\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}