\begin{frame} \frametitle{Exercises} \begin{alertblock}{} Explain the meaning of \;\;$\models$\;\; and \;\;$\vdash$\;\; for predicate logic. \end{alertblock} \pause \begin{exampleblock}{} $\phi_1,\ldots,\phi_n \models \psi$ means that all models $\mathcal{M}$ and environments $\ell$ that make $\phi_1,\ldots,\phi_n$ true, also make $\psi$ true. \medskip \pause $\phi_1,\ldots,\phi_n \vdash \psi$ means $\psi$ is derivable using natural deduction starting from premises $\phi_1,\ldots,\phi_n$. \end{exampleblock} \pause \begin{alertblock}{} Explain soundness/correctness and completeness. \end{alertblock} \pause \begin{exampleblock}{} Soundness means that everything derivable by natural deduction is also semantically entailed: \begin{talign} \Gamma \vdash \phi \;\implies\; \Gamma \models \phi \end{talign} \pause Completeness means that the derivation rules are strong enough to derive everything that is semantically entailed: \begin{talign} \Gamma \models \phi \;\implies\; \Gamma \vdash \phi \end{talign} \end{exampleblock} \end{frame}