Dynamic Epistemic Logic for Implicit and Explicit Beliefs

Abstract

Epistemic logic with its possible worlds semantic model is a powerful framework that allows us to represent an agent’s information not only about propositional facts, but also about her own information. Nevertheless, agents represented in this framework are logically omniscient: their information is closed under logical consequence. This property, useful in some applications, is an unrealistic idealisation in some others. Many proposals to solve this problem focus on weakening the properties of the agent’s information, but some authors have argued that solutions of this kind are not completely adequate because they do not look at the heart of the matter: the actions that allow the agent to reach such omniscient state. Recent works have explored how acts of observation, inference, consideration and forgetting affect an agent’s implicit and explicit knowledge; the present work focuses on acts that affect an agent’s implicit and explicit beliefs. It starts by proposing a framework in which these two notions can be represented, and then it looks into their dynamics, first by reviewing the existing notion of belief revision, and then by introducing a rich framework for representing diverse forms of inference that involve both knowledge and beliefs.

Appendix

Proposition 1 The fact that $R$ is a preorder is already given in both sides of the equivalence. Assuming this property, the remaining ones will be proved.

$\varvec{(\Longrightarrow )}$ For local connectedness, take any comparability class $C_{{R}}{(w)}$ and any $w_1, w_2$ in it. Since $R$ is a locally-well preorder, $\mathrm{Max }_R(\{ w_1, w_2 \})$ should be non-empty so it is either $\{ w_1 \}$ (i.e., $Rw_2w_1$), or else $\{ w_2 \}$ (i.e., $Rw_1w_2$) or else $\{ w_1, w_2 \}$ (i.e., $Rw_2w_1$ and $Rw_1w_2$); thus, local connectedness is obtained. For converse well-foundedness, proceed by contradiction. Suppose there is an infinite ${R}^{|}$-ascending chain $w_1w_2\ldots$; then ${R}^{|}w_iw_{i+1}$ for every $i \ge 1$. Such chain and ${R}^{|}$’s definition imply $Rw_iw_{i+1}$ for every $i \ge 1$; then reflexivity and transitivity imply $Rw_1w_i$ for every $i \ge 1$ and thus $\{ w_1, w_2, \ldots \} \subseteq C_{{R}}{(w_1)}$. Hence $\mathrm{Max }_R(\{ w_1, w_2, \ldots \})$ is empty, contradicting the definition of a locally-well preorder. Then, such infinite chain cannot exists, and $R$ is conversely well-founded.

$\varvec{(\Longleftarrow )}$ Proceed by contradiction: let $R$ be locally connected and conversely well-founded, and suppose there is a comparability class $C_{{R}}{(w)}$ and a non-empty $U \subseteq C_{{R}}{(w)}$ such that $\mathrm{Max }_R(U)=\varnothing$. Then, there is no $v$ in $U$ such that $Ruv$ for every $u \in U$, that is, for every $v$ in $U$ there is a $u \in U$ such that no $Ruv$. But then, since such $v$ and $u$ are in the same comparability class and $Ruv$ is not the case, local connectedness yields $Rvu$. Thus, for every $v \in U$ there is a $u \in U$ such that no $Ruv$ and $Rvu$, that is, every $v \in U$ is strictly below some $u \in U$. This produces an infinite ${R}^{|}$-ascending chain, contradicting converse well-foundedness. Hence, such $U$ and such $C_{{R}}{(w)}$ cannot exists, and therefore $R$ is a locally-well preorder.

Proposition 12 Following Proposition 1, it will be shown that if $\le$ is a locally connected and conversely well-founded preorder, so is $\le '$. A key observation is that $w \le ' u$ if and only if, in $M$, $w$ is a $\lnot \chi$-world or $u$ is a $\chi$-world.

• Reflexivity. Take any $w \in W$; by reflexivity of $\le$, $w \le w$. Now, if $w$ is a $\chi$-world in $M$, part (1) of the definition of $\le '$ implies $w \le ' w$. Otherwise, $w$ is a $\lnot \chi$-world in $M$; then part (2) of the definition implies $w \le ' w$.

• Transitivity. Suppose $w \le ' u$ and $u \le ' v$; then $w, u, v$ are in the same comparability class in $M$. Now, if $w$ is a $\chi$-world in $M$, then so is $u$ (otherwise there would not be a $\le '$-link from $w$ to $u$) and hence so is $v$ too; therefore, by part (1) of the definition, $w \le ' v$. Otherwise, $w$ is a $\lnot \chi$-world at $M$; then part (2) of the definition implies $w \le ' v$.

• Local connectedness. First it will be shown that for every $u_1, u_2$ in $W$, $u_1 \sim u_2$ if and only if $u_1 \sim ' u_2$.

  From left to right, suppose $u_1 \sim u_2$. Now, if $u_1 \le ' u_2$, then $u_1 \sim ' u_2$ and we are done. Otherwise, $u_1$ should be a $\chi$-world in $M$ and $u_2$ should be a $\lnot \chi$-world in $M$; this together with $u_1 \sim u_2$ implies $u_2 \le ' u_1$ by part (3) of the definition, and hence $u_1 \sim ' u_2$. From right to left, if $u_1 \sim ' u_2$, then $u_1 \le ' u_2$ or $u_2 \le ' u_1$. Assume the first case, $u_1 \le ' u_2$, and let us review the three possibilities. If $u_1 \le ' u_2$ holds because of part (1) of the definition of $\le '$, then $u_1 \le u_2$ and therefore $u_1 \sim u_2$. And if it is because of part (2), then $u_1 \le u_2$ and therefore $u_1 \sim u_2$. If it is because of part (3), then $u_1 \sim u_2$. In the three possibilities, the required $u_1 \sim u_2$ is obtained. The second case is analogous.

  Now, for local connectedness, take any $w \in W$ and pick $u_1, u_2$ in $C_{{\le '}}{(w)}$; then, by definition of $C_{{\le '}}{(w)}$, $w \sim ' u_1$ and $w \sim ' u_2$. By the just proved property, $w \sim u_1$ and $w \sim u_2$; by local connectedness of $\le$, $u_1 \sim u_2$ and then by the just proved property again, $u_1 \sim ' u_2$.

• Converse well-foundedness. The proof proceeds by contradiction. Suppose there is an infinite $\le '$-ascending chain. These worlds are either $\chi$ or $\lnot \chi$-worlds in the original model. Since the chain is infinite, there must be an infinite sub-chain of either $\chi$ or else $\lnot \chi$-worlds (an alternation from a $\chi$-world to a $\lnot \chi$-one is impossible because of the definition of $\le '$). But inside these areas, the new relation is the old one, contradicting the converse well-foundedness of $\le$. Then, such infinite chain cannot exists, and therefore $\le '$ is conversely well-founded.

Proposition 13 Again, following Proposition 1, it will be shown that if $\le$ and $\preccurlyeq$ are locally connected and conversely well-founded preorders, so is $\le '$.

• Reflexivity. Take any $(w,e) \in W'$. By reflexivity of $\le$ and $\preccurlyeq$, $w \le w$ and $e \preccurlyeq e$; then $w \le w$ and $e \approxeq e$ so $(w, e) \le ' (w, e)$ from part (2) of the definition of $\le '$.

• Transitivity. Suppose $(w_1, e_1) \le ' (w_2, e_2)$ and $(w_2, e_2) \le ' (w_3, e_3)$. From the definition of $\le '$, each one of these two inequalities has two possible reasons, and this produces four cases. One of them will be proved in detail; the other three can be proved in a similar way. Suppose that while $(w_1, e_1) \le ' (w_2, e_2)$ holds because of part (1) of the definition of $\le '$, $(w_2, e_2) \le ' (w_3, e_3)$ holds because of part (2). Then

  $$\begin{aligned} \begin{array}{llll} e_1 \prec e_2,&\quad w_1 \sim w_2,&\quad e_2 \approxeq e_3,&\quad w_2 \le w_3. \end{array} \end{aligned}$$

  By unfolding the definitions,

  $$\begin{aligned} \begin{array}{llll} e_1 \preccurlyeq e_2, e_2 \not \preccurlyeq e_1, &{}\quad \left\{ \begin{array}{l} w_1 \le w_2 \\ w_2 \le w_1 \end{array} \right. ,&\quad e_2 \preccurlyeq e_3, e_3 \preccurlyeq e_2,&\quad w_2 \le w_3. \end{array} \end{aligned}$$

  For the action part, $e_1 \preccurlyeq e_2$ and $e_2 \preccurlyeq e_3$ imply $e_1 \preccurlyeq e_3$. Also, $e_3 \not \preccurlyeq e_1$ for otherwise $e_2 \preccurlyeq e_3$ and $\preccurlyeq$’s transitivity would imply $e_2 \preccurlyeq e_1$, contradicting part of the assumptions. Then, $e_1 \prec e_3$. For the static part there are two possibilities. If $w_1 \le w_2$, then $w_2 \le w_3$ and $\le$’s transitivity imply $w_1 \le w_3$; hence $w_1 \sim w_3$. If $w_2 \le w_1$, then $w_2 \le w_3$ and $\le$’s local connectedness imply $w_1 \le w_3$ or $w_3 \le w_1$; hence $w_1 \sim w_3$. By putting the action and the static pieces together, part (1) of the definition implies $(w_1, e_1) \le ' (w_3, e_3)$.

• Local connectedness. First it will be shown that, for every $(w_1, e_1), (w_2, e_2)$ in $W'$, $w_1 \sim w_2$ and $e_1 \approx e_2$ if and only if $(w_1, e_1) \sim ' (w_2, e_2)$.

  $\varvec{(\Longrightarrow )}$ If $w_1 \sim w_2$ and $e_1 \approx e_2$, then $w_1 \le w_2$ or $w_2 \le w_1$, and $e_1 \preccurlyeq e_2$ or $e_2 \preccurlyeq e_1$. This produces four cases. Suppose, for example, $w_1 \le w_2$ and $e_2 \preccurlyeq e_1$. Now, if $e_1 \preccurlyeq e_2$ then $e_1 \approxeq e_2$; thus $(w_1, e_1) \le ' (w_2, e_2)$ by part (2) of the definition of $\le '$ and hence $(w_1, e_1) \sim ' (w_2, e_2)$. Otherwise, local connectedness of $\preccurlyeq$ yields $e_2 \prec e_1$, and $w_1 \le w_2$ yields $w_1 \sim w_2$; then, from part (1) of the definition, $(w_2, e_2) \le ' (w_1, e_1)$ and hence $(w_1, e_1) \sim ' (w_2, e_2)$. The other three cases can be proved in a similar way.

  $\varvec{(\Longleftarrow )}$ If $(w_1, e_1) \sim ' (w_2, e_2)$, then $(w_1, e_1) \le ' (w_2, e_2)$ or $(w_2, e_2) \le ' (w_1, e_1)$. In the first case, the definition of $\sim '$ implies either (1) $e_1 \prec e_2$ and $w_1 \sim w_2$, and hence $e_1 \approx e_2$ and $w_1 \sim w_2$, or else (2) $e_1 \approxeq e_2$ and $w_1 \le w_2$, hence $e_1 \approx e_2$ and $w_1 \sim w_2$. The second case is similar.

  To show local connectedness, take any $(w, e) \in W'$ and pick $(w_1, e_1), (w_2, e_2)$ in $C_{{\le '}}{(w,e)}$; then, by definition of $C_{{\le '}}{(w,e)}$, $(w, e) \sim ' (w_1, e_1)$ and $(w, e) \sim ' (w_2, e_2)$. Hence, by the just proved property, $w \sim w_1$, $e \approx e_1$, $w \sim w_2$ and $e \approx e_2$; by local connectedness of $\le$ and $\preccurlyeq$, $w_1 \sim w_2$ and $e_1 \approx e_2$ and then, by the just proved property again, $(w_1, e_1) \sim ' (w_2, e_2)$.

• Converse well-foundedness The proof proceeds by contradiction. Suppose there is an infinite $<'$-ascending chain $(w_1, e_1) <' (w_2, e_2) <' \cdots$. Consider the infinite chain $e_1, e_2, \ldots$. If there is an infinite number of pairs $e_i$ and $e_{i+1}$ for which the plausibility order is strict, that is, if $e_i \prec e_{i+1}$ happens infinitely often, then there is an infinite $\prec$-ascending chain in $E$, contradicting the converse well-foundedness of $\preccurlyeq$. On the other hand, if $e_i \prec e_{i+1}$ only happens finitely often, then from some moment on there are only $\preccurlyeq$-equally plausible worlds, that is, from some moment on, $e_i \approxeq e_{i+1}$ is the case. But then, from that moment on, $w_i <w_{i+1}$ is the case, that is, there is an infinite $<$-ascending chain in $W$, contradicting the converse well-foundedness of $\le$. Then, the infinite chain in $W'$ cannot exists, and hence $\le '$ is conversely well-founded.