So my question is situated in modal logic and everything is defined as usual. I'm reading volume 2 of logic, language, and meaning and on page 24 it says:
$V_{M,w_3}(\square p)= 1$
So the valuation function determined by the model makes the formula $\square p$ true at $w_3$. And this is their explanation: "since there are no worlds at all which are accessible from $w_3$ (so that p is true in all of the [nonexistent] worlds which are accessible)."
So they seem to be saying that if a world has no other world accessible from it a formula like $\square p$ is always true? Am I interpreting this right and if so then why is this true?
The keyword is vacuous truth
Since there is no accessible world where $p$ can be false, it is true in all accessible worlds. Thus it is necessary. $$V_{M,w_3}( \Box p )=1$$
On the other hand, since there is no accessible world where $p$ can be true, it is not true in any accessible world. Thus it is also not possible. $$V_{M,w_3}( \Diamond p )=0$$