\begin{frame}
   \frametitle{Formal Definition of Truth in Worlds}

  In Kripke model $\mathcal{M} = (W, R, L)$ we first define \emph{truth} \aemph{per world}.
  \bigskip\pause

  \begin{goal}{Definition of $\mathcal{M},x \fc \phi$} 
    \vspace{-2ex}
    \begin{eqnarray*}
      x \notfc \bot &&\\
      \mpause[1]{ x \fc p & \riff &  p\in L(x) } \\
      \mpause{ x \fc \neg \phi & \riff &  x\notfc \phi } \\
      \mpause{ x \fc  \phi \wedge \psi & \riff &  x \fc \phi  \;\;\mbox{and} \;\; x\fc \psi } \\
      \mpause{ x \fc  \phi \vee \psi & \riff &  x \fc \phi  \;\;\mbox{or} \;\; x\fc \psi } \\
      \mpause{ x \fc  \phi \to \psi & \riff & \mbox{if} \;\; x \fc \phi  \;\;\mbox{then also} \;\; x\fc \psi } \\
      \mpause{ x \fc \some \phi  &\riff& \mbox{there exists $y \in W$ with $R(x,y)$ and $y\fc\phi$} } \\
      \mpause{ x \fc \all \phi  &\riff& \mbox{for all $y\in W$ with $R(x,y) $ holds:\; $y\fc\phi $}}
    \end{eqnarray*}
    \vspace{-2ex}
  \end{goal}
  \mpause{
  \hint{Note the analogy with the truth definition in predicate logic.}
  }
\end{frame}