, in which case the undesirable inference becomes valid. This problem is known as Ross's paradox [31]. So far in this paper, I have assumed an informal interpretation of InqBK and InqBK where inquisitive disjunction is used to form alternative questions, and formulas in general are interpreted as sets of information states. However, there are also other informal interpretations available. In order to formalize the kind of free-choice reasoning considered above, it is more natural to adopt an action-oriented interpretation. The alternatives for a sentence can be construed as playing the role of the options available to an agent in a particular choice situation. A choice-offering permission can then be roughly analyzed as a permission to freely resolve a deliberative question, i.e. an issue of what to do. For example, (1a) can be analyzed as saying that Jane is permitted to freely resolve the issue of whether to buy a green car or a blue car. 11 There are several possible ways to spell out an interpretation of inquisitive semantics in terms of actions and choices in more detail; see e.g. [13,27] and [10, pp. 302-304]. Here, I will focus on an interpretation where non-modal formulas express properties of actions. In particular, a non-modal formula in the form of an inquisitive disjunction expresses a choice between different alternative actions. For example, if expresses the action of buying a green car and expresses the action of buying a blue car, then expresses the choice between buying a green car and buying a blue car. On the level of semantics, the domain of a model consists of action tokens, i.e. particular actions performed by a specific agent at a specific time. Sets of action tokens represent generic action types. The set of alternatives for a formula is taken to represent the action types available in the choice situation described by the formula. For example, the formula roughly describes the choice between the action type denoted by and the action type denoted by . The accessibility relation represents the legal status of action tokens, in the sense that means that the action token is legal given the realization of the action token . 12 If is a nonmodal formula, then -with the intended reading that is permitted -expresses 11 A strong reason for deviating from the standard interpretation of inquisitive semantics in terms of questions and information states in the context of deontic logic is that it is difficult to make intuitive sense of deontic operators operating on questions in general. For example, what does it mean to say that the question of whether it rains or not is permitted or obligatory? Perhaps it means that asking or answering the question is permitted or obligatory, but then the deontic operators operate on an action (i.e. the action of asking, respectively answering, the question), rather than a question. 12 It is natural to assume that the set of legal action tokens is constant in the model, in the sense that for all , . This assumption implies that what is legal to do is independent of what is actually being done. On the other hand, Anglberger et al. [4, p. 815, n. 10] argue that although this property is natural, it may fail e.g. in cases of what in decision theory is called 'state-act dependence' or 'moral hazard'. that there are legal action tokens instantiating any alternative action type in the choice described by .
As already seen, the following conditional free-choice principle holds: . By taking to express the action of buying a green car and to express the action of buying a blue car, (1a) above by can be formalized as while (1b) can be formalized as . Given this formalization, and assuming that the action types denoted by and are non-subordinate in the sense that neither is properly contained in the other, the inference from (1a) to (1b) above is predicted to be valid. Given the action-theoretic interpretation of the semantics, the intuitive reason for the validity of this inference is that is true only if there are legal action tokens instantiating both the action type of buying a green car and the action type of buying a blue car; in other words, both alternatives must constitute permitted courses of action.
The present account also has resources to avoid Ross's paradox. By defining as , with the intended reading that is obligatory, the following truth conditions can be derived: 13 iff ALT and for all ALT .
Thus, is true if and only if any legal action token instantiates an alternative for , and each alternative for is instantiated by some legal action token. To see how this notion of obligation avoids Ross's paradox, let express the action of posting the letter, let express the action of burning the letter, and consider the model where , , and . In this model, the only legal action token instantiates the action type of posting the letter, but not the action type of burning the letter. Hence, holds, but since not all alternatives for are instantiated by legal action tokens, it holds that . The account of free-choice inferences presented here is similar to that offered by alternative semantics (see e.g. [1][2][3]32]). In alternative semantics, every sentence denotes a set of classical propositions (sets of possible worlds), and deontic modalities are construed as operating on the alternatives of their argument clauses. A main difference between alternative semantics and inquisitive semantics is the treatment of disjunctions where one disjunct entails another, either logically or in a specific model [17].
In alternative semantics, disjunction is seen as a set-formation operator introducing a set containing all and only the classical propositions associated with each disjunct. On this account, the meaning of a disjunction ' or ' is generally different from the meaning of , even in cases where entails . In inquisitive semantics, 13 These truth conditions for obligation sentences agree with those suggested by Simons [32]. In fact, the truth conditions for formulas of the form and can be equivalently stated in terms of Simons's notion of a supercover. A set of sets is a supercover of a set if (i) and (ii) every element of has a non-empty intersection with [32, p. 276]. It is easily verified that iff there is a nonempty such that ALT is a supercover of ; and iff ALT is a supercover of . See also [26], where a deontic logic based on Simons's supercover semantics is developed. on the other hand, only maximal supporting states contribute to the meaning of a sentence. Hence, given that or is interpreted using , the meaning of the disjunction ' or ' coincides with the meaning of when entails . This feature of inquisitive semantics makes it possible to account for Hurford disjunctions. These are disjunctive sentences where one disjunct entails another, like 'John is an American or a Californian.' [22, p. 410]. Such disjunctions are generally taken to be infelicitous [22]. The standard explanation of this infelicity appeals to a more general ban on redundant operations [17,24,33]. In inquisitive semantics, this explanation is straightforward. However, according to the alternative semantics framework, no redundancy is involved in sentences like 'John is an American or a Californian', so the explanation from redundancy fails [17]. Recent research suggests that Hurford's constraint is operative in free-choice inferences as well [8,13,27]. This, I think, gives the present inquisitive semantics account an advantage over alternative semantics when it comes to accounting for free-choice inferences.
There are additional puzzles associated with free-choice reasoning. For example, Booth [8] argues that choice-offering permissions and obligations convey that each alternative is independently permitted, so that e.g. the sentence 'Jane may take an apple or a pear' implies that Jane is permitted to take an apple without taking a pear, and that Jane is permitted to take a pear without taking an apple. Another puzzle concerns the intuition that free-choice effects disappear under negation. Intuitively, 'Jane may not take an apple or a pear' implies that Jane is neither permitted to take an apple, nor permitted to take a pear (see e.g. [1,3]). Although I have not discussed these puzzles here, I believe that the present approach is a promising starting point for an analysis of free-choice reasoning.

Some Notes on Expressivity
In this section, I consider the expressivity of the languages and . In particular, I investigate the relation between the truth-conditional and classical fragments of both languages. I show that there are truth-conditional formulas of that are not equivalent to any classical formulas. As a consequence, can express statements that are neither expressible in the standard modal language, nor in the modal inquisitive language . Then, I show that the additional expressive power of the language restores the connection between truth-conditionality and classicality.

Truth-Conditionality and Classicality in L
In the languages InqB and , classicality and truth-conditionality coincide in the sense that any truth-conditional formula is equivalent to a classical one. This is not the case for the language . The reason for this is that certain formulas of can express global properties of models. For example, the truth of the formula in a world in a model crucially depends on the interpretation of atomic formulas at all worlds in the model , and not only on the interpretation of atomic formulas at worlds reachable from .
Consider the two models depicted in Fig. 3. The truth-conditional formula is true in 1 in model 1 , since the only alternative for is 1 . On the other hand, is false in 2 in model 2 , since there are two alternatives, 2 and 2 , but only one of them is consistent with the available information in 2 . However, the pointed models 1 1 and 2 2 are bisimilar in the standard modal language, so in light of Lemma 3.7 they are indistinguishable by any classical formula in . Hence, is equivalent to 1 , which is equivalent to 1 , which in turn is a formula of .
In fact, is more expressive than .

Proposition 5.3
There is no formula such that is equivalent to .
Proof Suppose towards contradiction that there is a formula such that is equivalent to . Then must be truth-conditional. By Proposition 2.6, there is a classical formula such that is equivalent to . Let be the formula Fig. 3 Two models that are bisimilar in the standard modal language, with the alternatives for highlighted obtained by substituting for any occurrences of in . Then , and and are equivalent by the same reasoning as used in the proof of Proposition 5.2. It follows that and are equivalent, so and are equivalent as well. However, this is impossible by Proposition 5.1.

Truth-Conditionality and Classicality in L
While truth-conditionality and classicality come apart in , the additional expressive power of restores the connection. In this section, I show that any declarative formula in is equivalent to a classical formula. In fact, I will prove a slightly stronger result: any declarative in is equivalent to a classical formula where all occurrences of are in the form of u . I call such formulas global-classical formulas, or g-classical formulas for short. .
As before, u is an abbreviation of . The g-classical fragment can be seen as the language of standard modal logic extended with the global modality, which can be interpreted using standard Kripke semantics [6,20]. Again, I write to mean that is true at in according to Kripke semantics; in particular, u iff for all worlds of .

Lemma 5.5 For any g-classical formula g-cl , any model and any state of , if and only if for all .
In the rest of this section, I will show that there is an equivalence-preserving mapping that maps declarative formulas to g-classical formulas.
For the sake of readability, I use the following abbreviation, where and are declarative formulas in : The meaning of is clarified in the following lemma. The above lemma provides a way to single out resolutions that denote alternatives, which in turn suggests a natural way to explicitly express the quantification over alternatives in the support conditions for modal formulas.
Again to increase readability, I use the following abbreviations, where are declarative formulas and are any formulas: 1 2 .
( 2 ) Proof First, I prove (1 Next, I will define an equivalence-preserving mapping t such that if is a declarative formula, then t is g-classical. The recursive definition of t and the subsequent inductive proofs rely on a complexity ordering of where the resolutions of the argument(s) of a modal formula are less complex than the modal formula itself (cf. [10, p. 208]). The modal depth of a formula is defined in the standard way, i.e. as the maximum number of nested occurrences of the modalities and in . Let . Then is less complex than , written , if either , or is a proper subformula of . It holds that is irreflexive and transitive. The ordering of formulas according to modal depth as well as the proper subformula relation are well-founded relations. It follows that is a well-founded strict partial ordering on (see e.g. [7]). Hence, is a suitable complexity ordering for inductive definitions and proofs. Now, the mapping t is specified as follows. t if Prop or , and t commutes with , and . The cases for formulas of the form and are given by the following clauses: The following lemma follows from the definition of t and Lemma 5.7.

Lemma 5.8 For any formula
, is equivalent to t .

Lemma 5.9
If is a declarative formula in , then t is a g-classical formula Proof The proof is by induction over the complexity of in terms of . The only interesting cases are the ones for and . In both cases, by the induction hypothesis it holds that t is a formula constructed from g-classical subformulas using the operations , , and u . Hence, t is g-classical.
The following proposition follows directly from Lemmas 5.8 and 5.9.

Proposition 5.10
Any declarative formula in is equivalent to a g-classical formula in g-cl .
By the above considerations, and the fact that and are equally expressive, adding the operator to results in a language with the same expressive power as . It can also be noted that the mapping t maps any formula to an equivalent formula where all occurrences of are in the form of u . Hence, can be expressed in terms of the global modality. However, the mapping t does not constitute a definition of in terms of u , since the translation is not uniform in the relevant sense (see e.g. [19, p. 492-493]).

Axiomatization
As seen previously, the truth conditions of certain formulas of depend on global features of models in a non-trivial way. For example, the truth-value of in a world does not only depend on properties of the set of worlds accessible from , but also on worlds that are not accessible from . The reason for this is that the truth conditions for depend on the set of alternatives for , and the set of alternatives, in turn, are defined globally in models. This feature makes it non-trivial to provide an axiomatization using only . Here, I focus on axiomatizing logics featuring also the modality .
Different entailment relations can be defined by restriction to models built over different frame classes. Let L be a normal modal logic. The modal inquisitive logic InqBL corresponding to L is obtained by restriction of the entailment relation to models built over L-frames, in the following sense: for , entails in InqBL (written

InqBL
) if and only if for all models built on an L-frame and all states of , if for all , then . The system InqBK , then, is the logic arising from the class of all Kripke frames.
I will first prove strong completeness of an axiomatization of InqBK , i.e. the logic of all Kripke frames. Then, I will show that if L is a canonical normal modal logic in the sense that its canonical frame is an L-frame (see [6, pp. 203-204]), adding the axioms of L to the axiomatization of InqBK results in a strongly complete axiomatization of InqBL . This is analogous to how adding the axioms of L to the axiomatization of Ciardelli's modal inquisitive logic InqBK results in a strongly complete axiomatization with respect to the class of L-frames [10, Chapter 6]. Finally, I show that the problem of deciding InqBL -validity can be reduced to deciding validity in L extended with the global modality.

Axiomatization of InqBK
The axiom system Ax InqBK consists of the following axioms and inference rules.
1. Axioms for inquisitive propositional logic InqB, where ranges over the declarative fragment : 14 2. Axioms for u : 3. Axioms for , where and range over the declarative fragment : 4. Axioms for :

Rules of inference:
(a) Modus ponens: from and infer (b) Necessitation for u : from infer u

Let
. I write InqBK (meaning is a theorem) if is derivable in the above system, and InqBK (meaning is derivable from ) if there are 1 such that 1 is derivable. 15 Say that and are provably equivalent, written InqBK , if is derivable. When there is no risk for confusion, I will drop the subscript and write and instead of InqBK and InqBK . Axioms (1a) to (1i) form a complete set of axioms for intuitionistic logic, with viewed as the primary disjunction connective [10,15]. Hence, all intuitionistic theorems are also theorems of Ax InqBK .

Proposition 6.1 (Deduction theorem) Let . Then if and only if .
Each formula is provably equivalent to its normal form . The proof is by induction on the structure of . Since modal formulas are their own resolutions, the proof is essentially the same as the one for InqB (see Lemma 3.3.4. in [10, p. 86]).

Proposition 6.2 For any formula
, .
Again, since modal formulas are their own resolutions, the following proposition can be proved in the same way as the corresponding result for InqB (see Lemma 3.3.7. in [10, pp. 87-88]).

Proposition 6.3 Let . If
, then there is a set of declarative formulas such that and contains a resolution for each .
Recall that global equivalences of the form u express identity of inquisitive propositions at the level of the object language. This feature is mirrored in the axiom system.

Proposition 6.4 (Replacement of global equivalents) Let
, and let be obtained from by replacing one or more occurrences of in by . Then u .

Proof
The following are theorems of intuitionistic logic, and hence also theorems of Ax InqBK : , for ; , for .

Declarative Theories and Theory-Filtered Resolutions
The completeness proof of InqBK is based on a canonical model construction, where the worlds in the canonical model are sets of declarative formulas that are consistent, complete, and closed under deduction of declaratives.

Definition 6.5 (Declarative theories) A set of declarative formulas
is a declarative InqBK -theory (or simply a declarative theory, if there is no risk for confusion) if for all , implies . A declarative InqBK -theory is complete if , and or for all .
The proof of the relevant version of Lindenbaum's lemma is standard and omitted here (see e.g. Lemma 4.17 in [6,pp. 197]).

Lemma 6.6
If is a consistent set of declarative formulas, then there is a complete declarative theory such that .
Using the notion of a complete declarative theory, the proof-theoretical counterpart to model-filtered resolutions can be defined. A key to the completeness proof is that the set of model-filtered resolutions for the canonical model coincides with the set of theory-filtered resolutions for any complete declarative theory figuring as a world in the canonical model. Then is a preorder (but not necessarily a partial order) on , and hence is the set of -maximal elements of . Such maximal elements exist since is finite. Since any element of a finite preordered set is less than or equal to a maximal element, the following results holds. for all and all . Then . By Axioms (3d) and (3b), it follows that . By Lemma 6.9 and replacement of global equivalents, .

Soundness and Completeness
Soundness of the axiom system is straightforward to establish. As usual, it must be shown that all axioms are valid and that the rules of inference preserve validity. The validity of the modal axioms has been discussed in Section 3.1 and in Lemmas 3.19 and 3.23. For the non-modal axioms I refer to [10,Chapter 3] and [11]. In the rest of this section, I will establish strong completeness for InqBK .

Theorem 6.15 (Completeness for InqBK ) For any set of formulas ,
The general proof strategy is an adaption of Ciardelli's completeness proof for InqBK [10,Chapter 6]. An important difference is that the proof by Ciardelli [10] relies on a general support lemma, which connects support at a state in the canonical model with derivability from the intersection of that state (Lemma 6.4.20. in [10, p. 220]). Such a general support lemma cannot be established in the present context since InqBK lacks the disjunction property, which means that the inductive case for inquisitive disjunction would not go through. Instead, the proof strategy I use here relies on a weaker truth lemma connecting truth of a declarative in a world in the canonical model with derivability of from .
Define the following relation on the set of complete declarative theories: 1 2 if and only if for all declarative formulas , if u 1 then 2 . The following result is easily verified using the S5 axioms for u , and I omit the proof here.

Lemma 6.16
The relation is an equivalence relation. The canonical frame for InqBK relative to is the pair .
Fix a complete declarative theory , and let be the canonical model for InqBK relative to .
The results in the following lemma are standard in modal logic (see e.g. Lemma 4.20 in [6, p. 198]); I include the proofs here for completeness. . Then .

Transfer of Completeness and Decidability
In this section, I show how to obtain axiomatizations for inquisitive extensions of canonical normal modal logics. This is done in a way analogous to Ciardelli's general recipe for axiomatizing extensions of InqBK obtained by restricting the semantics to particular frame classes [10,Chapter 6]. Let L be a normal modal logic in the language . Recall that InqBL is the logic characterized by the class of L-frames, and that InqBL -entailment is denoted InqBL . Let Ax InqBL be the axiom system obtained by adding the axiom schemas of L spread over g-cl to Ax InqBK . For example, the axiomatization of the logic over transitive frames is obtained by adding the schema , where ranges over g-cl , to Ax InqBK . I denote derivability in Ax InqBL by InqBL . I will show that if L is a canonical normal modal logic, in the sense that its canonical frame is an L-frame (see [6, p. 203-204]), then Ax InqBL is sound and strongly complete with respect to the class of L-frames. The key to the completeness result is to establish a connection between InqBL and the logic L extended with the global modality. This connection is also used to prove a general result about the decidability of InqBL -validity.
First, it can be noted that Ax InqBL is sound. The only new thing to check is the validity of the axiom schemas of L. This is straightforward, as InqBL is conservative over L. Recall the translation t defined in Section 5.2, which maps declarative formulas to their g-classical counterparts. Recall also Lemma 5.8 of the same section, stating that any declarative formula in is equivalent to t . The next result can be seen as the proof-theoretical counterpart to Lemma 5.8. Proof Let be a declarative formula. By Lemma 5.8, InqBK t , so by completeness of InqBK , InqBK t . Since Ax InqBL is a conservative extension of Ax InqBK , it follows that InqBL t .
Let L be a normal modal logic in the language . Let L be the minimal extension of L with the global modality, i.e. the smallest normal modal logic in the language g-cl containing all instances of the axiom schemas of L, all instances of the S5 axiom schemas for u , and all instances the axiom schema u ; for details, see [20, pp. 14-15] and [6,Chapter 7]. The notion of a complete L -theory is standard. Define the relation on the set of complete L -theories as follows: 1 2 if and only if for all g-cl , if u 1 then 2 . It is easily established that is an equivalence relation. Definition 6.25 (Canonical frame for L ) Let L be a normal modal logic, and let be a complete L -theory. The canonical frame for L relative to is the structure , where is the equivalence class of under , if and only if for all g-cl , if then .
The following result is proved in [20] (see also the proof of Theorem 7.3 in [6, p. 417-418]). Lemma 6.26 Let L be a canonical normal modal logic and let be a complete L -theory. Then the canonical frame for L relative to is an L-frame.
A declarative InqBL -theory is defined in a way analogous to Definition 6.5. Given a complete declarative InqBL -theory , the canonical frame for InqBL relative to is defined in a way analogous to Definition 6.17. Finally, I will utilize the connection between InqBL and L to prove a general result about the decidability of InqBL . Proof By Lemmas 6.29, 5.8 and 5.9, checking whether is InqBL -valid amounts to checking whether u t is L -valid. For the other direction, note that g-cl and that InqBL is conservative over L . Hence, checking whether g-cl is L -valid amounts to checking whether it is InqBL -valid.
By the above result, it follows that checking a formula for InqBLvalidity reduces to checking for validity in L . Thus, for example, it follows that InqBK , i.e. the logic of all Kripke frames, is decidable (see [6, p. 418]). This result is silent about the complexity of the decision problem. Indeed, the use of the translation t results in an exponential blowup of the size of formulas.

Conclusion
In this paper, I studied modal inquisitive logic interpreted on Kripke models, using a generalized existential modality as primitive modal operator. In particular, I considered the logic InqBK , obtained by adding inquisitive disjunction to normal modal logic K with the existential modality as primitive modal operator, and the logic InqBK obtained by extending InqBK with an additional modality . The main results of the paper are soundness and completeness results. I provided an axiom system for InqBK , and I proved its soundness and strong completeness with respect to the class of all Kripke frames. I also provided axiom systems for various extensions of InqBK obtained by restriction to specific frame classes. In particular, I proved that for any canonical normal modal logic L, the proposed axiom system for the inquisitive extension InqBL of L is sound and strongly complete with respect to the class of L-frames. Apart from these main results, I considered the decidability of InqBL , as well as the relationship between truth-conditionality and classicality in the languages and . I also discussed applications to deontic logic, and to reasoning about ignorance.
There are several topics for future research. One open problem is how to provide InqBK with a complete proof system without having the additional expressive power of available. Other topics concern expressivity: for instance, combining with makes it possible to express some global properties of models, but are and together expressive enough to completely capture the global modality? Another topic for future research is to extend other logics based on possible worlds semantics to the inquisitive semantics setting by quantifying over the alternatives for formulas. For example, Ciardelli [9] shows how to lift the semantics of a large class of conditional logics to the inquisitive semantics setting by quantifying over the alternatives of the antecedents and consequents of conditionals (see also [14,Chapter 7]). It would also be interesting to consider inquisitive extensions of non-normal modal logic based on neighborhood semantics. For example, a modal statement can be interpreted as expressing a relationship between the alternatives of the argument of the modal operator, and the neighborhoods of the world of evaluation. In particular, the operator seems like a very useful tool in this context.