diff --git a/content/normal-modal-logic/filtrations/euclidean-filtrations.tex b/content/normal-modal-logic/filtrations/euclidean-filtrations.tex index 162b2e26..e3dc71da 100644 --- a/content/normal-modal-logic/filtrations/euclidean-filtrations.tex +++ b/content/normal-modal-logic/filtrations/euclidean-filtrations.tex @@ -105,6 +105,8 @@ \begin{proof} \begin{enumerate} + \item Exercise. Use the fact that \Ax{B} and $\Ax{B_\Diamond}$ are + valid in all symmetric models. \item If $\mModel{M^*}$ is a coarsest filtration, then by definition $R^*[u][v]$ holds if and only if $C_1(u,v)$. For transitivity, suppose $C_1(u,v)$ and $C_1(v,w)$; we have to show @@ -121,8 +123,6 @@ $\mSat{M}{\Diamond!A}[u]$.}{} \item Exercise. Use the fact that both \Ax{5} and $\Ax{5_\Diamond}$ are valid in all euclidean models. - \item Exercise. Use the fact that \Ax{B} and $\Ax{B_\Diamond}$ are - valid in all symmetric models. \end{enumerate} \end{proof}