diff --git a/docs/model.tex b/docs/model.tex index 1064ea8f5abcecd32a89ba65103025bc1e42b921..8767494c21c21c0b032dee6554538fe8890c1481 100644 --- a/docs/model.tex +++ b/docs/model.tex @@ -98,20 +98,22 @@ We can now define \emph{semantic} logical entailment. \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] \MoveEqLeft \forall n \in \mathbb{N}.\; \forall \rs \in \textdom{Res}.\; \forall \gamma \in \Sem{\vctx},\; \\& -\bigl(\All \propB \in \pfctx. n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\rs)\bigr) -\Ra n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\rs) +n \in \Sem{\vctx \proves \prop : \Prop}_\gamma(\rs) +\Ra n \in \Sem{\vctx \proves \propB : \Prop}_\gamma(\rs) \end{aligned} -The soundness statement of the logic reads -$\vctx \mid \pfctx \proves \prop \Ra \Sem{\vctx \mid \pfctx \proves \prop}$ +The following theorem connects syntactic and semantic entailment: +$\vctx \mid \prop \proves \propB \Ra \Sem{\vctx \mid \prop \proves \propB}$ + +It now becomes trivial to show soundness of the logic. %%% Local Variables: %%% mode: latex