Inferential erotetic logic meets inquisitive semantics
Abstract
Inferential erotetic logic (IEL) and inquisitive semantics (INQ) give accounts of questions and model various aspects of questioning. In this paper we concentrate upon connections between inquisitiveness, being the core concept of INQ, and question raising, characterized in IEL by means of the concepts of question evocation and erotetic implication. We consider the basic system InqB of INQ, remain at the propositional level and show, inter alia, that: (1) a disjunction of all the direct answers to an evoked question is always inquisitive; (2) a formula is inquisitive if, and only if it evokes a yes–no question whose affirmative answer expresses a possibility for the formula; (3) inquisitive formulas evoke questions whose direct answers express all the possibilities for the formulas, and (4) each question erotetically implies a question whose direct answers express the possibilities for the direct answers to the implying question.
Keywords
Logic of questions Inferential erotetic logic Inquisitive semantics1 Introduction
This paper focuses on interrelations between two notions: question raising and inquisitiveness. The former is in the centre of attention of inferential erotetic logic (IEL), while the latter is the core concept of inquisitive semantics (INQ).
1.1 Inquisitive semantics
Inquisitive semantics originated from an analysis of questions, but currently evolves toward a general theory of meaning. The beginnings of INQ date to the late 1990s. INQ, however, is currently a research programme rather than a completed theory: alternative accounts are still being proposed.^{1}
The basic idea of INQ can be briefly expressed as follows: the meaning of a sentence comprises two components, informative content and inquisitive content. The former is the information provided by a sentence, the latter is the issue raised by the sentence. By and large, if the information provided is sufficient to settle the issue that is raised, the sentence is an assertion. If, however, the information provided is insufficient to settle the raised issue, the sentence is inquisitive.
Both types of contents, as well as the concepts of possibility, assertion and inquisitiveness are modelled formally (see below).
In what follows we concentrate upon the “basic”, most often used system of inquisitive semantics, labelled as InqB, and we remain at the propositional level only.
1.2 Inferential erotetic logic
Generally speaking, IEL is a logic that analyses inferences in which questions play the role of conclusions and proposes criteria of validity for these inferences. The idea originates from the late 1980s.^{2}
Clearly, (I), (II) and (III) pass the test, in contradistinction to (IV) or (V). However, there are cases in which one does not get indisputable results. On the other hand, validity is a normative notion, and, as long as erotetic inferences are concerned, is given neither by God nor by Tradition. So some arbitrary decisions have to be made.
IEL proceeds as follows. First, some criteria of validity are proposed, separately for each kind of erotetic inference (for details see, e.g., Wiśniewski 2013, Chapter 5). Then two semantic relations are defined: evocation of questions by sets of declarative sentences/formulas, and erotetic implication of a question by a question together with a set of declarative sentences/formulas. Validity of erotetic inferences of the first kind is defined in terms of evocation; at the same time the definition of evocation provides an explication of the intuitive notion “a question \(Q\) arises from a set of declarative sentences \(X\)”. As for erotetic inferences of the second kind, their validity is defined in terms of erotetic implication (of the question being the conclusion by the question which is the premise on the basis of the declarative premise(s) involved). The proposed definition of erotetic implication is an explication of the intuitive notion “a question \(Q\) arises from a question \(Q_1\) on the basis of a set of declarative sentences \(X\)”.^{3} In view of IEL question raising (explicated in terms of evocation and erotetic implication) and validity of erotetic inferences are two sides of the same coin.
Both erotetic implication and question evocation are purely semantic concepts. In order to define them one needs, inter alia, a certain concept of entailment for declaratives. However, in its general setting IEL is neutral in the controversy concerning what “The Logic” of declaratives is. One can use Classical Logic, but nonclassical logics are also permitted. In this paper we will be dealing with the simplest case, that is, with evocation and erotetic implication based on Classical Propositional Logic.
1.3 IEL versus InqB
IEL and InqB are conceptually incommensurable. IEL distinguishes questions from declaratives syntactically, while InqB does not do this. InqB defines the basic semantic categories in terms of support, being a relation between a formula and a set of possible worlds/an information state. IEL, in turn, operates with the concept of truth as the basic one. However, despite conceptual differences, both theories give accounts of the phenomenon of question/issue raising, though the phenomenon is viewed by them from different angles. InqB explicates the intuitive notion of a possibility “offered” by a sentence, distinguishes sentences which offer at least two possibilities and, in effect, gives a semantic characteristics of expressions that call for supplementary information. However, InqB does not provide a systematic account of arriving at issues/questions. IEL, in turn, focuses on semantic relations between questions and the contexts of their appearance. So it seems that IEL and INQ can benefit from their meeting, both conceptually and in further developments. In this paper we present some formal results which may be viewed as arguments in favour of the above claim.
1.4 Outline of the paper
The perspective adopted in this paper is the following. As in IEL, and unlike InqB, we consider a (propositional) language which has two disjoint categories of wellformed expressions: declarative wellformed formulas and questions. The latter are defined purely syntactically. In other words, the language has a “declarative part” and an “erotetic part”. We supplement the declarative part (but not the erotetic part!) with an inquisitive semantics which is, basically, InqB. Since the standard concepts of truth and entailment are definable in InqB, this step enables us to operate with inquisitive as well as standard semantic concepts, in particular to define “classical” question evocation and erotetic implication.
In Sect. 2 we provide an exposition of the basics of InqB and we introduce the semantic concepts used throughout the paper. In Sect. 3 we describe how questions are conceptualized in InqB and in IEL, and we point out some affinities between the socalled proper questions and inquisitive formulas of a special form. Section 4 is devoted to question evocation and its connections with inquisitiveness. We show that a disjunction of all the direct answers to an evoked question is always inquisitive, and that the evoked questions are just the questions whose disjunctions of asserted direct answers are inquisitive. We prove that a formula is inquisitive if, and only if the formula evokes a yes–no question whose affirmative answer expresses a possibility for the formula. We also show that an inquisitive formula evokes a question whose direct answers express all the possibilities for the formula. In Sect. 5 we turn to erotetic implication, explain the underlying intuitions and present the definition. Then we prove that each question (of the language considered) erotetically implies a question whose direct answers express all the possibilities for the direct answers to the implying question.
2 InqB
2.1 Possible worlds, states, and support
As for InqB, we consider a (nonmodal) propositional language \(\mathcal {L}_{\mathcal {P}}\) over a nonempty set of propositional variables \(\mathcal {P}\), where \(\mathcal {P}\) is either finite or countably infinite. The primitive logical constants of the language are: \(\bot , \vee , \wedge , \rightarrow \). Wellformed formulas (wffs) of \(\mathcal {L}_{\mathcal {P}}\) are defined as usual.
The letters \(A, B, C, D\), with or without subscripts, are metalanguage variables for wffs of \(\mathcal {L}_{\mathcal {P}}\), and the letters \(X, Y\) are metalanguage variables for sets of wffs of the language. The letter \(\mathtt p \) is used below as a metalanguage variable for propositional variables.
\(\mathcal {L}_{\mathcal {P}}\) is supposed to be associated with the set of suitable possible worlds, \(\mathcal {W}_{\mathcal {P}}\), being the model of \(\mathcal {L}_{\mathcal {P}}\). A possible world, in turn, is conceived either as a subset of \(\mathcal {P}\) or as a valuation of \(\mathcal {P}\). Regardless of which of these solutions is adopted, \(\mathcal {W}_{\mathcal {P}}\) is uniquely determined. When possible worlds are conceptualized as sets of propositional variables, \(\mathcal {W}_{\mathcal {P}} = \wp (\mathcal {P})\), that is, \(\mathcal {W}_{\mathcal {P}}\) is the power set of \(\mathcal {P}\). If, however, possible worlds are identified with indices, that is, valuations of \(\mathcal {P}\), then \(\mathcal {W}_{\mathcal {P}}\) is the set of all indices.
Subsets of \(\mathcal {W}_{\mathcal {P}}\) are called states. States are thus sets of possible worlds. One can think of such sets as modelling information states. Singleton sets/states correspond to information states of maximal consistent information, while \(\mathcal {W}_{\mathcal {P}}\) corresponds to the ignorant state, i.e. an information state in which no possible world is excluded. \(\emptyset \) represents the absurd state.
The letters \(\sigma , \tau , \gamma \), with or without subscripts, will refer to states.
The most important semantic relation between states and wffs is that of support. In the case of InqB support, \(\vDash \), is defined by:
Definition 1
 1.
\(\sigma \vDash \mathtt p \) iff for each \(w \in \sigma \): \(\mathtt p \) is true in \(w\),
 2.
\(\sigma \vDash \bot \) iff \(\sigma = \emptyset \),
 3.
\(\sigma \vDash (A \wedge B)\) iff \(\sigma \vDash A\) and \(\sigma \vDash B\),
 4.
\(\sigma \vDash (A \vee B)\) iff \(\sigma \vDash A\) or \(\sigma \vDash B\),
 5.
\(\sigma \vDash (A \rightarrow B)\) iff for each \(\tau \subseteq \sigma \): if \(\tau \vDash A\) then \(\tau \vDash B\).
When \(\mathcal {W}_{\mathcal {P}}\) comprises indices, “\(\mathtt p \) is true in \(w\)” means “the value of \(\mathtt p \) under \(w\) equals \(\mathbf {1}\)”. When \(\mathcal {W}_{\mathcal {P}} = \wp (\mathcal {P})\), the meaning is: “\(\mathtt p \in w\)”.

(neg) \(\sigma \vDash \lnot A\) iff for each \(\tau \subseteq \sigma \) such that \(\tau \ne \emptyset : \tau \not \vDash A\).
Lemma 1
(Persistence) If \(\sigma \vDash A\), then \(\tau \vDash A\) for any \(\tau \subseteq \sigma \).
As a consequence we get:
Corollary 1
If \(\sigma \vDash A\), then \(\{w\} \vDash A\) for each \(w \in \sigma \).
However, the converse of Corollary 1 does not hold. As an illustration, consider a state \(\{w, v\}\) such that \(p\) is true in \(w\), \(q\) is false in \(w\), \(p\) is false in \(v\) and \(q\) is true in \(v\). The wff \(p \vee q\) is true both in \(w\) and in \(v\), but, due to clauses (1) and (4) of Definition 1, \(\{w, v\}\) does not support \(p \vee q\).
The truth set of a wff \(A\), in symbols: \(\vert A \vert \), is the set of all the worlds from \(\mathcal {W}_{\mathcal {P}}\) in which \(A\) is true, where the concept of truth is understood classically. More formally, \(\vert A\vert = \{w \in \mathcal {W}_{\mathcal {P}} : \{w\} \vDash A\}.\)
2.2 Meaning, inquisitiveness, and possibilities
In view of INQ the meaning of a sentence has two aspects: informative content and inquisitive content. Roughly, the former is identified with the information provided, while the latter is the issue raised.
The informative content of a wff \(A\), \(info(A)\), is defined as follows:
Definition 2
(Informative content) \(info(A) = \bigcup \{\sigma \subseteq \mathcal {W}_{\mathcal {P}}: \sigma \vDash A\}\)
The inquisitive content of a wff \(A, [A]\), is defined by:
Definition 3
(Inquisitive content) \([A] = \{\sigma \subseteq info(A): \sigma \vDash A\}\)
When one starts with the concept of proposition as the basic one (i.e. the proposition expressed by \(A\) construed as a nonempty downward closed set of subsets of \(\vert A \vert \)), the structure \(\langle \Gamma , \subseteq \rangle \), where \(\Gamma \) is the set of all propositions that belong to \(\wp (\mathcal {W}_{\mathcal {P}})\), is a Heyting algebra with a (pseudo)complementation operation \(^{\circ }\) characterized by: \(\mathcal {A}^{\circ } =_{df} \{\beta : \beta \cap \bigcup \mathcal {A} = \emptyset \)}, and relative pseudocomplementation, \(\Longrightarrow \), defined by: \(\mathcal {A} \Longrightarrow \mathcal {B} =_{df} \{\alpha : for\; every\;\beta \subseteq \alpha , if\; \beta \in \mathcal {A} then\;\beta \in \mathcal {B}\}\). This provides an algebraic justification for the clauses of the definition of support. For details see Roelofsen (2013).
A wff \(A\) is inquisitive when, and only when its informative content is not an element of its inquisitive content, that is, the (full) informative content of \(A\) does not support \(A\). To be more precise:
Definition 4
(Inquisitiveness) A wff \(A\) is inquisitive iff \(info(A) \not \vDash A\).
When we remain at the propositional level, \(\mathsf {InqB}\)inquisitive contents of wffs have maximal elements, called possibilities. To be more precise:
Definition 5
(Possibility) A possibility for wff \(A\) is a state \(\sigma \subseteq \mathcal {W}_{\mathcal {P}}\) such that \(\sigma \vDash A\) and for each \(w \not \in \sigma : \sigma \cup \{w\} \not \vDash A\).
If a wff is inquisitive, the set of possibilities for the wff comprises at least two elements.
It can be proven that each state supporting a wff \(A\) is a subset of some possibility for \(A\).^{7} Hence it follows that, in particular, each world that belongs to the truth set of a wff \(A\) belongs to some possibility for \(A\) (see also Lemma 5 below).
2.3 Entailment
Entailment is defined in INQ in terms of inclusion: \(A\) (inquisitively) entails \(B\) iff \(info(A) \subseteq info(B)\) and \([A] \subseteq [B]\). As for InqB, however, \([A] \subseteq [B]\) yields \(info(A) \subseteq info(B)\). Thus in the case of InqB inquisitive entailment, \(\models _{inq}\), can be defined by:
Definition 6
(Inquisitive entailment) \(X \models _{inq} A \; \text{ iff } \bigcap \limits _{B \in X} [B] \subseteq [A]\).
Note that the underlying idea is the transmission of support. Needless to say, classical entailment, \(\models _{cl}\), can be defined as follows:
Definition 7
(Classical entailment) \(X \models _{cl} A \; \text{ iff } \bigcap \limits _{B \in X} \vert B \vert \subseteq \vert A \vert \).
Clearly, the following holds:
Corollary 2
\(\models _{inq} \;\subset \;\models _{cl}\).
We also need the concept of multipleconclusion entailment,^{8} being a relation between sets of wffs. Roughly, \(X\) multipleconclusion entails (mcentails for short) \(Y\) if, and only if the truth of all the wffs in \(X\) warrants that at least one wff in \(Y\) is true. In the case of Classical Propositional Logic we express the idea by:
Definition 8
(Classical mcentailment) \(X \mid \models _{cl} Y \; \text{ iff } \bigcap \limits _{B \in X} \vert B \vert \; \subseteq \; \bigcup \limits _{A \in Y} \vert A \vert \).
3 Questions
Generally speaking, questions/interrogatives can be incorporated into a formal language in two ways.
1. (The “Define within” approach.) One can embed questions into a language. To be more precise, one can regard as questions some already given wellformed formulas differentiated by some semantic feature(s), or construe questions as meanings of some specific (but already given) wellformed formulas.
2. (The “Enrich with” approach.) One can enrich a language with questions/interrogatives. In order to achieve this, one adds to the vocabulary some questionforming expressions and then introduces questions/interrogatives syntactically, as a new category of wellformed formulas. The new category is disjoint with the remaining categories. This way of proceeding is natural when questions are conceived in accordance with the independent meaning thesis, according to which the meaning/semantic content of an interrogative sentence cannot be adequately characterized in terms of semantics of expressions that belong to other categories.
IEL proceeds in accordance with the “enrich with” approach, while InqB adopts the “define within” approach.^{10} However, InqB postulates no paraphrase of questions in terms of expressions of other categories; questions and assertions are distinguished semantically among expressions which are traditionally construed as declaratives (see below).
3.1 Questions in InqB
The language \(\mathcal {L}_{\mathcal {P}}\) does not include a separate syntactic category of questions/interrogatives. However, some wffs are regarded as having the property of being a question, or \(\mathcal {Q}\)property for short. Recall that \(\mathcal {W}_{\mathcal {P}}\) stands for the model of \(\mathcal {L}_{\mathcal {P}}\), and \(\vert A\vert \) for the truth set of wff \(A\) in \(\mathcal {W}_{\mathcal {P}}\) (cf. Sect. 2.1).
Definition 9
(Being a question) A wff \(A\) of \(\mathcal {L}_{\mathcal {P}}\) has the \(\mathcal {Q}\)property iff \(\vert A \vert = \mathcal {W}_{\mathcal {P}}\).
The \(\mathcal {Q}\)property is simply the complement of informativeness defined as follows:
Definition 10
(Informativeness) A wff \(A\) of \(\mathcal {L}_{\mathcal {P}}\) is informative iff \(\vert A \vert \ne \mathcal {W}_{\mathcal {P}}\).
Hence a wff \(A\) is (i.e. has the property of being) a question when, and only when \(A\) is not informative, that is, \(A\) is true in each possible world of \(\mathcal {W}_{\mathcal {P}}\). Thus the wffs having the \(\mathcal {Q}\)property are just classical tautologies. Yet, as we have already observed, \(\mathsf {InqB}\) enables a differentiation among classical tautologies: some of them are inquisitive, and some are not. So there exist inquisitive questions and noninquisitive questions.
Recall that the set of possibilities for an inquisitive wff has at least two elements. So if an inquisitive wff, \(A\), has the \(\mathcal {Q}\)property (i.e. “is” a question), then there exist at least two possibilities for \(A\) and, by noninformativeness, the truth set of \(A\) equals \(\mathcal {W}_{\mathcal {P}}\).
3.2 Questions in IEL
IEL adopts the “enrich with” approach to questions, although in its general setting does not predetermine how to shape them. On the level of a logical analysis it is assumed that questions are specific expressions of a formal language, and that to each question considered there is assigned an at least twoelement set of direct answers. Direct answers are supposed to be declarative formulas (sentences in the first or higherorder case). In the general considerations it is also assumed that each finite and at least twoelement set of declarative formulas/sentences constitutes the set of direct answers to a question of a formal language under consideration. One can build formal languages having the above features according to different patterns, for example by using Belnap’s (Belnap and Steel 1976) or Kubiński’s (Kubiński 1980) accounts of questions/interrogatives, or according to the semireductionistic schema proposed in Wiśniewski (1995) or Wiśniewski (2013).
In this paper we apply a formal account of propositional questions which is presumably the simplest one possible.^{12}
3.2.1 Questions as erotetic wffs
We augment the vocabulary of \(\mathcal {L}_{\mathcal {P}}\) with the following signs: ?, {, }, and the comma. We get a new language, \(\mathcal {L}_{\mathcal {P}}^{?}\), which has two categories of wellformed formulas: declarative wellformed formulas (dwffs) and erotetic wellformed formulas, that is, questions.
A dwff of \(\mathcal {L}_{\mathcal {P}}^{?}\) is a wff of \(\mathcal {L}_{\mathcal {P}}\).
Each question of \(\mathcal {L}_{\mathcal {P}}^{?}\) has at least two direct answers. At the same time the set of direct answers to a question is always finite. Note that questions of \(\mathcal {L}_{\mathcal {P}}^{?}\) are wellformed expressions of the language. One cannot identify a question with the set of direct answers to it. For instance, \(\mathord {?}\{p, q\} \ne \; \mathord {?}\{q, p\}\).
The schema (11) is general enough to capture most (if not all) of propositional questions studied in the literature.
Notation. We will use \(Q, Q^{*}, Q_1, \ldots \) as metalanguage variables for questions of \(\mathcal {L}_{\mathcal {P}}^{?}\). The set of direct answers to a question \(Q\) will be referred to as \(\mathsf {d}Q\).
3.3 “Translating” InqBquestions into questions of \(\mathcal {L}_{\mathcal {P}}^{?}\)
Let \(A\) be a dwff having the \(\mathcal {Q}\)property, that is, a question of \(\mathcal {L}_{\mathcal {P}}\). One can always find a question \(Q\) of \(\mathcal {L}_{\mathcal {P}}^?\) such that the family of all the truth sets of direct answers to \(Q\) is just the set of possibilities for \(A\).
3.4 Proper questions of \(\mathcal {L}_{\mathcal {P}}^{?}\) and inquisitiveness
In this section we point out some close connections between the socalled proper questions and inquisitiveness. Proper questions constitute the most important semantic category of questions in IEL.
We need some auxiliary notions.
Following Belnap (see Belnap and Steel 1976, pp. 119–120), we say that a dwff \(B\) is a presupposition of a question \(Q\) iff \(B\) is entailed by each direct answer to \(Q\). By \(\mathsf {Pres}Q\) we designate the set of all the presuppositions of \(Q\).
Definition 11
(Selfrhetorical question) A question \(Q\) is selfrhetorical iff \(\mathsf {Pres}Q\) entails some direct answer(s) to \(Q\).
The underlying idea is: a selfrhetorical question is a question that is already resolved by its presupposition(s).
As long as the language \(\mathcal {L}_{\mathcal {P}}^{?}\) is concerned, ‘entails’ means ‘CLentails’. We have:
Corollary 3
Let \(\mathsf {d}Q = \{A_1, \ldots , A_n\}\). \(Q\) is selfrhetorical iff \(\vert A_1 \vee \cdots \vee A_n\vert \subseteq \vert A\vert \) for some \(A \in \mathsf {d}Q\).
In the case of \(\mathcal {L}_{\mathcal {P}}^?\) proper questions are defined by:^{17}
Definition 12
(Proper question of \(\mathcal {L}_{\mathcal {P}}^?\)) A question \(Q\) of \(\mathcal {L}_{\mathcal {P}}^?\) is proper iff \(Q\) is not selfrhetorical.
One can prove:
Lemma 2
If \(\mathord {?}\{A_1, \ldots , A_n\}\) is proper, then \(A_1 \vee \cdots \vee A_n\) is inquisitive.
Proof
Corollary 4
\(\sigma \vDash \, !A \; \text{ iff } \; \{w\} \vDash \,!A \; \text{ for } \text{ each } \; w \in \sigma \).
We have:^{18}
Lemma 3
\(\mathord {?}\{A_1, \ldots , A_n\}\) is proper iff \(!A_1 \vee \cdots \vee \,!A_n\) is inquisitive.
Proof
(\(\Rightarrow \)) Suppose that \(!A_{(1,n)}\) is not inquisitive. Hence \(\vert !A_{(1,n)}\vert \vDash \,\,!A_{(1,n)}\). Thus \(\vert !A_{(1,n)} \vert \vDash \, !A_i\) for some \(i\), where \(1 \le i \le n\), and therefore \(!A_{(1,n)} \models _{cl}\, !A_i\). It follows that \(A_1 \vee \cdots \vee \,A_n \models _{cl} A_i\), that is, \(\mathord {?}\{A_1, \ldots , A_n\}\) is not proper.
(\(\Leftarrow \)) If \(!A_{(1,n)}\) is inquisitive, then \(\vert !A_{(1,n)}\vert \not \vDash \,\, !A_{(1,n)}\). Hence \(\vert !A_{(1,n)} \vert \not \vDash \,\, !A_i\) for \(i = 1, \ldots , n\). Thus, by Corollary 4, for each \(i\), where \(1 \le i \le n\), there exists \(w \in \vert !A_{(1,n)} \vert \) such that \(\{w\} \not \vDash A_i\). Therefore \(A_1 \vee \cdots \vee A_n \not \models _{cl} A_i\) for \(i = 1, \ldots , n\), that is, \(\mathord {?}\{A_1, \ldots , A_n\}\) is proper. \(\square \)
Lemma 3 shows that a question of \(\mathcal {L}_{\mathcal {P}}^?\) is proper if, and only if a disjunction of all its “asserted” direct answers is inquisitive. This sheds new light on the concept of proper question.
As for transitions from questions of \(\mathcal {L} _{\mathcal {P}}^?\) to questions of \(\mathcal {L} _{\mathcal {P}}\), Lemma 3 yields that a proper question of \(\mathcal {L}_{\mathcal {P}}^? \) is akin to an inquisitive question of \( \mathcal {L}_{\mathcal {P}}\) of the form (27) provided that the wff (27) is not informative (since each question of \(\mathcal {L}_{\mathcal {P}}\) is, by definition, noninformative).^{19} Needless to say, in the case of some questions of \(\mathcal {L}_{\mathcal {P}}^{?}\) the respective dwffs falling under the schema (27) are informative (a simple example: \(\mathord {?}\{p, q\}\)).
Terminology. From now on, unless otherwise stated, by questions we will mean questions of \(\mathcal {L}_{\mathcal {P}}^{?}\), and similarly for dwffs.
4 Evocation of questions
4.1 Definition of evocation
Evocation of questions is a binary relation between sets of dwffs and questions. The idea underlying the definition of evocation proposed in IEL is very simple. A question \(Q\) is evoked by a set of dwffs \(X\) when, and only when the truth of all the dwffs in \(X\) warrants the existence of a true direct answer to \(Q\), but does not warrant the truth of any single direct answer to \(Q\). In other words, if \(X\) consists of truths, at least one direct answer to \(Q\) must be true, but the issue “which one” remains unresolved.
The notion of multipleconclusion entailment allows us to express the above intuition in exact terms. Let \(\mid \models \) stand for mcentailment (see Sect. 2.3).
Definition 13
 1.
\(X \mid \models \mathsf {d}Q\), and
 2.
\(X \mid \not \models \{A \}\) for each \(A \in \mathsf {d}Q\).
By putting \(\mid \models _{cl}\) in the place of \(\mid \models \) we get the definition of CLevocation. It will be referred to as \(\mathbf {E}_{cl}\).
In the case of \(\mathcal {L}_{\mathcal {P}}^{?}\) (but not in the general case!) we can get rid of mcentailment.
Corollary 5
\(\mathbf {E}_{cl}(X, \mathord {?}\{A_1, \ldots A_n\})\) iff \(X \models _{cl} A_1 \vee \cdots \vee A_n\) and \(X \not \models _{cl} A\) for each \(A \in \{A_1, \ldots A_n\}\).
Let us note:
Lemma 4
A question \(Q\) of \(\mathcal {L}_{\mathcal {P}}^?\) is proper iff \(Q\) is CLevoked by a certain set of dwffs of \(\mathcal {L}_{\mathcal {P}}^?\).
Proof
Let \(Q = \mathord {?}\{A_1, \ldots , A_n\}\).
Clearly, if \(Q\) is proper, then \(\mathbf {E}_{cl}(A_1 \vee \cdots \vee A_n, Q)\).
If \(\mathbf {E}_{cl}(X, Q)\), then \(X \models _{cl} A_1 \vee \cdots \vee A_n\) and \(X \not \models _{cl} A\) for each \(A \in \mathsf {d}Q\). Suppose that \(Q\) is not proper. Therefore \(A_1 \vee \cdots \vee A_n \models _{cl} A\) for some \(A \in \mathsf {d}Q\) and hence \(X \models _{cl} A\). A contradiction. \(\square \)
4.1.1 Examples of evocation
4.2 Evocation and inquisitiveness
One can easily prove that a disjunction of all the direct answers to an evoked question is always inquisitive. By Lemma 2 and Lemma 4 we immediately get:
Corollary 6
If \(\mathord {?} \{A_1, \ldots , A_n\}\) is CLevoked by a certain set of dwffs, then \(A_1 \vee \cdots \vee A_n\) is inquisitive.
Corollary 6 cannot be strengthened to equivalence. But we have:^{20}
Theorem 1
Let \(\mathord {?}\{A_1, \ldots , A_n\}\) be a question of \(\mathcal {L}_{\mathcal {P}}^?\). \(\mathord {?}\{A_1, \ldots , A_n\}\) is CLevoked by a certain set of dwffs iff \(!A_1 \vee \cdots \vee \,!A_n\) is inquisitive.
4.3 Inquisitiveness and evocation
As for propositional InqB, each inquisitive wff “offers” at least two possibilities, that is, maximal states that support the wff. Moreover, the following hold:
Lemma 5
Let \(A\) be a dwff. If \(w \in \vert A \vert \), then there is a possibility for \(A\) such that \(w\) belongs to this possibility.
As we have already observed (see footnote 7), Lemma 5 follows from some results established in Ciardelli (2009) and Ciardelli and Roelofsen (2011).
Lemma 6
If \(A\) is an inquisitive dwff, then: (a) the set of possibilities for \(A\) is finite, and (b) for each possibility \(\gamma \) for \(A\) there exists a dwff \(B\) such that \(\vert B\vert \) equals \(\gamma \).
The above lemma follows from the existence of disjunctive normal forms in InqB. For details see Ciardelli (2009, p. 39), or Ciardelli and Roelofsen (2011, p. 69).
Let us first prove that a dwff is inquisitive if, and only if it CLevokes a yes–no question whose affirmative answer expresses a possibility for the dwff.
Theorem 2
Let \(\vert B \vert \) constitute a possibility for a dwff \(A\). \(A\) is inquisitive iff \(\mathbf {E}_{cl}(A, \mathord {?}B)\).
Proof
(\(\Rightarrow \)) Clearly \(A \mid \models _{cl} \{B, \lnot B\}\).
Since \(\vert B \vert \) is a possibility for \(A\), we have \(\vert B \vert \vDash A\). If \(\vert A \vert \subseteq \vert B\vert \), then, by Lemma 1, \(\vert A\vert \vDash A\). It follows that \(A\) is not inquisitive. A contradiction. Therefore \(A \not \models _{cl} B\) and hence \(A \mid \not \models _{cl} \{B\}\).
Now observe that when \(A\) is inquisitive, each possibility for \(A\) must be nonempty (because if \(\emptyset \) is a possibility for \(A\), it is the only possibility for \(A\)).
Since \(\vert B\vert \) is a possibility for \(A\), we have \(\vert B\vert \vDash A\). Suppose that \(A \models _{cl} \lnot B\). Thus \(B \models _{cl} \lnot A\). Since \(\vert B \vert \ne \emptyset \), it follows that there exists \(w \in \mathcal {W}_{\mathcal {P}}\) such that \(A\) is true in \(w\) and its classical negation, \(\lnot A\), is true in \(w\), which is impossible. Hence \(A \not \models _{cl} \lnot B\) and thus \(A \mid \not \models _{cl} \{\lnot B\}\).
(\(\Leftarrow \)) If \(\mathbf {E}_{cl}(A, \mathord {?}B)\), then \(A \not \models _{cl} B\) and \(A \not \models _{cl} \lnot B\). Hence \(\vert A \vert \ne \emptyset \) as well as \(\vert B \vert \ne \emptyset \). Thus, by Lemma 5, there exist nonempty possibilities for \(A\) and for \(B\).
Assume that \(\vert B \vert \) is the only possibility for \(A\). Hence \(info(A) = \vert B \vert \) and thus \(\vert A \vert = \vert B \vert \). It follows that \(A \models _{cl} B\) and therefore \(A\) does not CLevoke \(\mathord {?}B\). A contradiction. Thus \(\vert B \vert \) is not the only possibility for \(A\). It follows that \(A\) is inquisitive. \(\square \)
Comment. Observe that \(\vert B\vert \) in the assumption of Theorem 2 is an arbitrary possibility for \(A\). Thus a philosophical comment on Theorem 2 is this: a dwff is inquisitive if, and only if it raises (in the sense of CLevocation) yes–no questions about possibilities for the dwff. This sheds new light on the concept of inquisitiveness (in InqB).
Next, it can be proven that an inquisitive dwff raises (i.e. CLevokes) a question whose direct answers express all the possibilities for the dwff.
Theorem 3
Let \(A\) be an inquisitive dwff. If \(\{\vert B_1\vert , \ldots , \vert B_n \vert \}\) is the set of possibilities for \(A\), then \(\mathbf {E}_{cl}(A, \mathord {?}\{B_1, \ldots , B_n\})\).
Proof
If \(A\) is inquisitive, then the set of possibilities has at least two elements and, by Lemma 6, is a finite set. Moreover, again by Lemma 6, each possibility for \(A\) is the truth set of some dwff.
Suppose that \(\{\vert B_1\vert , \ldots , \vert B_n \vert \}\) is the set of possibilities for \(A\).
Let \(w \in \vert A\vert \). Hence \(\{w\} \vDash A\). By Lemma 5, \(w\) belongs to some possibility for \(A\). Since \(\vert B_1\vert , \ldots , \vert B_n\vert \) exhaust the set of possibilities for \(A, w\) belongs to some \(\vert B_i\vert \) and hence to \(\vert B_1 \vee \cdots \vee B_n\vert \). Therefore \(A \models _{cl} B_1 \vee \cdots \vee B_n\) and thus \(A \mid \models _{cl} \{B_1, \ldots , B_n\}\).
If \(A\) is inquisitive and \(\vert B_i\vert \), for \(1 \le i \le n\), is a possibility for \(A\), then \(\vert A \vert \not \subseteq \vert B_i\vert \), and hence \(A \mid \not \models _{cl} \{B_i\}\).
Thus \(\mathbf {E}_{cl}(A, \mathord {?}\{B_1, \ldots , B_n\})\) holds. \(\square \)
Comment. One can expect an inquisitive wff to raise a question “about” all the possibilities for the wff, that is, a question whose direct answers express all the possibilities for the formula. The claim of Theorem 3 amounts exactly to this, but the concepts used belong to IEL.
Let us now prove:
Theorem 4
Let \(A\) be an inquisitive dwff, and let \(\vert B_1\vert , \vert B_2 \vert \) constitute distinct possibilities for \(A\). Then \(\mathbf {E}_{cl}(A, \mathord {?}\pm \vert B_1, B_2\vert )\).
Proof
Since \(A\) is inquisitive and \(\vert B_1\vert , \vert B_2\vert \) are possibilities for \(A\), it follows that \(\vert A \vert \not \subseteq \vert B_i\vert \) for \(i=1,2\). Hence \(A \mid \not \models _{cl} \{B_i\}\).
For conciseness, let us designate ‘\(\mathord {?}\pm \vert B_1, B_2 \vert \)’ by \(Q^{*}\). Recall that \(\mathsf {d}Q^{*} = \{B_1 \wedge B_2, B_1 \wedge \lnot B_2, \lnot B_1 \wedge B_2, \lnot B_1 \wedge \lnot B_2\}\). It is easily seen that a disjunction of all the elements of \(\mathsf {d}Q^{*}\) is a classical tautology. Therefore \(A \mid \models _{cl} \mathsf {d}Q^{*}\).
Since \(A \mid \not \models _{cl} \{B_i\}\) for \(i=1,2\), we have \(A \mid \not \models _{cl} C\) for any \(C \in \mathsf {d}Q^{*}\) different from \(\lnot B_1 \wedge \lnot B_2\). Suppose that the latter is CLentailed by \(A\). It follows that \(A \models _{cl} \lnot B_i\) for \(i=1,2\), and hence \(B_i \models _{cl} \lnot A\). On the other hand, \(\vert B_i\vert \vDash A\) and thus \(B_i \models _{cl} A\). Therefore \(\vert B_1 \vert = \emptyset \) and \(\vert B_2 \vert = \emptyset \). Hence \(\vert B_1 \vert = \vert B_2 \vert \), which contradicts the assumption. Therefore \(A \mid \not \models _{cl} \{\lnot B_1 \wedge \lnot B_2\}\). Thus for each \(C \in \mathsf {d}Q^{*} : A \mid \not \models _{cl} \{C\}\). \(\square \)
Generally speaking, \(\mathord {?}\pm \vert B_1, B_2 \vert \) is a “partition” question that asks which of the possibilities offered by \(A\) and expressed by \(B_1, B_2\) holds, and allows both of them as well as neither of them to hold.
5 Erotetic implication
Validity of erotetic inferences which have questions as premises and conclusions is defined in IEL by means of erotetic implication, being a ternary semantic relation between a question, a (possibly empty) set of dwffs, and a question. By defining erotetic implication we explicate the intuitive notion “a question arises from a question on the basis of a set of declarative sentences”.
5.1 Intuitions
Let us start with naturallanguage examples.
Example 1
Example 2
Example 3

(F \(_1\)) If at least one direct answer to an implying question is true and the set of declarative premises consists of truths, it is impossible that every direct answer to the implied question is false.

(F \({_1}^{*})\) If the implying question is sound and the declarative premises are true, it is impossible that the implied question is unsound.

(F \(_2\)) Any expansion of the set of declarative premises by a direct answer to the implied question results in narrowing down the initial search space determined by (the set of direct answers to) the implying question.
IEL defines erotetic implication in such a way that the effects sketched above occur due to purely semantic reasons.
5.2 Definition of erotetic implication
As in Sect. 4, \(\mid \models \) stands for mcentailment. Let us abbreviate ‘\(X \cup \{C\}\)’ as ‘\(X, C\)’. Recall that \(\mathsf {d}Q\) designates the set of direct answers to \(Q\). Here is the schema of definition of erotetic implication:
Definition 14
 1.
for each \(A \in \mathsf {d}Q: X, A \mid \models \mathsf {d}Q_1\), and
 2.
for each \(B \in \mathsf {d}Q_1\) there exists a nonempty proper subset \(Y\) of \(\mathsf {d}Q\) such that \(X, B \mid \models Y\).
Note that the respective \(Y\)’s can (but need not) be singleton sets. Moreover, we do not assume that \(X \ne \emptyset \).
By putting \(\mid \models _{cl}\) for \(\mid \models \) we get the definition of classical erotetic implication, \(\mathbf {Im}_{cl}\).
Notation. For \(Z = \{C_1, \ldots , C_n\}\), ‘\(\bigvee Z\)’ abbreviates ‘\(C_1 \vee \cdots \vee C_n\)’. Of course, \(\bigvee \{C_1\} = C_1\).
As for \(\mathcal {L}_{\mathcal {P}}^?\) (but, again, not in the general case!), one can define erotetic implication without making use of the concept of mcentailment. This is due to:
Corollary 7
 1.
\(X, \bigvee \mathsf {d}Q \models _{cl} \bigvee \mathsf {d}Q_1\), and
 2.
for each \(B \in \mathsf {d}Q_1\) there exists a nonempty proper subset \(Y\)of \(\mathsf {d}Q\) such that \(X, B \models _{cl} \bigvee Y\).
It is easily seen that clause (1) of Definition 14 warrants the fulfilment of condition (\(\mathbf F _1\)). Concerning condition (\(\mathbf F _2\)): due to clause (2) of Definition 14, the set of declarative premises extended by any direct answer to the implied question mcentails a proper subset of the set of direct answers to the implying question (or, to put it differently, entails either a direct answer or a disjunction of only some direct answers to the question). So if a given direct answer to the implied question is true and \(X\) consists of truths, the initial class of options offered by the implying question becomes narrowed down, that is, roughly, there is a warranty that a true answer to the implying question belongs to a certain class that is “smaller” than the initial class.^{21} What is important, this condition holds for any direct answer to the implied question. Needless to say, the respective proper subsets related to different answers are usually distinct.
5.2.1 Examples of erotetic implication
For further examples, properties, and types of erotetic implication see Wiśniewski (1994, 1995, 2001, 2013). Let us add that erotetic implication links questions in the socalled erotetic search scenarios^{22} and constitutes the background of the method of Socratic proofs.^{23}
5.3 Erotetic implication and possibilities
It can be proven that each question erotetically implies a question whose direct answers express all the possibilities for the direct answers to the implying question.
Notation. Let \(Poss(A)\) stand for the set of possibilities for a dwff \(A\).
Recall that when \(A\) is inquisitive, then, by Lemma 6, \(Poss(A)\) is a finite at least twoelement set which comprises truth sets of some dwffs. If \(A\) is not inquisitive, then \(Poss(A) = \{\vert A\vert \}\).
Let us observe that each truth set may be expressed by more than one dwff (in fact, by infinitely many dwffs). For if \(A\) and \(B\) are CLequivalent, then \(\vert A \vert = \vert B \vert \), even if \(A\) and \(B\) are syntactically distinct. Thus we may assume without loss of generality that dwffs \(B_{1_1}, \ldots , B_{1_{k_1}}, \ldots , B_{n_1}, \ldots , B_{n_{k_n}}\) mentioned in the statement of Theorem 5 below are syntactically distinct.
Theorem 5
Let \(\mathsf {d}Q = \{A_1, \ldots , A_n\}\) and let \(\text{ Poss }(A_i) = \{\vert B_{i_{1}} \vert , \ldots , \vert B_{i_{k_{i}}}\vert \}\). Then \(\mathbf {Im}_{cl}(Q, Q^{\lozenge })\), where \(\mathsf {d}Q^{\lozenge } = \{B_{1_{1}}, \ldots , B_{1_{k_{1}}}, \ldots , B_{n_{1}}, \ldots B_{n_{k_{n}}}\}\).
Proof
If \(A_i\) (\(1 \le i \le n\)) is inquisitive, then for reasons indicated in the proof of Theorem 3 we have \(A_i \mid \models _{cl} \{B_{i_{1}}, \ldots , B_{i_{k_{i}}}\}\) and hence \(A_i \mid \models _{cl} \mathsf {d}Q^{\lozenge }\). If \(A\) is not inquisitive, then \(A_i \mid \models _{cl} \{B_{i_{1}}\}\) and therefore, again, \(A_i \mid \models _{cl} \mathsf {d}Q^{\lozenge }\).
Since each \(\vert B_{i_j}\vert \) is a possibility for \(A_i\) (where \(1 \le i \le n\), \(1 \le j \le k_i\)), we have \(\vert B_{i_j}\vert \vDash A_i\) and therefore \(B_{i_j} \mid \models _{cl} \{A_i\}\). \(\square \)
Thus a (propositional) question is, in a sense, reducible to a question whose direct answers display all the possibilities for the direct answers to the initial question. Recall that if \(\vert B \vert \) is a possibility for \(A\), then \(B\) CLentails \(A\). Thus when the implied question is answered, the implying question is answered as well.
Needless to say, the “questions about all the possibilities” are not the only questions erotetically implied.
6 Concluding remarks
As we have shown, enriching the conceptual apparatus of IEL with the notion of inquisitiveness results in revealing some important features of CLevoked questions: disjunctions of all their direct answers are inquisitive (cf. Corollary 6) and CLevoked questions are just these questions whose disjunctions of asserted direct answers are inquisitive (Theorem 1). On the other hand, when we “liberalize” the concept of question present in InqB, inquisitiveness of a formula amounts to evoking by the formula a yes–no question with the affirmative answer expressing a possibility for the formula (Theorem 2), and inquisitive formulas CLevoke, among others, questions about all the possibilities for the formulas (Theorem 3). Moreover, a question is, in a sense, reducible to a question whose direct answers express all the possibilities for the direct answers to the “initial” question (Theorem 5). What is more important than the above formal results, however, is the inferential perspective gained: we are able to model the phenomenon of arriving at inquisitive questions and/or questions about possibilities.
Footnotes
 1.
 2.
 3.
 4.
 5.
In more traditional terms: the informative content of \(A\) amounts to the proposition expressed by \(A\).
 6.
As for (3), \(\vert p \vee q \vert = \{w \in \mathcal {W}_{\mathcal {P}} : w(p) = \mathbf {1} \; \text{ or } \; w(q) = \mathbf {1}\}\). But \(\vert p \vee q \vert \not \vDash p \vee q\), since it is neither the case that \(\vert p \vee q \vert \vDash p\) nor it is the case that \(\vert p \vee q \vert \vDash q\). (Recall that \(\vert p \vee q \vert \vDash p\) iff \(w(p) = \mathbf {1}\) for each \(w \in \vert p \vee q \vert \), and similarly for \(q\). Neither of these cases hold.) Concerning (4): clearly \(\vert p \vee \lnot p \vert = \mathcal {W}_{\mathcal {P}}\). But, again, we neither have \(w(p) = \mathbf {1}\) for every \(w \in \mathcal {W}_{\mathcal {P}}\) nor we have \(w(p) \ne \mathbf {1}\) for each world \(w\) belonging to a nonempty subset of \(\mathcal {W}_{\mathcal {P}}\). Hence \(\vert p \vee \lnot p \vert \) does not support \(p \vee \lnot p\) and thus \(\vert p \vee \lnot p \vert \notin [p \vee \lnot p]\), that is, (4) is inquisitive.
 7.
 8.
For this concept see, e.g., Shoesmith and Smiley (1978).
 9.
(8) is a Hintikkastyle paraphrase, while (9) agrees with Vanderveeken’s approach to questions. The survey paper Harrah (2002) provides a comprehensive exposition of logical theories of questions elaborated up until late 1990s. Supplementary information about more linguistically oriented approaches can be found, e.g., in Groenendijk and Stokhof (1997) (reprinted as Groenendijk and Stokhof 2011) and Krifka (2011). The paper Ginzburg (2011) provides a survey of recent developments in the research on questions, both in logic and in linguistics. A general overview of approaches to questions and their semantics can also be found in Wiśniewski (2015).
 10.
 11.
This holds for the socalled presuppositional inquisitive semantics; see Ciardelli et al. (2013b, Section 6).
 12.
 13.For example, any of the following:$$\begin{aligned}&\mathord {?}\{p, q\}\end{aligned}$$(13)$$\begin{aligned}&\mathord {?}\{p \wedge \lnot q, \lnot p \wedge q\}\end{aligned}$$(14)$$\begin{aligned}&\mathord {?}\{p \wedge \lnot q, \lnot p \wedge q, p \wedge q\}\end{aligned}$$(15)corresponds to a question expressed by the interrogative sentence:$$\begin{aligned}&\mathord {?}\{p, q, \lnot (p \vee q)\} \end{aligned}$$(16)If (17) is understood so that a single choice among the alternatives given is sufficient for answering the relevant question, the question is represented by (13). If an exclusive choice is requested, the representation is (14). If a complete choice is requested, the question is represented by (15). If, however, the question expressed by (17) is construed as allowing that neither of the alternatives holds, it has (16) as its representation.$$\begin{aligned} \text{ Is } \text{ John } \text{ a } \text{ philosopher, } \text{ or } \text{ a } \text{ logician? } \end{aligned}$$(17)
 14.
“Any formula can be represented as a disjunction of assertions in such a way that its possibilities coincide with the classical meaning of the disjuncts” (Ciardelli 2009, pp. 19–20.) The result is explicitly proven in Ciardelli (2009) for finite \(\mathcal {P}\)’s, but, obviously, the finiteness assumption is dispensable. See also Lemma 6 in Sect. 4.3.
 15.
Observe that \(\vert \lnot p\vert \cap \vert \lnot q\vert \) is the truth set of both \(\lnot (p \vee q)\) and \(\lnot p \wedge \lnot q\). In general, a possibility is always the truth set of many CLequivalent wffs.
 16.
We owe this observation to an anonymous referee.
 17.
Proper questions are defined in IEL as normal questions which are not selfrhetorical; a question is normal iff the set of its presuppositions is nonempty and mcentails the set of direct answers to the question. Since each question of \(\mathcal {L}_{\mathcal {P}}^?\) is normal, in the case of \(\mathcal {L}_{\mathcal {P}}^?\) the definition simplifies.
 18.
We owe this observation to a remark of an anonymous referee.
 19.
But recall that there are systems of inquisitive semantics in which this requirement is abandoned; see footnote 11.
 20.
Similarly as in the case of Lemma 3, we owe this observation to an anonymous referee.
 21.
Let us stress that narrowing down is not tantamount to elimination. For example, let \(Z = \{p, p \rightarrow q \vee r\}\) and \(W = \{q, r, t\}\). Clearly, \(Z \mid \models _{cl} \{q, r\}\) and \(\{q, r\}\) is a proper subset of \(W\), but \(Z\) does not eliminate any dwff in \(W\). Similarly, elimination need not yield narrowing down. For instance, \(\lnot q\) eliminates the dwff \(q\) of \(W\), but does not mcentail (in CL) any proper subset of \(W\).
 22.
 23.
For most recent developments of the method see LeszczyńskaJasion et al. (2013).
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