Abstract
Inferential Erotetic Logic (IEL) gives an account of inferences in which questions play the role of conclusions, and proposes criteria of validity for these inferences. We show that some tools elaborated within IEL are useful for the formal modeling of: (a) replying with questions that are not clarification requests, and (b) question answering based on additional information actively sought for.
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Notes
 1.
 2.
See Hintikka (1999).
 3.
Added in 2013. For a taxonomy of questionreplies (QR) see Łupkowski and Ginzburg (2013). A corpus study reveals that clarification requests constitute about 80 % of QR, while about 10 % of QR falls into the category of ‘dependent questions’. QR, in turn, constitute slightly more than 20 % of all responses to queries found in the spoken part of the British National Corpus.
 4.
See Grice (1975).
 5.
The logic of questions is also called erotetic logic (from Greek erotema that means “question”). Roughly, a direct answer is a possible justsufficient answer, i.e. a possible answer which provides neither less nor more information than it is requested by the question.
 6.
Clearly, it suffices to suppose that question (5) is sound and premise (1) is true. But a stronger assumption, according to which all the premises are true, does not make any harm. We aim at a semantic relation between a question, a set of declarative sentences/formulas, and a question.
 7.
 8.
In the first case none of the answers is potentially useful. As for the second case, the negative answer is useful, whereas the affirmative answer is useless. Needless to say, in any of the above cases condition (I) is satisfied for a trivial reason, due to the structure of the “questionconclusion” only.
 9.
 10.
In the case of dialogues, as Ginzburg (2010) observes, when query responses are erotetically implied, relevance (in a dialogue) is retained. Of course, relevance of query responses can be retained in other ways as well.
 11.
 12.
For simplicity, we operate on the propositional level. Letters p, q, r, t, … are propositional variables.?{A _{1}, …, A _{ n }} is a question whose direct answers are exactly the explicitly listed formulas A _{1}, …, A _{ n }.
 13.
 14.
In IEL based on Classical Logic we do not have Im(?{A, ¬A}, B → A, ?{B, ¬B}). However, we have both Im(?{A, ¬A}, B → A, ?{A, ¬A, B}) and Im(?{A, ¬A, B}, ?{B, ¬B}). So, although Im is not “transitive”, it is possible to reach?{B, ¬B} from?{A, ¬A} and B → A, but in two steps (recall that Im is monotone with respect to sets of declaratives).
 15.
Again, there are nonqueries involved. As long as Classical Logic constitutes the background, we do not have Im(?{A, ¬A}, A → B, ?{B, ¬B}). But we do have Im(?{A, ¬A}, A → B, ?{A, ¬A, ¬B}) and Im(?{A, ¬A, ¬B}, ?{B, ¬B}).
 16.
? ±∣p, q∣ abbreviates the conjunctive question?{p ∧ q, p ∧¬q, ¬p ∧ q, ¬p ∧¬q}.
 17.
It can happen that a given question labels more than one node of an escenario. However, dec(ϕ _{ γ }) is always unique, since γ refers to a node.
 18.
For simplicity, we remain at the propositional level.
 19.
 20.
I owe this formulation of DP to Mariusz Urbański.
 21.
It may be of interest that there also exists a second IELbased approach to problem solving. This time the underlying idea is: transform a question into consecutive questions until a question which can be answered in only one rational way is arrived at. This is modelled by means of the socalled erotetic calculi. Rules of these calculi operate on questions only; a rule transforms a question into a further question. A Socratic transformation is a sequence of questions, starting with a question about entailment/derivability/theoremhood. This question is then transformed, step by step, into consecutive questions according to the rules of a calculus. Answers play no role in the process. There are successful and unsuccessful transformations; a successful transformation ends with a question of a required final form (the details depend on the logic under consideration). A successful transformation is a Socratic proof. The rules are designed in such a way that once a successful transformation is accomplished, the initial issue is affirmatively resolved and there is no need for performing any further deductive moves. Moreover, each step in a Socratic transformation is an IELvalid inference from a question to a question. So far erotetic calculi have been developed for Classical Logic (see Wiśniewski 2004b; Wiśniewski and Shangin 2006), some paraconsistent propositional logics (see Wiśniewski et al. 2005), and normal modal propositional logics (Leszczyńska 2004, 2007, 2008, 2009). An approach to Intuitionistic Propositional Logic based on a similar idea can be found in Skura (2005).
 22.
Cf. also Peliš and Majer (2011).
 23.
 24.
For this concept see Shoesmith and Smiley (1978).
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Appendix
Appendix
14.1.1 Erotetic Implication
Let L be an arbitrary but fixed formal language such that the following conditions are satisfied:

(a)
The set D _{ L } of declarative wellformed formulas (dwffs) of L is defined;

(b)
The set Ψ _{ L } of questions of L is defined, where \(\mathbf{D}_{L} \cap \varPsi _{L} = \varnothing \);

(c)
If Q is a question of L, then there exists an at least twoelement set \(\mathbf{d}Q \subseteq \mathbf{D}_{L}\) of direct answers to Q;

(d)
(The declarative part of) L is supplemented with a semantics rich enough to define the concept of truth for dwffs, and the class of admissible partitions.
A partition of D _{ L } is an ordered pair \(\text{P} =\langle \mathbf{T}_{\text{P}},\mathbf{U}_{\text{P}}\rangle\), where \(\mathbf{T}_{\text{P}} \cap \mathbf{U}_{\text{P}} = \varnothing \), and \(\mathbf{T}_{ \text{P}} \cup \ \mathbf{U}_{\text{P}} = \mathbf{D}_{L}\). Intuitively, T _{ P } consists of all the dwffs which are true in P, and U _{P} is made up of all the dwffs which are untrue in P. For brevity, we will be speaking about truths and untruths of a partition.
By a partition of L we simply mean a partition of D _{ L }.
Note that we have used the term “partition” as pertaining to the set of dwffs only. What is “partitioned” is neither the “logical space” nor the set of questions. Recall that \(\mathbf{D}_{L} \cap \varPsi _{L} = \varnothing \). Thus when we have a partition \(\langle \mathbf{T}_{\text{P}},\mathbf{U}_{\text{P}}\rangle\) of L and a question of L, the question is neither in T _{P} nor in U _{P}.
A question Q is sound in a partition \(\langle \mathbf{T}_{\text{P}},\mathbf{U}_{\text{P}}\rangle\) iff \(\mathbf{d}Q \cap \mathbf{T}_{\text{P}}\neq \varnothing \).
The concept of partition is very wide and admits partitions which are rather odd from the intuitive point of view. For example, there are partitions in which T _{P} is a singleton set, or in which U _{P} is the empty set. In order to avoid oddity on the one hand, and to reflect some basic semantic facts about the language just considered on the other, we should distinguish a class of admissible partitions, being a nonempty subclass of the class of all partitions of the language.
Admissible partitions are defined either directly or indirectly. In the former case one imposes some explicit conditions on the class of all partitions. In the latter case one uses a previously given semantics of dwffs. For example, when D _{ L } is the set of wellformed formulas of Classical Propositional Calculus (CPC), a partition \(\langle \mathbf{T}_{ \text{P}},\mathbf{U}_{\text{P}}\rangle\) is called admissible iff for some CPCvaluation v, \(\mathbf{T}_{ \text{P}} =\{ A \in \mathbf{D}_{L}: v(A) = \mathbf{1}\}\), and \(\mathbf{U}_{ \text{P}} =\{ A \in \mathbf{D}_{L}: v(A) = \mathbf{0}\}\).
In what follows it is assumed that we are dealing with expressions of L and admissible partitions of L. For brevity, the specifications “in L” and “of L” are omitted.
Let X stand for a set of dwffs and let A be a dwff. Entailment, symbolized by ⊧, is defined by:
Definition 1.
X⊧A iff there is no admissible partition \(\langle \mathbf{T}_{\text{P}},\mathbf{U}_{\text{P}}\rangle\) such that \(X \subseteq \mathbf{T}_{\text{P}}\) and A ∈ U _{P}.
We also need multipleconclusion entailment (mcentailment for short).^{Footnote 24} This is a relation between sets of dwffs. Mcentailment, ∣⊧, is defined as follows:
Definition 2.
X∣⊧Y iff there is no admissible partition \(\langle \mathbf{T}_{\text{P}},\mathbf{U}_{\text{P}}\rangle\) such that X ⊆ T _{P} and Y ⊆ U _{P}.
Thus X mcentails Y if there is no admissible partition in which X consists of truths and Y consists of untruths. In other words, mcentailment between X and Y holds just in case the truth of all the dwffs in X warrants the presence of some truth(s) among the elements of Y: whenever all the dwffs in X are true in an admissible partition P, then at least one dwff in Y is true in the partition P.
Erotetic implication is defined by:
Definition 3.
A question Q implies a question Q _{1} on the basis of a set of dwffs X (in symbols: Im(Q, X, Q _{1})) iff

1.
For each \(A \in \mathbf{d}Q: X \cup \{ A\}\mid \models \mathbf{d}Q_{1}\), and

2.
For each B ∈ d Q _{1} there exists a nonempty proper subset Y of d Q such that \(X \cup \{ B\}\mid \models Y\).
14.1.2 EScenarios as Labelled Trees
Escenarios have been defined in Wiśniewski (2003) (cf. also Wiśniewski 2001, 2004a) as families of interconnected sequences of questions and dwffs, the socalled erotetic derivations. In this paper, however, we give an equivalent definition in terms of trees. Escenarios will be defined here as labelled trees, where the labels are dwffs and questions.
Definition 4.
A finite labelled tree Φ is an erotetic search scenario for a question Q relative to a set of dwffs X iff

1.
The nodes of Φ are labelled by questions and dwffs; they are called enodes and dnodes, respectively;

2.
Q labels the root of Φ;

3.
Each leaf of Φ is labelled by a direct answer to Q;

4.
\(\mathbf{d}Q \cap X = \varnothing \);

5.
For each dnode γ _{ d } of Φ: if A is the label of γ _{ d }, then

A ∈ X, or

A ∈ d Q ^{∗}, where Q ^{∗} ≠ Q and Q ^{∗} labels the immediate predecessor of Φ, or

{B _{1}, …, B _{ n }}⊧A, where B _{ i } (1 ≤ i ≤ n) labels a dnode of Φ that precedes the dnode γ _{ d } in Φ;


6.
Each dnode of Φ has at most one immediate successor;

7.
There exists at least one enode of Φ which is different from the root;

8.
For each enode γ _{ e } of Φ different from the root: if Q ^{∗} is the label of γ _{ e }, then d Q ^{∗} ≠ d Q and

Im(Q ^{∗∗}, Q ^{∗}) or Im(Q ^{∗∗}, {B _{1}, …, B _{ n }}, Q ^{∗}), where Q ^{∗∗} labels an enode of Φ that precedes γ _{ e } in Φ and B _{ i } (1 ≤ i ≤ n) labels a dnode of Φ that precedes γ _{ e } in Φ, and

An immediate successor of γ _{ e } is either an enode or is a dnode labelled by a direct answer to the question that labels γ _{ e }, moreover

If an immediate successor of γ _{ e } is an enode, it is the only immediate successor of γ _{ e },

If an immediate successor of γ _{ e } is not an enode, then for each direct answer to the question that labels γ _{ e } there exists exactly one immediate successor of γ _{ e } labelled by the answer.


A query of an escenario Φ can be defined as a question that labels an enode of Φ which is different from the root and whose immediate successor is not an enode. Paths of escenarios are construed in the standard manner; a branch is a maximal path which originates from the root. By dwffs of a branch we mean the dwffs which are labels of dnodes of the branch, and similarly for questions.
The following holds:
Theorem 1 (Golden Path Theorem).
Let Φ be an escenario for Q relative to X. Let \(\text{P} =\langle \mathbf{T}_{\text{P}},\mathbf{U}_{\text{P}}\rangle\) be an admissible partition such that X ⊆ T _{P} and Q is sound in P. Then the escenario Φ has at least one branch ϕ such that:

1.
Each dwff of ϕ is in T _{P} , and

2.
Each question of ϕ is sound in P, and

3.
The leaf of ϕ is (labelled by) a direct answer to Q which is in T _{P} .
The core of the proof lies in the following observations. Erotetic implication preserves soundness given that the relevant declarative premises are true. Needless to say, entailment preserves truth. On the other hand, by the clause (8) of Definition 4, a query has all the direct answers “as” immediate successors, and thus also the true answer(s).
Although the theorem speaks about a (“golden”) branch, it is called a Golden Path Theorem because in the original setting (see Wiśniewski 2003) escenarios are not defined as trees and what is called a path of an escenario in the “old” setting corresponds to a branch of an escenario in the current setting.
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Wiśniewski, A. (2014). Answering by Means of Questions in View of Inferential Erotetic Logic. In: Weber, E., Wouters, D., Meheus, J. (eds) Logic, Reasoning, and Rationality. Logic, Argumentation & Reasoning, vol 5. Springer, Dordrecht. https://doi.org/10.1007/9789401790116_14
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