Genericity is still a little bit mysterious to me, although not as much as it used to be.

Here is a rough paraphrase of Theorem 3.5 of Kunen's *Set Theory: an Introduction to Independence Proofs*

Let $M$ be a countable transitive model, $\langle\mathbb P,\leq\rangle\in|M|$ a partial order and $G\subseteq{\mathbb P}$ $M$-generic. For any $\phi$ it is the case that $M[G]\vDash\phi$ if and only if $(\exists p\in G) p\Vdash\phi$.

I will refer to the right-to-left direction of this implication as "things which are forced are true in the extension".

Assume that $\mathbb P$ is splitting and we've chosen some $\phi$ such that $M\nvDash\phi$ and there is some $p\in\mathbb P$ such that $p\Vdash\phi$. Let $\hat p$ be the principal $\mathbb P$-filter generated by $p$. Among the $M$-generic filters $G$, those for which $M[G]\vDash\phi$ are exactly those for which $\hat p\subseteq G$.

But what about $\hat p$ itself? Well, all the principal filters of $\mathbb P$ are sets of the ground model, so $\hat p\in|M|$. So forcing with $\hat p$ will give us back an extension model which is isomorphic to $M$ -- no new sets. So $M[\hat p]\nvDash\phi$.

So, if I haven't screwed up so far, we know that for generic filters "things which are forced are true in the extension", and for filters which happen to be elements of $|M|$ it is not necessarily the case that "things which are forced are true in the extension". This leaves a gap: filters which are not elements of $|M|$, yet are not $M$-generic either.

**Question:** does "things which are forced are true in the extension" hold for $\{any,all\}$ filters $F\supseteq\hat p$ such that $F\notin|M|$ yet $F$ is also not $M$-generic?

Phrased another way: does the right-to-left implication rely on the filter being generic, or only on the fact that it isn't an $M$-set?

I can see quite clearly why the left-to-right implication ("things which are true in the extension are forced") relies on the fact that the filter is generic, and couldn't possibly work without "intersects every dense set". I'm having a harder time seeing why genericity (rather than simply "not in $|M|$") is necessary in the other direction, though. Kunen uses it in his proof (the case where $\phi$ is $\tau_1=\tau_2$), but it's not as easy to see why (or if) the use is essential.

This isn't a homework question (or even close to any of them). Hope you'll trust me.