Answering by Means of Questions in View of Inferential Erotetic Logic

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.

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 well-formed formulas (d-wffs) 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 two-element 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 d-wffs, 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 d-wffs which are true in P, and U _P is made up of all the d-wffs 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 d-wffs 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 non-empty 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 d-wffs. For example, when D _L is the set of well-formed formulas of Classical Propositional Calculus (CPC), a partition $\langle \mathbf{T}_{ \text{P}},\mathbf{U}_{\text{P}}\rangle$ is called admissible iff for some CPC-valuation 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 d-wffs and let A be a d-wff. 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 multiple-conclusion entailment (mc-entailment for short).²⁴ This is a relation between sets of d-wffs. Mc-entailment, ∣⊧, 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 mc-entails Y if there is no admissible partition in which X consists of truths and Y consists of untruths. In other words, mc-entailment between X and Y holds just in case the truth of all the d-wffs in X warrants the presence of some truth(s) among the elements of Y: whenever all the d-wffs in X are true in an admissible partition P, then at least one d-wff 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 d-wffs 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 non-empty proper subset Y of d Q such that $X \cup \{ B\}\mid \models Y$.

14.1.2 E-Scenarios as Labelled Trees

E-scenarios have been defined in Wiśniewski (2003) (cf. also Wiśniewski 2001, 2004a) as families of interconnected sequences of questions and d-wffs, the so-called erotetic derivations. In this paper, however, we give an equivalent definition in terms of trees. E-scenarios will be defined here as labelled trees, where the labels are d-wffs and questions.

Definition 4.

A finite labelled tree Φ is an erotetic search scenario for a question Q relative to a set of d-wffs X iff

• 1. The nodes of Φ are labelled by questions and d-wffs; they are called e-nodes and d-nodes, 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 d-node 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 d-node of Φ that precedes the d-node γ _d in Φ;

• 6. Each d-node of Φ has at most one immediate successor;

• 7. There exists at least one e-node of Φ which is different from the root;

• 8. For each e-node 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 e-node of Φ that precedes γ _e in Φ and B _i (1 ≤ i ≤ n) labels a d-node of Φ that precedes γ _e in Φ, and

  • An immediate successor of γ _e is either an e-node or is a d-node labelled by a direct answer to the question that labels γ _e , moreover

    • If an immediate successor of γ _e is an e-node, it is the only immediate successor of γ _e ,

    • If an immediate successor of γ _e is not an e-node, 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 e-scenario Φ can be defined as a question that labels an e-node of Φ which is different from the root and whose immediate successor is not an e-node. Paths of e-scenarios are construed in the standard manner; a branch is a maximal path which originates from the root. By d-wffs of a branch we mean the d-wffs which are labels of d-nodes of the branch, and similarly for questions.

The following holds:

Theorem 1 (Golden Path Theorem).

Let Φ be an e-scenario 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 e-scenario Φ has at least one branch ϕ such that:

• 1. Each d-wff 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) e-scenarios are not defined as trees and what is called a path of an e-scenario in the “old” setting corresponds to a branch of an e-scenario in the current setting.