Commit 7b92a6d7 authored by Ralf Jung's avatar Ralf Jung
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docs: update semantic entailment

parent 8bfac1ad
...@@ -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.
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