Syntactic awareness in logical dynamics

Abstract

The paper develops an interface between syntax-based logical models of awareness and dynamic epistemic logic. The framework is shown to be able to accommodate a variety of notions of awareness and knowledge, as well as their dynamics. This, it is argued, offers a natural formal environment for the analysis of epistemic phenomena typical of multi-agent information exchange, such as how agents become aware of relevant details, how they perform inferences and how they share their information within a group. Technically, the logics presented are all simple refinements of the logic of public announcements.

Appendix

Proof of Theorem 1 (sketch)

Soundness is proved by showing that axioms are valid and rules preserve validity. Completeness is proved by showing that every L-consistent set of $\mathcal {L}$-formulas is satisfiable in a pointed $\mathbf{M}_K$-model. For the latter, construct the standard modal canonical model $M^c$ and, for each maximal L-consistent set of formulas $w$, define the propositional availability and acknowledgement set functions in the following way:

With these definitions, it is easy to show that the new type of formulas also satisfy the Truth Lemma, that is,

For the first case, availability of formulas, an induction aided by the axioms describing their behaviour does the work. The second case, that for acknowledgement formulas, is straightforward.

Thus, any L-consistent set of formulas $\Sigma$ can be extended into a maximal L-consistent set $\Sigma ^*$ that is satisfiable in $(M^c, \Sigma ^*)$. Moreover, by standard results on canonicity and modal correspondence (Chapter 4 of Blackburn et al. 2001), axioms T, 4 and 5 make the accessibility relations of the canonical model equivalence relations. Then completeness follows. $\square$

Proof of Proposition 2

For the validity of the formula, suppose $(M, w) \models {{\text {Imp}}_i}(\varphi \rightarrow \psi ) \wedge {{\text {Imp}}_i}(\varphi )$. Then $(M, w) \models \Box _i{(\varphi \rightarrow \psi )} \wedge \Box _i{\varphi }$ and hence $(M, w) \models \Box _i{\psi }$. But $(M, w) \models \Box _i{\,^{av(i)}(\varphi \rightarrow \psi )}$ holds too, so $(M, w) \models \Box _i{\,^{av(i)}\psi }$. Therefore, $(M, w) \models {{\text {Imp}}_i}(\psi )$.

For the validity of the rule, take any pointed semantic model $(M, w)$. Since $\varphi$ is a validity, $(M, w) \models \Box _i{\varphi }$. From $(M, w) \models {{\text {Ao}}_i}{(\varphi )}$ it follows that $(M, w) \models \Box _i{\,^{av(i)}\varphi }$. Hence $(M, w) \models \Box _i{\varphi } \wedge \Box _i{\,^{av(i)}\varphi }$, that is, $(M, w) \models {{\text {Imp}}_i}(\varphi )$. $\square$

Proof of Proposition 3

Suppose ${{\text {Ao}}_i}{(\varphi )}$ holds at some world in a given model. From Proposition 1 it follows that the agent is aware of all atoms in $\varphi$; from the same proposition it follows that she is also aware of every formula built from such atoms. In particular, she is aware of ${{\text {Ao}}_i}{(\varphi )}$.

$\square$

Proof of Proposition 4

Suppose $\lnot {{\text {Ao}}_i}{(\varphi )}$ holds at some world in a given model. From Proposition 1 it follows that the agent is not aware of at least one atom of $\varphi$; then from the same proposition it follows that she cannot be aware of any formula involving such atom. In particular, she cannot be aware of $\lnot {{\text {Ao}}_i}{(\varphi )}$.

$\square$

Proof of Proposition 7

By unfolding the definitions, the formula becomes

$$\begin{aligned} \Box _i{\,^{ack(i)}\varphi } \wedge \Box _i{\varphi } \;\;\rightarrow \;\; \Box _i{\,^{ack(i)}{{\text {Inh}}_i}(\varphi )} \,\wedge \, \Box _i{\Box _i{\,^{ack(i)}\varphi }} \,\wedge \, \Box _i{\Box _i{\varphi }} \end{aligned}$$

The first conjunct of the antecedent, the extra requirement and the K axiom yield the first conjunct of the consequent. Transitivity plus the first and second conjunct of the antecedent yield the second and third conjunct of the consequent, respectively. $\square$

Proof of Proposition 8

Suppose $\lnot {{\text {Inh}}_i}(\varphi ) \wedge {{\text {At}}_i}{(\varphi )}$; then $\lnot \Box _i{\varphi }$. This and euclideanity yields $\Box _i{\lnot \Box _i{\varphi }}$, which implies $\Box _i{\lnot (\Box _i{\,^{ack(i)}\varphi } \wedge \Box _i{\varphi })}$ or, shortening, $\Box _i{\lnot {{\text {Inh}}_i}(\varphi )}$. Moreover, $\lnot \Box _i{\varphi }$ and the extra requirement imply $\Box _i{\,^{ack(i)}\lnot {{\text {Inh}}_i}(\varphi ))}$. These two pieces yield ${{\text {Inh}}_i}(\lnot {{\text {Inh}}_i}(\varphi ))$.

$\square$

Proof of Proposition 9

By unfolding the definitions, the formula becomes

$$\begin{aligned} \Box _i{\,^{av(i)}\varphi } \wedge \Box _i{\varphi } \;\;\rightarrow \;\; \Box _i{\,^{av(i)}\big (\Box _i{(\,^{av(i)}\varphi \wedge \varphi )}\big )} \,\wedge \, \Box _i{\Box _i{\,^{av(i)}\varphi }} \,\wedge \, \Box _i{\Box _i{\varphi }} \end{aligned}$$

By Proposition 1, the first conjunct of the antecedent implies the first one of the consequent; by transitivity, the first and second conjunct of the antecedent imply the second and third ones of the consequent. $\square$

Proof of Proposition 10

Suppose $\lnot {{\text {Imp}}_i}(\varphi ) \wedge {{\text {Ao}}_i}{(\varphi )}$; then $\Box _i{\,^{av(i)}\varphi }$ and $\lnot \Box _i{\varphi }$. From the first and Proposition 1 it follows that $\Box _i{\,^{av(i)}(\lnot {{\text {Imp}}_i}(\varphi ))}$. From the second and euclideanity it follows that $\Box _i{\lnot \Box _i{\varphi }}$, which implies $\Box _i{\lnot (\Box _i{\,^{av(i)}\varphi } \wedge \Box _i{\varphi })}$ or, shortening, $\Box _i{\lnot {{\text {Imp}}}(\varphi )}$. These two pieces yield ${{\text {Imp}}_i}(\lnot {{\text {Imp}}_i}(\varphi ))$. $\square$

Proof of Proposition 11

By unfolding the definitions, the formula becomes

By Proposition 1, the first conjunct of the antecedent implies the first of the consequent. By transitivity, the first, second and third conjunct of the antecedent imply the second, third and fourth of the consequent, respectively. Finally, the extra condition, the K axiom and the third conjunct of the antecedent imply the fifth conjunct of the consequent. $\square$

Proof of Proposition 12

The antecedent of the implication implies $\Box _i{\,^{av(i)}\varphi }$, $\Box _i{\varphi }$ and $\lnot \Box _i{\,^{ack(i)}\varphi }$. From the first it follows that $\Box _i{\,^{av(i)}(\lnot {{\text {Exp}}_i}(\varphi ))}$ as before. From the first, second and third together with transitivity and euclideanity it follows that $\Box _i{\Box _i{\,^{av(i)}\varphi }}$, $\Box _i{\Box _i{\varphi }}$ and $\Box _i{\lnot \Box _i{\,^{ack(i)}\varphi }}$, that is, $\Box _i{\lnot (\Box _i{\,^{av(i)}\varphi } \wedge \Box _i{\varphi } \wedge \Box _i{\,^{ack(i)}\varphi })}$ or, shortening, $\Box _i{\lnot {{\text {Exp}}_i}(\varphi )}$. The third and the further assumption implies $\Box _i{\,^{ack(i)}\lnot {{\text {Exp}}_i}(\varphi )}$. These three pieces yield ${{\text {Exp}}_i}(\lnot {{\text {Exp}}_i}(\varphi ))$.

$\square$

Proof of Proposition 13

Similar to those of Propositions 9 and 10. $\square$

Proof of Proposition 15

Define $\sim _{\mathsf {PA}_i(w)}$ as in Definition 6; the desired implication is proved by a simple induction on the complexity of $\gamma$. The implication in the other direction does not hold because the fact that $\gamma$’s truth-value is uniform through $\sim _{\mathsf {PA}_i(w)}$ does not imply that every atom in $\gamma$ is in $\mathsf {PA}_i(w)$. For a simple counter-example, take a model in which the truth-value of a given $p$ is the same in all worlds, and pick an evaluation point $w$ in which $p \not \in \mathsf {PA}_i(w)$; the formula $\left[ \sim _{\mathsf {PA}_i(w)}\right] p \vee \left[ \sim _{\mathsf {PA}_i(w)}\right] \lnot p$ holds at $w$, but $\,^{av(i)}p$ does not. $\square$

Proof of Proposition 17

Pick any pointed semantic model $(M, w)$. The first property is straightforward: the operation puts $q$ in the $\mathsf {PA}_j$-set of every world in the model, so $\,^{av(j)}q$ and hence $\Box _j{\,^{av(j)}q}$ (i.e., ${{\text {Ao}}_j}{(q)}$) becomes true everywhere.

For the second one, the three ingredients for explicit knowledge are verified. After the inference operation the agent is aware of $\chi$ because the precondition of the operation says she was already aware of $\eta \rightarrow \chi$ and the operation does not remove atoms from the propositional availability sets; thus, $\Box _j{\,^{av(j)}\chi }$. Moreover, after the operation, $\chi$ is in the $\mathsf {AC}_j$-set of every world that already had $\eta \rightarrow \chi$ and $\eta$, so in particular it is in every world $R_j$-accessible from $w$ since the precondition of the operation requires that the implication and its antecedent were already there; thus, $\Box _j{\,^{ack(j)}\chi }$. Finally, because of the precondition, $\chi$ holds in every world $R_j$-accessible from $w$ in the original model $M$ (explicit knowledge is also implicit knowledge, which is closed under logical consequence). But the operation affects only the truth-value of formulas containing $\,^{ack(j)}\chi$; hence, $\chi$ itself cannot be affected and therefore it will still be true at every world $R_j$-accessible from $w$ in the resulting model $M_{{\overset{\eta \rightarrow \chi }{\hookrightarrow }_{j}}}$; thus, $\Box _j{\chi }$. Therefore, ${{\text {Exp}}_j}(\chi )$ holds at $w$ in $M_{{\overset{\eta \rightarrow \chi }{\hookrightarrow }_{j}}}$.

The third property is also straightforward, and again the three ingredients are verified. The operation guarantees that, after it, $\Box _i{(\,^{av(i)}\chi \wedge \,^{ack(i)}\chi )}$ will be true at $w$. Moreover, the new model contains only worlds that satisfied ${{\text {Exp}}_j}(\chi )$ in the original model $M$ and, by truthfulness, they all also satisfy $\chi$ in $M$. Now, propositional formulas depend just on the atomic valuation, which is not affected by the announcement operation; then the surviving worlds will still satisfy $\chi$ in the resulting model $M_{^{j}:\chi !}$. Hence, $\Box _i{\chi }$ is true at $w$ and therefore ${{\text {Exp}}_i}(\chi )$ is true at $w$ in $M_{^{j}:\chi !}$.

$\square$