A Dynamic Logic of Interrogative Inquiry

Abstract

We propose a dynamic-epistemic analysis of the different epistemic operations constitutive of the process of interrogative inquiry, as described by Hintikka’s Interrogative Model of Inquiry (IMI). We develop a dynamic logic of questions for representing interrogative steps, based on Hintikka’s treatment of questions in the IMI, along with a dynamic logic of inferences for representing deductive steps, based on the tableau method. We then merge these two systems into a dynamic logic of interrogative inquiry which articulates a joint treatment of questions and inferences, providing thereby a unified framework representing the informational dynamics of interrogative inquiry. We provide sound and complete axiomatic systems for the three dynamic logics that we introduce, we compare our framework with existing approaches, and we finally propose several directions for further work.

Technical Appendix

1.1 A Epistemic Logic

We provide here the formal bases of epistemic logic.²⁵ We first define the language of epistemic logic $\mathcal{E}$ as follows:

Definition 31 (Epistemic language $\mathcal{E}$).

Let P be a countable set of atomic propositions. The epistemic language $\mathcal{E}$ is given by

$$\displaystyle{\varphi::= p\ \vert \ \neg \varphi \ \vert \ \varphi \wedge \varphi \ \vert \ K\varphi \quad \mbox{ where }p \in \mathsf{P}.}$$

In this language, formulas of the form $K\varphi$ are read as “the agent knows that $\varphi$”. We will write $\perp$ for $p \wedge \neg p$ and $\top$ for $\neg \perp$. We now define the notion of epistemic model:

Definition 32 (Epistemic model).

Let P be a countable set of atomic propositions. An epistemic model is a tuple $M =\langle W,\sim,V \rangle$ where:

• W is a non-empty set of worlds,

• $\sim \ \subseteq W \times W$ is a binary equivalence relation representing the epistemic indistinguishability relation of the agent,²⁶

• $V: W \rightarrow \mathcal{P}(\mathsf{P})$ is an atomic valuation function indicating the atomic propositions that are true at each world.

We refer to pairs (M, w), where M is an epistemic model and w is a world in M, as pointed epistemic models. The intuitive idea behind the use of the epistemic indistinguishability relation is the following: if w denotes the actual world and u is a world such that $u \sim w$, then this means that, given all what the agent knows, she cannot tell between w and u which one is the actual world. Finally, the epistemic language $\mathcal{E}$ is interpreted on epistemic models as follows:

Definition 33 (Semantics for $\mathcal{E}$).

Let $M =\langle W,\sim,V \rangle$ be an epistemic model. The semantics for the epistemic language $\mathcal{E}$ is given by

$$\displaystyle\begin{array}{rcl} M,w\models p& \mbox{ iff }& p \in V (w) {}\\ M,w\models \neg \varphi & \mbox{ iff }& \mbox{ not }M,w\models \varphi {}\\ M,w\models \varphi \wedge \psi & \mbox{ iff }& M,w\models \varphi \mbox{ and }M,w\models \psi {}\\ M,w\models K\varphi & \mbox{ iff }& \mbox{ for all }u\mbox{ such that }u \sim w\mbox{ we have }M,u\models \varphi. {}\\ \end{array}$$

The set of valid formulas of $\mathcal{E}$ on the class of epistemic models can be axiomatized using the following axiomatic system EL:

Definition 34 (Logic EL).

The logic EL is given by the following axiomatic system:

• 1. all classical propositional tautologies

• 2. $K(\varphi \rightarrow \psi ) \rightarrow (K\varphi \rightarrow K\psi )$

• 3. $K\varphi \rightarrow \varphi$

• 4. $K\varphi \rightarrow KK\varphi$

• 5. $\neg K\varphi \rightarrow K\neg K\varphi$

• 6. from $\varphi$ and $\varphi \rightarrow \psi$, infer ψ

• 7. from $\varphi$, infer $K\varphi$

Then, we have the following completeness result for EL with respect to the class of epistemic models:

Theorem 6 (Completeness for EL).

EL is strongly complete with respect to the class of epistemic models.

Proof.

See Blackburn et al. (2002). □ 

1.2 B Proofs of Propositions 1, 2 and Lemma 1

Proof (Proposition 1).

Assume that $M,w\models R\left (\bigwedge \varGamma \wedge \neg \gamma \right ) \wedge \mathsf{closed}(\varGamma,\gamma )$. By the structural property for semantic trees, we directly have that $\mathsf{T}(w) \in \mathsf{STrees(\mathcal{I})}_{\varGamma,\gamma }$. Since n _Γ, γ + 1 is the maximal number of branches of a semantic tree in $\mathsf{STrees(\mathcal{I})}_{\varGamma,\gamma }$, we have to show that for any $i \in [\![0,n_{\varGamma,\gamma }]\!]$, $\mathcal{B}_{i}(w)$ is either closed or empty. Let $i \in [\![0,n_{\varGamma,\gamma }]\!]$. Since $M,w\models \mathsf{closed}(\varGamma,\gamma )$, we have $M,w\models \mathsf{closed}(\mathcal{B}_{i})_{\varGamma,\gamma } \vee \mathsf{empty}(\mathcal{B}_{i})_{\varGamma,\gamma }$. If $M,w\models \mathsf{closed}(\mathcal{B}_{i})_{\varGamma,\gamma }$, this means that $M,w\models Br_{i}\gamma ' \wedge Br_{i}\neg \gamma '$ for some γ′ ∈ T(Γ, γ), and we get that $\mathcal{B}_{i}(w)$ is closed. If $M,w\models \mathsf{empty}(\mathcal{B}_{i})_{\varGamma,\gamma }$, this means that no element of T(Γ, γ) is in $\mathcal{B}_{i}(w)$, and we get that $\mathcal{B}_{i}(w)$ is necessarily empty. We conclude that $\mathsf{T}(w) \in \mathsf{STrees(\mathcal{I})}_{\varGamma,\gamma }$ and T(w) is closed. Now assume that $\mathsf{T}(w) \in \mathsf{STrees(\mathcal{I})}_{\varGamma,\gamma }$ and T(w) is closed. We directly have that $M,w\models R\left (\bigwedge \varGamma \wedge \neg \gamma \right )$. Then, since T(w) is closed, we have that for all $i \in [\![0,n_{\varGamma,\gamma }]\!]$, $\mathcal{B}_{i}(w)$ is either closed or empty, and thereby that $M,w\models \mathsf{closed}(\varGamma,\gamma )$. □ 

Proof (Proposition 2).

Assume that $M,w\models R\gamma$. Let χ ∈ ImpConf(γ). We want to show that $M,w \nvDash \chi$. Assume towards a contradiction that $M,w\models \chi$. Since $M,w\models R\gamma$, we have by the structural property of semantic trees that $\mathsf{T}(w) =\{ \mathcal{R},\mathcal{B}_{i}\}_{i\in \mathbb{N}}$ is a semantic tree with root γ. Since χ ∈ ImpConf(γ), this means that there exist $\gamma ' \in \mathcal{I}$ and $i \in \mathbb{N}$ such that (i) $\gamma ' \in \mathcal{B}_{i}$ and $\neg Br_{i}\gamma '$ is one of the conjuncts of χ or (ii) $\gamma '\notin \mathcal{B}_{i}$ and Br _i γ′ is one of the conjuncts of χ. Since we assumed that $M,w\models \chi$, this means that (i) $\gamma ' \in \mathcal{B}_{i}$ and $M,w\models \neg Br_{i}\gamma '$ or (ii) $\gamma '\notin \mathcal{B}_{i}$ and $M,w\models Br_{i}\gamma '$, which is a contradiction given the semantics of the operators Br _i . We conclude that $M,w\models \neg \chi$, and thereby that $R\gamma \rightarrow \neg \chi$ for χ ∈ ImpConf(γ) is a valid principle on the class of models TE. □ 

Proof (Lemma 1).

Let $\mathcal{T} =\{ \mathcal{R},\mathcal{B}_{i}\}_{i\in \mathbb{N}} \in \mathcal{P}(\mathcal{I})^{\mathbb{N}}$ s.t. $\mathcal{T}$ is not a semantic tree with root γ. Then, for every semantic tree $\mathcal{T}^{{\ast}} =\{ \mathcal{R}^{{\ast}},\mathcal{B}_{0}^{{\ast}},\ldots,\mathcal{B}_{n}^{{\ast}}\}$ with root γ, there exist $\gamma ' \in \mathcal{I}$ and $i \in \mathbb{N}$ such that (i) $\gamma ' \in \mathcal{B}_{i}^{{\ast}}$ and $\gamma '\notin \mathcal{B}_{i}$, or (ii) $\gamma '\notin \mathcal{B}_{i}^{{\ast}}$ and $\gamma ' \in \mathcal{B}_{i}$. We construct X as follows: for each semantic tree $\mathcal{T}^{{\ast}} =\{ \mathcal{R}^{{\ast}},\mathcal{B}_{0}^{{\ast}},\ldots,\mathcal{B}_{n}^{{\ast}}\}$ with root γ, (i) if there exists γ′ such that $\gamma ' \in \mathcal{B}_{i}^{{\ast}}$ and $\gamma '\notin \mathcal{B}_{i}$ we let $\neg Br_{i}\gamma ' \in X$, and (ii) if there exists γ′ such that $\gamma '\notin \mathcal{B}_{i}^{{\ast}}$ and $\gamma ' \in \mathcal{B}_{i}$ we let Br _i γ′ ∈ X. By construction, we have that X is a γ-tree impossible configuration such that (i) if $\neg Br_{i}\gamma ' \in X$ then $\gamma '\notin \mathcal{B}_{i}$ and (ii) if Br _i γ′ ∈ X then $\gamma ' \in \mathcal{B}_{i}$. □