\begin{frame}{Modification of Environments} In order to determine the truth values for \alert{$\forall x$}- en \alert{$\exists x$}-formulas\\ we must be able to modify the interpretation of the variable \alert{$x$}. \begin{definition}[Modified Environment] Let $\saluf \funin \textbf{var} \to \model{\adomain}$ be an environment, and $\avarsynvar$ a variable. \smallskip By $\saluf[\avarsynvar\mapsto \forestgreen{a}]$ we denote the environment that interprets $\avarsynvar$ as $\forestgreen{a}$, and that for all other variables acts in the same way as $\saluf$: \begin{talign} \fap{\saluf[\avarsynvar\mapsto \forestgreen{a}]}{\bvarsynvar} = \begin{cases} \forestgreen{a} & \text{if } \bvarsynvar = \avarsynvar \\ \aluf{\bvarsynvar} & \text{if } \bvarsynvar \neq \avarsynvar \end{cases} \end{talign} \end{definition}\pause{} \begin{exampleblock}{} More modifications are possible. \smallskip For $\salufacc = \saluf[x\mapsto \forestgreen{a}][z\mapsto \forestgreen{b}]$ we have: \begin{talign} \alufacc{\bvarsynvar} = \fap{\saluf[x\mapsto \forestgreen{a}][z\mapsto \forestgreen{b}]}{\bvarsynvar} = \begin{cases} \forestgreen{a} & \text{if } \bvarsynvar = x \\ \forestgreen{b} & \text{if } \bvarsynvar = z \\ \aluf{\bvarsynvar} & \text{if } \bvarsynvar \neq x \text{ and } \bvarsynvar \neq z \end{cases} \end{talign} \end{exampleblock} \end{frame}