\begin{frame} \frametitle{Modal Logic Equivalence} \begin{goal}{We define \;\;$ \phi \equiv \psi $\;\; as} In every world $w$ in every Kripke model $\mathcal{M}$ \begin{talign} \mathcal{M},w \models \phi \;\riff \; \mathcal{M},w \models \psi \end{talign} \end{goal} \pause \begin{block}{Alternative definition of modal equivalence} \begin{malign} \phi \equiv \psi \;\;\;\;\riff \;\;\; \; \phi\models\psi \;\;\mbox{ \aemph{and} } \;\; \psi \models \phi \end{malign} \end{block} \pause \begin{exampleblock}{} \vspace{-3ex} \begin{eqnarray*} \mpause[1]{ \all \phi & \equiv & \neg \some \neg \phi } \\ \mpause{ \some \phi & \equiv & \neg \all \neg \phi } \\ \mpause{ \some \neg \phi & \equiv & \neg \all \phi } \\ \mpause{ \all( \phi \wedge\psi) & \equiv & \all \phi \wedge \all \psi } \\ \mpause{ \all( \phi \vee\psi) & \not\equiv & \all \phi \vee \all \psi } \\ \mpause{ \phi \vee \psi & \equiv & \neg \phi \to\psi \;\;\mpause{^{\hint{$\star$}}}} \end{eqnarray*} \vspace{-3ex} \mpause[7]{\hint{$^{\hint{$\star$}}$: all equivalences from propositional logic hold also modal !}} \end{exampleblock} \end{frame}