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\begin{frame}
  \frametitle{Semantic and Syntactic Reasoning}

  \begin{goal}{Semantic implication (or semantic entailment)}
    \vspace{-1ex}
    \begin{talign}
      \phi_1,\ldots,\phi_n \;\models \; \psi
    \end{talign}
    means
    \begin{center}
      Every valuation that makes $\phi_1,\ldots,\phi_n$ true,\\
      also makes $\psi$ true.
    \end{center}
  \end{goal}
  (Recall, valuation is interpretation of the propositional letters.)
  \medskip\pause
  
  \begin{goal}{Syntactic derivability}
    \vspace{-1ex}
    \begin{talign}
      \phi_1,\ldots,\phi_n \;\vdash \; \psi
    \end{talign}
    means
    \begin{center}
      There is a natural deduction derivation of $\psi$ \\ starting from premises $\phi_1,\ldots,\phi_n$.
    \end{center}
  \end{goal}
\end{frame}