\begin{frame} \frametitle{Correctness Theorem} \begin{goal}{Soundness / Correctness} \vspace{-1ex} \begin{talign} \phi_1,\ldots,\phi_n \;\vdash \; \psi \quad\Rightarrow\quad \phi_1,\ldots,\phi_n \;\models \; \psi \end{talign} \end{goal} \pause\bigskip \begin{block}{Explanation} \pause In a slogan: everything derivable is true. \bigskip\pause If $\psi$ is syntactically derivable from $\phi_1,\ldots,\phi_n$,\\ then every valuation that makes $\phi_1,\ldots,\phi_n$ true, makes $\psi$ true. \bigskip\pause Thus truth in a model (valuation) is preserved under derivation. \bigskip\pause The syntactic deduction rules are \emph{correct} in the sense that it is not possible to derive \emph{false conclusions} from \emph{true premises}. \end{block} % \mpause[1]{In a slogan: everything true is derivable.\\ % {\small (Every semantical conclusion is also syntactically derivable.)}} \end{frame}