The Dynamic Logic of Stating and Asking: A Study of Inquisitive Dynamic Modalities

Abstract

Inquisitive dynamic epistemic logic (IDEL) extends public announcement logic incorporating ideas from inquisitive semantics. In IDEL, the standard public announcement action can be extended to a more general public utterance action, which may involve a statement or a question. While uttering a statement has the effect of a standard announcement, uttering a question typically leads to new issues being raised. In this paper, we investigate the logic of this general public utterance action. We find striking commonalities, and some differences, with public announcement logic. We show that dynamic modalities admit a set of reduction axioms, which allow us to turn any formula of IDEL into an equivalent formula of static inquisitive epistemic logic. This leads us to establish several complete axiomatizations of IDEL, corresponding to known axiomatizations of public announcement logic.

Appendices

Appendix 1. Proof of the reduction law for E

To prove Proposition 7, we first need some lemmata. As a first step, we provide a characterization of the updated inquisitive state $\varSigma _a^\varphi (w)$ in terms of resolutions.

Lemma 2

Let $\varphi \in \mathcal {L}^\textsf {IDEL}$ and let $\mathcal {R}(\varphi )=\{\alpha _1,\dots ,\alpha _n\}$. Given any M and w:

Proof

Theorem 1 ensures that . Given the support clause for this implies $[\varphi ]_M=[\alpha _1]_M\cup \dots \cup [\alpha _n]_M$. Thus, we have:

$$\begin{aligned} \varSigma _a^\varphi (w)= & {} \varSigma _a(w)\cap [\varphi ]_M=\varSigma _a(w)\,\cap \,([\alpha _1]_M\cup \dots \cup [\alpha _n]_M)\\= & {} \bigcup _{1\le i\le n}\!\!(\varSigma _a(w)\cap [\alpha _i]_M)=\varSigma _a^{\alpha _1}(w)\cup \dots \cup \varSigma _a^{\alpha _n}(w) \end{aligned}$$

Since resolutions are declaratives and thus truth-conditional, we have by Fact 3 that . Thus, we have .    $\square$

We will also make use of the next lemma, stating that whenever the antecedent of an implication is a truth-conditional formula $\alpha$, the clause for implication can be simplified: $\alpha \rightarrow \psi$ is supported at s iff $\psi$ is supported at the state $s\cap |\alpha |_M$.

Lemma 3

If $\alpha \in \mathcal {L}^\textsf {IDEL}$ be truth-conditional. Then for any M, s, and $\psi$:

$M,s\,\models \,\alpha \rightarrow \psi \iff M,s\cap |\alpha |_M\,\models \,\psi$

Proof

If $\alpha$ is truth-conditional, then the subsets of s that support $\alpha$ are all and only the subsets of $s\cap |\alpha |_M$. Using this fact, the claim follows straightforwardly by the persistence of support.    $\square$

Finally, we will make use of the following equivalence, which can be established simply by spelling out the support conditions for the two formulas.

Lemma 4

For any $\varphi ,\psi ,\chi \in \mathcal {L}^\textsf {IDEL}$,

Proof of Proposition 7. Let $\mathcal {R}(\varphi )=\{\alpha _1,\dots ,\alpha _n\}$. First, notice that, by Lemma 2, the information states $s\in \varSigma _a^\varphi (w)$ are all and only those of the form $s=t\cap |\alpha _i|_M$ for some $t\in \varSigma _a(w)$ and some $\alpha _i\in \mathcal {R}(\varphi )$. Since $\alpha _i\in \mathcal {R}(\varphi )$, it follows from Theorem 1 that $|\alpha _i|_M\subseteq |\varphi |_M$, whence $t\cap |\alpha _i|_M=t\cap |\alpha _i|_M\cap |\varphi |_M$.

Now suppose that $M,w\,\models \,\varphi$, so that w survives in the updated model $M^\varphi$. Making use of these facts, of Theorem 1, Lemmas 3 and 4, we have:

Finally, using this equivalence we get, for any model M and world w:

$$\begin{aligned} M,w\,\models \,[\varphi ]E_a\psi\iff & {} M,w\,\models \,\varphi \text { implies }M^\varphi ,w\,\models \,E_a\psi \\\iff & {} M,w\,\models \,\varphi \text { implies }M,w\,\models \,E_a(\varphi \rightarrow [\varphi ]\psi )\\\iff & {} M,w\,\models \,\varphi \rightarrow E_a(\varphi \rightarrow [\varphi ]\psi ) \end{aligned}$$

We have thus proved that $[\varphi ]E_a\psi$ and $\varphi \rightarrow E_a(\varphi \rightarrow [\varphi ]\psi )$ have the same truth-conditions. Since both formulas are declaratives, and thus truth-conditional by Proposition 1, this ensures that these formulas are equivalent.   $\square$

Appendix 2. Proof of completeness via !Comp and !Mon

Proof of Theorem 4. The proof is analogous to the one in Sect. 7.4 of [10] for PAL. We only provide a proof sketch. We first define a complexity measure as follows:

• $c(p)=c(\bot )=1$

• 

• $c(K_a\varphi )=c(E_a\varphi )=1+c(\varphi )$

• $c([\varphi ]\psi )=(4+c(\varphi ))\cdot c(\psi )$

By recursion on this notion of complexity, we define a map $(\cdot )^*:\mathcal {L}^\textsf {IDEL}\rightarrow \mathcal {L}^\textsf {IEL}$:

We can then easily prove $\varphi \dashv \vdash _{\textsf {IDEL}^{\textsf {!Comp}}}\varphi ^*$, using the reduction rules and !Comp. Completeness then follows since $\vdash _{\textsf {IDEL}^{\textsf {!Comp}}}$ includes a complete system for IEL.   $\square$

Proof of Theorem 5. The proof is similar to the previous one, and to the proof of the analogous result for PAL in [14]. We modify the above definition of $(\cdot )^*$ by setting $([\varphi ][\psi ]\chi )^*=([\varphi ]([\psi ]\chi )^*)^*$. By induction on the complexity of a formula (as defined above), we show that (i) $\varphi ^*$ is well-defined; (ii) if $\varphi \not \in \mathcal {L}^\textsf {IEL}$, then $c(\varphi ^*)<c(\varphi )$; and (iii) $\varphi \dashv \vdash _{\textsf {IDEL}^{\textsf {!Mon}}}\varphi ^*$. The only case which is not straightforward is the inductive step for a formula $[\varphi ][\psi ]\chi$, which I will spell out in detail.

For (i), notice that since $[\psi ]\chi$ is less complex than $[\varphi ][\psi ]\chi$, by induction hypothesis we have that $([\psi ]\chi )^*$ is well-defined and less complex than $[\psi ]\chi$. It follows that $[\varphi ]([\psi ]\chi )^*$ is less complex than $[\varphi ][\psi ]\chi$. So, the induction hypothesis implies that $([\varphi ]([\psi ]\chi )^*)^*$ is well-defined, i.e., that $([\varphi ][\psi ]\chi )^*$ is well-defined.

For (ii), as both $[\psi ]\chi$ and $[\varphi ]([\psi ]\chi )^*$ are less complex than $[\varphi ][\psi ]\chi$, using the induction hypothesis we have $c(([\varphi ][\psi ]\chi )^*)=c(([\varphi ]([\psi ]\chi )^*)^*)<c([\varphi ]([\psi ]\chi )^*)<c([\varphi ][\psi ]\chi )$. So, $c(([\varphi ][\psi ]\chi )^*)<c([\varphi ][\psi ]\chi )$.

For (iii), as $[\psi ]\chi$ is less complex than $[\varphi ][\psi ]\chi$, the induction hypothesis gives $[\psi ]\chi \dashv \vdash _{\textsf {IDEL}^{\textsf {!Mon}}}([\psi ]\chi )^*$. By two applications of the rule !Mon we get $[\varphi ][\psi ]\chi \dashv \vdash _{\textsf {IDEL}^{\textsf {!Mon}}}[\varphi ]([\psi ]\chi )^*$. Now, since $[\varphi ]([\psi ]\chi )^*$ is less complex than $[\varphi ][\psi ]\chi$, the induction hypothesis applies, and gives $[\varphi ]([\psi ]\chi )^*\dashv \vdash _{\textsf {IDEL}^{\textsf {!Mon}}}([\varphi ]([\psi ]\chi )^*)^*$. Putting things together, we have obtained $[\varphi ][\psi ]\chi \dashv \vdash _{\textsf {IDEL}^{\textsf {!Mon}}}([\varphi ]([\psi ]\chi )^*)^*$ which is what we need, since by definition $([\varphi ]([\psi ]\chi )^*)^*=([\varphi ][\psi ]\chi )^*$.   $\square$