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\begin{frame}{Consistency and Syntactical Consistency}
We already know a \emph{semantical} notion of consistency.

\begin{definition}[reminder]
A set $\asetforms$ of formulas is \emph{consistent} (also: \emph{satisfiable})
if there is a model $\model{\amodel}$ and an environment $\saluf$
such that $\model{\amodel} \satisfies \aform$ for all $\aform\in\asetforms$:
\begin{talign}
\exists \model{\amodel}\; \exists \saluf\;\; \forall \aform\in \asetforms\;
(\, \model{\amodel} \satisfieslookup{\saluf} \aform \,)
\end{talign}
\pause
or equivalently
\begin{talign}
\exists \model{\amodel} \exists \saluf\;\; \model{\amodel} \satisfieslookup{\saluf} \asetforms
\end{talign}
\end{definition}
\pause\medskip

There is also a \emph{syntactical}\/ variant:
\begin{definition}
A set $\asetforms$ of formulas is \emph{syntactically consistent} if:
\begin{talign}
\asetforms & \derivesnot \formula{\false}
\end{talign}
(That is, there is no derivation of $\formula{\bot}$ from $\asetforms$.)
\end{definition}
\end{frame}