Working in modal logic, if $W=\{w\}$ for a model $M=(W, R, V)$, do we have that $wRw$?

Question

If our set of worlds $W$ is a singleton, that is $W=\{w\}$, can we automatically conclude that this world $w$ is accessible from itself?

Answer 1

Nope. Any relation between the worlds is acceptable as an accessibility relation ... including the empty relation (nothing sees anything). Now the empty relation does make for a rather silly logic, regardless of how many worlds there are - any statement of the form "$\Diamond\varphi$" is vacuously false at every world, and any statement of the form "$\Box\varphi$" is vacuously true at every world - but it is allowed.

That said, we're often interested only in frames satisfying some additional conditions, either explicitly graph-theoretic (e.g. "the accessibility relation is transitive") or logical (e.g. "the frame validates the sentence $p\rightarrow \Diamond p$"). In such a case a "silly" accessibility relation like the empty relation may indeed be ruled out. But we don't do that automatically.