\frametitle{Correctness Theorem}

  \begin{goal}{Soundness / Correctness}
      \phi_1,\ldots,\phi_n \;\vdash \; \psi 
      \phi_1,\ldots,\phi_n \;\models \; \psi 

    In a slogan: everything derivable is true.
    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.
    Thus truth in a model (valuation) is preserved under derivation.

    The syntactic deduction rules are \emph{correct} in the sense that
    it is not possible to derive \emph{false conclusions} from \emph{true premises}.
%             \mpause[1]{In a slogan: everything true is derivable.\\
%             {\small (Every semantical conclusion is also syntactically derivable.)}}