\begin{frame} \frametitle{Truth of Boxes: $\all \phi$ } \begin{goal}{$w \fc \all \phi$} The formula \;\;$\all \phi$ is true in world $w$\;\; if $\phi$ is true in all worlds $w'$ with $R(w,w')$. \end{goal} \pause \begin{block}{} As a formula:\quad $w \fc \all \phi \quad\DARKRED{\iff}\quad \myall{w'}{\big( R(w,w') \;\to\; w' \fc \phi \big)}$ \end{block} \pause \begin{exampleblock}{} \exampleA For example, \begin{talign} \mpause[1]{ w_1 &\mpause{\notfc} \all\lab{p} } & \mpause{ w_3 &\mpause{\fc} \all\lab{q} } & \mpause{ w_1 &\mpause{\fc} \all\lab{(q \vee r)} } & \mpause{ w_2 &\mpause{\fc} \all\lab{\bot} } \end{talign} \end{exampleblock} \pause\pause\pause\pause\pause\pause\pause\pause\pause \hint{Note that $\all\lab{\bot}$ holds only in worlds without outgoing edges!} \end{frame}