Kripke Models - evaluating the meaning of $\Box\Box p$
This is an unfortunate use of the word &ldquo;reachable&rdquo;, in that Kripke structures are graphs but &ldquo;reachable&rdquo; in a Kripke structure is not the same thing as &ldquo;reachable&rdquo; in a graph.  Let us avoid this confusion by saying that a state&nbsp;$y$ is a successor of state&nbsp;$x$ if there is a (directed) edge $(x,y)$ in the structure.

Now, the semantics of modal logic says that $\Box\varphi$ is true at state&nbsp;$x$ if, and only if, $\varphi$ is true at every successor of&nbsp;$x$. Informally, $\Box\varphi$ means, &ldquo;$\varphi$&nbsp;is true everywhere I can get to from here in one step.&rdquo; So, to understand the meaning of $\Box\Box p$, just substitute $\Box p$ for&nbsp;$\varphi$:


$\Box p$ is true at every successor
$p$ is true at every successor of every successor.


In other words, $p$&nbsp;is true everywhere I can get in two steps.  Note that this is not, in general, the same thing as $\Box p$, which means that $p$&nbsp;is true after one step. Indeed, one can show that $(\Box \varphi)\rightarrow (\Box\Box\varphi)$ is a tautology in a particular Kripke frame if, and only if, the successor relation is transitive.

Note that, without assuming transitivity, basic modal logic with only $\Box$ and&nbsp;$\Diamond$ has no way of expressing &ldquo;$\varphi$&nbsp;is true everywhere that can be reached from here, in any number of steps&rdquo; (i.e., the usual graph-theoretic meaning of &ldquo;reachable&rdquo;).