Deduction and Reduction Theorems for Inferential Erotetic Logic

The concepts of question evocation and erotetic implication play central role in Inferential Erotetic Logic. In this paper, deduction theorems for question evocation and erotetic implication are proven. Moreover, it is shown how question evocation by a finite non-empty set of declaratives can be reduced to question evocation by the empty set, and how erotetic implication based on a finite non-empty set of declaratives can be reduced to a relation between questions only.


Introduction
The logic of questions, sometimes called erotetic logic, 1 is a branch of philosophical logic. Many prominent logicians, for example Nuel D. Belnap, Jaakko Hintikka or Johan van Benthem-to mention only a few-devoted books or their substantive chapters to the field. (See [1,[8][9][10], and [17], respectively.) The interest in the logic of questions is currently growing, as witnessed, for instance, by [2,[12][13][14], or the special issue of Synthese [6] published in 2015. However, no commonly accepted theory has been worked out so far. The main approaches still differ conceptually and, what is more important, in focussing their interests on different aspects of questions and questioning. Comparing these approaches is not an easy task, which has been accomplished only partially (cf. [5,14,22]). Given this, it seems quite justified to address problems by means of the conceptual apparatus of a preferred paradigm, leaving apart the ongoing foundational dispute. This is how we are going to proceed here. In this paper we will be working within Inferential Erotetic Logic (IEL for short). IEL focusses its attention on inferential aspects of questioning. The idea of IEL originates from the late 1980s. The monograph [18] summarizes results obtained until the early 1990s. The book [19] presents IEL in its current form.

Erotetic Inferences and IEL: Aims of This Paper
In many cases arriving at questions resembles coming to conclusions: there are premises involved and some inferential thought processes take place. In other words, there exist erotetic inferences, that is, thought processes in which one arrives at a question on the basis of some previously accepted declarative sentence(s) and/or a previously posed question.
IEL offers an account of validity of erotetic inferences. It proceeds as follows. First, some criteria of validity are proposed, separately for erotetic inferences that involve only declarative premises and for these in which an interrogative premise occurs (for details see, e.g., [19], Chapter 5). Then two semantic relations are defined: evocation of questions by sets of declarative formulas, and erotetic implication of a question by a question together with a (possibly empty) set of declarative formulas. Validity of erotetic inferences of the consecutive kinds is defined in terms of question evocation and erotetic implication, respectively.
The role played by question evocation and erotetic implication in IEL resembles that performed by entailment in a logic of statements. Thus the properties of question evocation and erotetic implication are worth being studied. As a matter of fact, a lot is known about them so far (cf., e.g., [18,19]). Our aim is to make a further step in this direction. First, we prove some theorems which seem to deserve the label deduction theorems for question evocation and erotetic implication. Second, we prove some reduction theorems. As for question evocation, we address the following issue. Suppose that a question Q is evoked by a non-empty set of declarative formulas X. How can one reduce this to evocation of a question (related both to Q and X) by the empty set? Concerning erotetic implication, the issue is: how can one reduce erotetic implication based on a non-empty set of declarative formulas to erotetic implication between questions only, that is, to pure erotetic implication?

The Logical Basis
We will be working here with IEL in its most general setting presented in [19], based on Minimal Erotetic Semantics (MiES for short).
Generally speaking, MiES enables an introduction of some important semantic notions pertaining to questions regardless of whether-and if so, how-the semantics of questions has been previously elaborated in detail. Moreover, MiES relies upon only few assumptions concerning the syntax of questions considered and, at the level of declaratives, can be conjoined with (semantics of) both Classical Logic and a wide class of non-classical logics. 2

Syntax
Let L be a formal language, in which at least two categories of well-formed expressions occur: declarative well-formed formulas (hereather: d-wffs) and interrogative formulas (hereafter: questions). 3 Generally speaking, L has thus (possibly among others) a "declarative part" and an "erotetic part." We do not assume in advance what formal language (a non-modal propositional language, a modal propositional language, a first-order language, etc.) performs the role of the declarative part of L. We also stay (almost) neutral concerning the form of questions of L. 4 We assume, however, the existence of an assignment of direct answers to questions. More precisely, we stipulate that for each question Q of L there exists an at least two-element set, dQ, of d-wffs of L, called the set of direct answers to Q. Formally, d is a function that assigns a set of d-wffs to a question. Intuitively, dQ comprises these possible answers to Q which provide neither less no more information than it is requested by Q. Being true is not a prerequisite for being a direct answer.

Semantics
In order to define question evocation and erotetic implication we need the concept of multiple-conclusion entailment (see, e.g., [15]), or mc-entailment for short. Mc-entailment is a semantic relation between sets of d-wffs, where an entailed set is allowed to contain more than one element. Intuitively speaking, a set of d-wffs X mc-entails a set of d-wffs Y just in case the hypothetical truth of all the d-wffs in X warrants the existence of at least one true d-wff in Y .
Mc-entailment in a language depends on the underlying logic of d-wffs of the language and/or their semantics. But IEL in its general form is neutral in the controversy as to what "The Logic" of declaratives is. Below we present a construction which outputs the semantic concept of mc-entailment from the proof-theoretic concept of consequence relation determined by a logic. The construction is not applicable to all cases, but its area of applicability is wide.

Consequence and Mc-entailment.
Let a logic , where is either Classical Logic or a non-classical logic, be the logic of the declarative part of L. In what follows we assume that is arbitrary but fixed. The logic determines the corresponding consequence relation among d-wffs of L. Let stand for the consequence relation determined by , satisfying, possibly among others, the standard conditions (cf. [15]): We additionally assume that is neither universal nor empty. We now define the notion of a -maximally consistent set. Let D L stand for the set of d-wffs of L, and In what follows we assume that has the following property:

Lb: For any set of d-wffs X of L and any d-wff
The property Lb can be called Lindenbaum feature 5 or simply Lb-feature.
Remark that Lb is a property of some, but not all consequence relations. In order to pass to the semantic level, we first introduce the concept of partition of the set D L of d-wffs of L; the idea comes from [15]. A partition 5 We use here the term "Lindenbaum feature", instead of the more common term "Lindenbaum property", for two reasons. First, we do not consider consequence operations, but consequence relations. Second, we would like to put a stress on the fact that the d-wffs that are not consequences of the "initial" set X are not consequences of the relevant maximal consistent extension(s) of X. (Thus Lb is akin to the concept of Lindenbaum property analysed in [23, pp. 92-93], defined in terms of relative maximality, but with respect to consequence operation and sets closed under the operation; cf. also [24, pp. 26-28].) of D L is an ordered pair T P , U P , where T P ∩ U P = ∅ and T P ∪ U P = D L . Intuitively, a partition divides d-wffs into "true" (i.e. belonging to T P ) and "untrue" (that is, elements of U P ). However, we are not interested in partitions which do it at random, but only in these partitions of D L which, to speak generally, comply with the logic of d-wffs by making the consequence relation, , truth preserving. We call these partitions admissible.
By assumption, has the Lb-feature. -admissible partitions are defined by: We get: Proof. (⇒) Assume that there exists an -admissible partition, P, of D L such that X ⊆ T P and A / ∈ T P . As P is an -admissible partition, we have Since, by assumption, has the Lb-feature, there exists an -maximally consistent set of d-wffs, Y , such that The class of admissible partitions of D L is defined by: We have assumed that has the Lb-feature. Given this, and due to Proposition 1, the following can be regarded as the semantic counterpart of the consequence relation .
Clearly, we have: The semantic relation = L of mc-entailment in L can now be defined by: Remark As it is common in MiES, we defined entailment and mc-entailment by referring to the class of admissible partitions of (the declarative part of) a language. However, the method of defining the class out of a consequence relation presented above has not been used in MiES so far. Instead, some other methods were employed (cf. [19], Chapter 3). Admissible partitions constructed in the way presented in this section are basically proof-theoretic entities. There is no accident in that. When one speaks about deduction theorems, he usually has in mind proof-theoretic results. Although IEL in its current form has a semantic bent, we want to stay as close as possible to proof theory.

Question Evocation and Erotetic Implication
We are now ready to introduce the concepts of question evocation and erotetic implication. There is no room for an extensive presentation (and discussion) of the underlying intuitions; an interested reader is advised to consult, e.g., [18], Chapter 1, or [19], Chapters 6 and 7. A question is sound iff at least one direct answer to the question is true. Generally speaking, the first clause of Definition 6 amounts to transmission of truth into soundness: if only X consists of truths, the question Q must be sound. The second clause amounts to the claim that no single direct answer to Q is entailed by X. The first clause of Definition 7 warrants the transmission of soundness and truth into soundness. The intuition that underlies the second clause is: each direct answer to an implied question narrows down, together with the respective set X, the class of "possibilities" offered by the whole set of direct answers to the implying question. Validity of erotetic inferences which do not involve declarative premises can be defined in terms of pure erotetic implication (pure e-implication for short). Formally, pure e-implication is erotetic implication based on the empty set of d-wffs.

The Method
We aim at generality here, so neither syntax nor semantics of a language considered will be described below in detail. In the consecutive sections we will be listing only these specific semantic and syntactic assumptions concerning language L which enable us to prove the consecutive lemmas and theorems. In other words, we will be indicating minimal specific assumptions on which the lemmas and theorems rely upon.

The Case of Question Evocation
Minimal assumptions In this section, as well as in Section 4, we assume that the vocabulary of language L includes the implication connective, →. We also assume that the following conditions are met: CD → : For each admissible partition P = T P , U P of the language: Comments Neither condition CQ 1→ nor condition CQ 2→ claim that a question can occur as an antecedent or as a consequent of an implication. The conditions only warrant the existence of the corresponding questions, where the correspondence takes place at the level of the form of direct answers. Given the generality of the approach presented in Section 2, it cannot be said that condition CD → always holds. There are languages of the considered kind in which the implication connective does not occur. More importantly, the declarative part of a language can be supplemented with a logic in view of which condition CD → does not hold. For instance, when we operate with Classical Propositional Logic, the condition holds, but Intuitionistic Propositional Logic provides a counterexample.
The minimal assumptions specified above perform the role of assumptions of lemmas and theorems presented in this section, and in Section 4. For brevity, however, these assumptions will not be explicitly listed each time. Instead, we will be using the label L1. When writing = L1 , we indicate that we are considering mc-entailment in a language fulfilling the above minimal assumptions, and similarly for E L1 , Im L1 , and Im L1 . By admissible partitions we will mean admissible partitions of the language just considered.

A Deduction Theorem for Question Evocation
For conciseness, we introduce the following notational convention: Let us stress that an inscription |C → Y | is not a metalanguage schema of a d-wff, but refers to a set of d-wffs.
One can prove: Proof. (⇒) Assume that X = / L1 |C → Y |. It follows that there exists an admissible partition, P, such that X ⊆ T P and |C → Y | ⊆ U P . Each element of the set |C → Y | is of the form C → A, where A ∈ Y . Therefore, by condition CD → , C ∈ T P and Y ⊆ U P . It follows that Thus for some admissible partition, P, we have X ⊆ T P , C ∈ T P , and Y ⊆ U P . Hence, by condition CD → , 'C → A' ∈ U P for every A ∈ Y . It follows that X = / L1 |C → Y |.
Note that Lemma 1 yields a deduction theorem for entailment: X ∪{C} |= L1 A iff X |= L1 C → A. Moreover, we get: Proof. By Lemma 1. Recall that, by assumption, dQ = ∅. As for the implication from left to right, the existence of question Q * is warranted by condition CQ 1→ . In the case of the implication from right to left, question Q exists due to condition CQ 2→ .
According to Theorem 1, a question Q is evoked by a set of d-wffs, Z, that includes a d-wff C if, and only if a "conditionalized" (with regard to C) counterpart of the question Q is evoked by the set Z\{C}.

A Reduction Theorem for Question Evocation
For brevity, let us introduce the following notational convention: By applying Theorem 1 n times we get: Thus question evocation by a finite non-empty set of d-wffs reduces, in a sense, to question evocation by the empty set. The price for the reduction is a switch from the "originally" evoked question to its conditionalized counterpart of the above form.

The Case of Erotetic Implication
Let us now turn to erotetic implication. The general assumptions we rely on in this section are these specified at the beginning of Section 3 above.
In order to prove a deduction theorem for erotetic implication we need: Thus there exists an admissible partition, P, such that X ⊆ T P , 'C → D' ∈ T P , C ∈ T P , and Y ∩ T P = ∅. Hence, by condition CD → , D ∈ T P and therefore X ∪ Thus for some admissible partition P we have: X ⊆ T P , D ∈ T P , and |C → Y | ∩ T P = ∅. By condition Proof. By Lemma 2 and the fact that, due to Lemma 1,

A Deduction Theorem for Erotetic Implication
The following holds: Proof. (⇒) The existence of questions Q * and Q * 1 is warranted by condition CQ 1→ . Recall that dQ = ∅ and dQ 1 = ∅.
Assume that Im L1 (Q, X ∪ {C}, Q 1 ). By clause (1) of Definition 7 and Lemma 3, Let B be an element of dQ 1 , and Y be a non-empty proper subset of dQ such that ( is a non-empty proper subset of dQ * . Therefore Im L1 (Q * , X, Q * 1 ). (⇐) If Im L1 (Q * , X, Q * 1 ), then, first, X ∪ {C → A} = L1 |C → dQ 1 | for any A ∈ dQ (note that questions Q and Q 1 exist since condition CQ 2→ holds). Thus, by Lemma 3, (X ∪ {C}) ∪ {A} = L1 dQ 1 for each A ∈ dQ. Second, Im L1 (Q * , X, Q * 1 ) yields that for each d-wff of the form C → B, where B ∈ dQ 1 , there exists a non-empty set Y of d-wffs of the form C → A, where A ∈ dQ, such that Y is a proper subset of dQ * and X ∪ {C → B} = L1 Y . Observe that Y equals |C → Y |, where Y ⊆ dQ and, as Y is a non-empty proper subset of dQ * , Y is a non-empty proper subset of dQ. By Lemma 3, According to Theorem 3, a question Q 1 is implied by a question Q on the basis of a set of d-wffs Z that includes a d-wff C if, and only if, a conditionalized (with regard to C) counterpart of the question Q 1 is implied by the respective conditionalized counterpart of the question Q on the basis of the set of d-wffs Z\{C}. Remark that both an implied question and an implying question has to be conditionalized in this way.

A Reduction Theorem for Erotetic Implication
As in the case of question evocation, the deduction theorem yields the corresponding reduction theorem.
Theorem 4. (Reduction 1 for erotetic implication) Let X = {C 1 , . . . , C n }, where n ≥ 1. Let C i 1 , . . . , C i n be a sequence without repetitions of all the elements of X. Then Im L1 (Q, X, Proof. By Theorem 3 and Definition 8. More precisely, we apply Theorem 3 n times, starting with C i n . Recall that {C 1 , . . . , Thus one can transform erotetic implication based on a finite non-empty set of d-wffs into pure e-implication, yet on the price of conditionalizing the implying question as well as the implied question. Since Theorem 4 has the form of a biconditional, a move in the other direction is also possible.

Another Reduction Theorem for Erotetic Implication
Belnap introduces in [1, p. 98] the concepts of "added-condition questions" and "added-conjunct questions." The former have direct answers generated by uniformly adding a condition to a given set of direct answers. An addedconjunct question, in turn, has direct answers which result from direct answers to some other question by adding to them the content of the respective "given-that" clause. 8 More precisely, Q * is an added-conjunct question iff for some d-wff A, dQ * = {A∧B : B ∈ dQ}, where Q is a question for which the following condition holds: there is no d-wff D such that each direct answer to Q has the form of a conjunction with D as a conjunct.
The deduction and reduction theorems presented above rely, among others, on condition CQ 1→ . What the condition means in view of Belnap's proposal is this: an added-condition question exists if the respective "basic" question exists. Let us now consider a similar yet different condition: CQ 1∧ : If C is a d-wff and Q is a question which is not an added-conjunct question, then there exists a question Q * such that dQ * = {C ∧ B : B ∈ dQ}.
Condition CQ 1∧ warrants the existence of an added-conjunct question based on a question which itself is not an added-conjunct question.
Let us now assume that the conjunction connective, ∧, occurs in L, and that, besides condition CQ 1∧ , the following conditions are met: CD ∧ : For each admissible partition P = T P , U P of the language: 'A ∧ B' ∈ T P iff A ∈ T P and B ∈ T P . Similarly as in the previous sections, we will be using a specific label, namely L2, to indicate that the results presented in this section rely on the above minimal assumptions. We adopt the following notational conventions: We use {C 1 , . . . , C n } for a conjunction of all the d-wffs in {C 1 , . . . , C n }. Needless to say, {C} = C. One can easily prove: The following is true: Theorem 5. (Reduction 2 for erotetic implication) Let X = {C 1 , . . . , C n }, where n ≥ 1. Let C i 1 , . . . , C i n be a sequence without repetitions of all the elements of X. Assume that neither Q nor Q 1 is an added-conjunct question.
Observe that Z is a non-empty proper subset of dQ iff | C i 1 , . . . , C i n ∧ Z| is a non-empty proper subset of dQ • . If questions Q and Q 1 exist, then, by condition CQ 1∧ , questions Q • and Q • 1 exist as well. If questions Q • and Q • 1 exist, then Q and Q 1 exist as well due to condition CQ 2∧ .
As previously, it cannot be said that the minimal assumptions specified above hold universally. However, Theorem 5 can be viewed as describing reduction results for logics in which implication does not fulfil condition CD → , but, nevertheless, conjunction behaves in the standard manner.

Summary and a Remark
Generally speaking, we have shown that, provided that some assumptions are met, question evocation is retained when: (a) removing a d-wff from an evoking set is accompanied by conditionalizing the evoked question by the removed d-wff, that is, forming the corresponding hypothetical question, (b) removing the condition of an evoked hypothetical question is accompanied by enriching the evoking set with the removed condition (cf. Theorem 1). Erotetic implication, in turn, is retained when: (a) removing an element from an implying set of d-wffs is accompanied by conditionalizing both the implying question and the implied question by the removed d-wff, (b) assuming that the implied and the implying questions share a condition, removing it is accompanied by enriching the implying set of d-wffs with the condition (cf. Theorem 3). Question evocation by a finite non-empty set of d-wffs coincides with evocation of the corresponding 'conditional' question(s) by the empty set (cf. Theorem 2), while erotetic implication based on a finite non-empty set of d-wffs is reducible to pure e-implication among the corresponding questions (cf. Theorems 4 and 5).
The final remark is this. Recall that the role played in IEL by the concepts of question evocation and erotetic implication resembles that played by entailment in a logic of statements. Given the analogy, it seems worthwhile to build, at the proof-theoretic level, formal systems whose theorems describe what questions are evoked by what sets of d-wffs, and what questions are implied by what questions on the basis of what sets of d-wffs. So far there exist systems of this kind for question evocation (for the classical propositional case only; see [16,21]), but there is no system for erotetic implication. The reduction theorems for erotetic implication presented above open a new perspective: in order to build a system for erotetic implication one may consider first-or even only-the conceptually (though not computationally) simpler case of pure e-implication. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.