Back when I was first learning about forcing and trying to understand the need to consider *generic* filters, I came up with the following question. Suppose we have a countable transitive model $M$. Let's say that "$p$ pseudoforces $\phi$" if for every filter (not necessarily generic) $G\in P$, $p\in G$ implies that $\phi$ is true in $M[G]$. Is pseudoforcing definable in $M$? I'll allow a little wiggle room about what $M[G]$ means when $G$ is not generic, but I suspect that the answer is no, regardless. Is that correct?

The reason I was led to ask this was that one intuitive justification for why forcing (as standardly defined) is definable in $M$ is that one doesn't need to know anything "specific" about any particular $G$ to decide whether $p$ forces $\phi$. But this level of handwaving would seem to apply to pseudoforcing as well, so I think would be illuminating to understand exactly how genericity comes into play here.

**EDIT** in response to Joel David Hamkins's request that I clarify what I mean by $M[G]$ when $G$ is not generic: I'm going to forget about Boolean-valued models and follow the approach in Kunen's textbook. We have an arbitrary poset $P$ in our countable transitive model $M$ of ZFC. We take the usual definition of a $P$-name: $\tau$ is a $P$-name if and only if $\tau$ is a relation and for all $\langle \sigma,p\rangle \in \tau$, $\sigma$ is a $P$-name and $p\in P$. Next we have define how to evaluate $\tau$ at $G\subseteq P$, but again we can just use the standard definition:
$$\tau_G = \{ \sigma_G \mid \exists p\in G : \langle\sigma,p\rangle \in \tau\}.$$
Then $M[G]$ is defined to be the set of all $\tau_G$ as $\tau$ ranges over all $P$-names in $M$. This is already a non-trivial construction because Kunen shows that if $G$ is *any* nonempty filter then $M[G]$ satisfies Extensionality, Foundation, Pairing, and Union.

Now it seems to me that I can define $p \mathrel{?\mathord{\vdash}} \phi$ (read "$p$ pseudoforces $\phi$") analogously to $p\Vdash\phi$ simply by dropping the word "generic" from the definition—instead of "for all generic filters $G$" we say "for all filters $G$." What goes wrong? My guess, based on something Andreas Blass once told me, is that we run into trouble when we try to prove the definability of pseudoforcing.

A related question is this. Kunen proves two crucial facts about forcing; (1) it's definable, and (2) every $\phi$ that is true in $M[G]$ is forced by some $p\in G$. Suppose I formulate the conjecture that these two facts are also true of pseudoforcing. Could I then deduce that $M[G]$ satisfies ZFC from this conjecture? Since the conjecture is false (see Goldstern's comment about part (2)), the answer to this question is yes for trivial reasons, but what I'm trying to get at is whether the genericity of $G$ is primarily needed in order to prove these two crucial facts, and that the rest of the proof that $M[G]$ satisfies ZFC follows "formally" from them.