Accessibility relation

Summary

An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a possible world $w$ can depend on what is true at another possible world $v$, but only if the accessibility relation $R$ relates $w$ to $v$. For instance, if $P$ holds at some world $v$ such that $wRv$, the formula $\Diamond P$ will be true at $w$. The fact $wRv$ is crucial. If $R$ did not relate $w$ to $v$, then $\Diamond P$ would be false at $w$ unless $P$ also held at some other world $u$ such that $wRu$.