\begin{frame} \frametitle{Completeness Theorem} \begin{goal}{Completeness} \vspace{-1ex} \begin{talign} \phi_1,\ldots,\phi_n \;\models \; \psi \quad\Rightarrow\quad \phi_1,\ldots,\phi_n \;\vdash \; \psi \end{talign} \end{goal} \pause\bigskip \begin{block}{Explanation} \pause In a slogan: everything true is derivable. \bigskip\pause If $\psi$ follows semantically from $\phi_1,\ldots,\phi_n$,\\ then $\psi$ can be derived syntactically from premises $\phi_1,\ldots,\phi_n$. \bigskip\pause This means that the syntactic derivation rules are strong enough to derive every semantic conclusion. \bigskip\pause Thus the system is \emph{complete}: no more rules are needed. \end{block} \end{frame}