Commit 7b92a6d7 by Ralf Jung

### docs: update semantic entailment

 ... @@ -98,20 +98,22 @@ We can now define \emph{semantic} logical entailment. ... @@ -98,20 +98,22 @@ We can now define \emph{semantic} logical entailment. \typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : \mProp} \typedsection{Interpretation of entailment}{\Sem{\vctx \mid \pfctx \proves \prop} : \mProp} \[ \Sem{\vctx \mid \pfctx \proves \prop} \eqdef \Sem{\vctx \mid \prop \proves \propB} \eqdef \begin{aligned}[t] \begin{aligned}[t] \MoveEqLeft \MoveEqLeft \forall n \in \mathbb{N}.\; \forall n \in \mathbb{N}.\; \forall \rs \in \textdom{Res}.\; \forall \rs \in \textdom{Res}.\; \forall \gamma \in \Sem{\vctx},\; \forall \gamma \in \Sem{\vctx},\; \\& \\& \bigl(\All \propB \in \pfctx. n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\rs)\bigr) n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\rs) \Ra n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\rs) \Ra n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\rs) \end{aligned} \end{aligned} \] The soundness statement of the logic reads The following theorem connects syntactic and semantic entailment: $\vctx \mid \pfctx \proves \prop \Ra \Sem{\vctx \mid \pfctx \proves \prop}$ $\vctx \mid \prop \proves \propB \Ra \Sem{\vctx \mid \prop \proves \propB}$ It now becomes trivial to show soundness of the logic. %%% Local Variables: %%% Local Variables: %%% mode: latex %%% mode: latex ... ...