docs: plainly consistency

parent ce91f292
 ... @@ -432,13 +432,14 @@ The premise in \ruleref{upd-update} is a \emph{meta-level} side-condition that h ... @@ -432,13 +432,14 @@ The premise in \ruleref{upd-update} is a \emph{meta-level} side-condition that h The consistency 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*} \begin{align*} \lnot(\TRUE \proves (\upd\later)^n\spac\FALSE) \lnot(\TRUE \proves (\later)^n\spac\FALSE) \end{align*} \end{align*} where $(\upd\later)^n$ is short for $\upd\later$ being nested $n$ times. where $(\later)^n$ is short for $\later$ 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. 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. For $\always$ and $\plainly$, this follows from the elimination rules. Hence we declare that it is impossible to derive a contradiction below any combination of these two modalities. For updates, we use the fact that $\upd\FALSE \proves \upd\plainly\FALSE \proves \FALSE$. However, there is no elimination rule for $\later$, so we declare that it is impossible to derive a contradiction below any number of laters. %%% Local Variables: %%% Local Variables: ... ...
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