I should derive $\lozenge$ from the semantics for $\square, \neg$ with classical reasoning at the meta-level:
Using the definitions:
$v_{\mathcal{M}}(\square F,w) = 1 \text{ if } \forall u (wRu \Rightarrow v_{\mathcal{M}}(F,u) = 1)$ and 0 otherwise
$v_{\mathcal{M}}(\lozenge F,w) = 1 \text{ if } \exists u (wRu \text{ and } v_{\mathcal{M}}(F,u) = 1)$ and 0 otherwise
Starting with:
$\forall u (wRu \Rightarrow v_{\mathcal{M}}(F,u) = 1)$ 1. definition of $\square$
$\neg \forall u (wRu \Rightarrow \neg v_{\mathcal{M}}(F,u) = 1)$ 2. introducing $\neg$
$\neg \forall u (\neg wRu \text{ or } \neg v_{\mathcal{M}}(F,u) = 1)$ 3. using $p \Rightarrow q \Leftrightarrow \neg p\lor q$
$ \exists u ( wRu \text{ and } v_{\mathcal{M}}(F,u) = 1)$ 4. solving $\neg$
Is that correct reasoning on the meta-level? I am not sure but I think my Professor mentioned that this procedure should take 8 steps. Am I missing something?
What you need to show is that: when $\nu_\mathcal M(\square F, w)=1$ then $\nu_\mathcal M(\lozenge\lnot F, w)=0$.
Start by making the assumptions that $\nu_\mathcal M(\square F, w)=1$ and $\nu_\mathcal M(\lozenge\lnot F, w)=1$. Next apply the definitions and thus show that these assumptions are contradictory.