 This truth-preserving translation, whose precise definition can be found in van Ditmarsch, van der Hoek, and Kooi (2008: Section 7.4), shows that the public announcement modality can also be seen as having a syntactic definition: any formula involving \([\chi{!}]\) can be rewritten within \(\cL_{\left\{ K  \right\}}\).[11]  Nevertheless, this is not a ‘one line’ translation, as it is the case, e.g., for the ‘everybody knows’ modality E. The translation is given by a recursive approach, with the modality defined in a different way depending on the formula one needs to place under its scope. This leads us to the second twist: because of this recursive definition, even though adding \([\chi{!}]\) does not increases the language’s expressivity, its addition does change the properties of the logical system. Indeed, in \(\cL_{\left\{ K, {!} \right\}}\), the rule of uniform substitution of atomic propositions by arbitrary formulas is not validity-preserving anymore. Consider the following formula, stating that “after the public announcement of p, the agent will know that p is the case”:   \[ \Vdash [p{!}]Kp. \]   The formula is valid: a truthful public announcement of p discards worlds from the original model M where p was not the case. Hence, the resulting \(M_{p{!}}\) will have only worlds satisfying p, thus making \(Kp\) true Now consider the formula below, which results from substituting p by \(p \land \lnot Kp\) in the previous validity:   \[ [(p \land \lnot Kp){!}] K(p \land \lnot Kp). \]   The above formula states now that 