modal logic - examples for it may be supposed that/it is compulsory that

Question

Could someone give me example with modal logic ? $\diamond X$ it may be supposed that
$\Box$ it is neccessary that.
I mean some example with worlds and arrows between them.
Why am I asking about it ? Simply, I can't understand it.

Answer 1

Here is some random Kripke model from Google. According to the semantics:

$(M,w) \models \square \varphi$ iff for any $v \in W: (w,v) \in R$ implies $(M,v) \models \varphi$

$(M,w) \models \diamond \varphi$ iff there is a $v \in W: (w,v) \in R$ and $(M,v) \models \varphi$

Informally, the first clause says that in every world, reachable from the given one, $\varphi$ holds. The second $-$ there is at least one world, reachable from the given one, where $\varphi$ holds. For example, $(M, w_3) \models \diamond \varphi$ (since there is a transition from $w_3$ to a $\varphi$-world, namely $w_3$), $(M, w_3) \not \models \square \varphi$ (since $\varphi$ is not true in $w_5$), $(M,w_5) \not \models \diamond \varphi$ (since there are no transitions), $(M, w_5) \models \square \varphi$ ($\square$ is vacuous here (see the implication in the definition)).