\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}