For $\Box$, do you read it as “for all states. . .”, that which one, maybe, cannot withdraw from?
Does this, $\Box$, not request proving of itself?
$ \Box P \leftrightarrow \neg \Diamond \neg P$
How do you use it in English?
If, for all states $P$, if and only if, not in some states, not $P$?
On
There are several ways to interpret modal symbols in the normal langage. The common one is the following : $\Box$ means "it is necessary" and $\Diamond$ means "it is possible". Thus in english the formula $$\Box P \iff \neg \Diamond\neg P$$ can be read "It is necessary to have $P$ if and only if it is not possible to not have $P$." I guess you will agree that makes sense.
You shouldn't quantify over all states, merely the accessible ones. The most precise translation of $\Box$ is "For all accessible states," and of $\Diamond$ is "For some accessible state." So "$\neg \Diamond \neg P$" means "There is no accessible state such that $\neg P$ holds," which is equivalent to saying "In every accessible state, $P$ holds" - which is "$\Box P$."
More colloquially, "$\Box$" and "$\Diamond$" are usually translated as "Necessarily" and "Possibly," respectively. So "$\neg\Diamond\neg P$" means "It is not possible that $P$ fails," or equivalently "$P$ is necessarily true" - i.e. "$\Box P$."
Keep in mind that the accessibility relation between states may have nonintuitive properties - e.g. maybe $\alpha$ is accessible from $\beta$ and $\beta$ is accessible from $\gamma$, but $\alpha$ is not accessible from $\gamma$.