\begin{frame}{Satisfiability, Validity, Consistency}
  \begin{definition}[Satisfiability, validity of formulas]
    Let $\aform$ be a formula, and $\asetforms$ be a set of formulas.
    \begin{itemize}
      \medskip
      \item
        $\aform$ is \emph{satisfiable} $\iff$
        there is \structure{some} model $\model{\amodel}$ and\\ \structure{some} environment $\saluf$ 
        such that $\model{\amodel} \satisfieslookup{\saluf} \aform$.   
      \medskip
      \item
        $\aform$ is \emph{valid} $\iff$
        $\model{\amodel} \satisfieslookup{\saluf} \aform$
        holds for \structure{all} models $\model{\amodel}$ and\\ 
        \structure{all} environments $\saluf$
        in which $\aform$ can be checked.
        \vspace*{0.75ex}
      \medskip
      \item
        $\asetforms$ is \emph{consistent} or \emph{satisfiable} $\iff$
        there is \structure{some} model $\model{\amodel}$ 
        and \structure{some} environment $\saluf$ such that
        \begin{center}
          $\model{\amodel} \satisfieslookup{\saluf} \bform$ \;\;for all $\bform\in\asetforms$.
        \end{center}   
      \medskip
    \end{itemize}
  \end{definition}
\end{frame}