\begin{frame} \frametitle{Examples} \begin{alertblock}{} Give a counter-model for: \begin{talign} &\myall{x}{\myex{y}{ R(x,y)}}, \\ &\myall{x}{\myall{y}{\myall{z}{\big(R(x,y) \wedge R(y,z) \to R(x,z)\big)}}} \\ &\quad \models\;\; \myex{x}{R(x,x)} \end{talign} \end{alertblock} \pause\bigskip What do the premises it mean? \begin{enumerate}[(a)] \pause \item Every object has a successor. \pause \item The successor-relation is transitive. \\\pause Hence any $n$-step successor is an immediate successor. \end{enumerate} \pause What does the conclusion mean? \begin{itemize} \pause \item There is an object that is its own successor. \end{itemize} \pause\bigskip \begin{alertblock}{} Can there be finite counter-models? \end{alertblock} \pause No, because by (a) there would by cycles, and by (b) every element on a cycle would be its own successor. \vspace{10cm} \end{frame}