The soundness statement of the logic reads as follows: For any $n$, we have
The consistency statement of the logic reads as follows: For any $n$, we have
\begin{align*}
\lnot(\TRUE\vdash (\upd\later)^n\FALSE)
\lnot(\TRUE\proves (\upd\later)^n\spac\FALSE)
\end{align*}
where $(\upd\later)^n$ is short for $\upd\later$ being nested $n$ times.
The reason we want a stronger consistency than the usual $\lnot(\TRUE\proves\FALSE)$ is our modalities: it should be impossible to derive a contradiction below the modalities.
For $\always$, this follows from the elimination rule, but the other two modalities do not have an elimination rule.
Hence we declare that it is impossible to derive a contradiction below any combination of these two modalities.
