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.