Generic Absoluteness restricted to formulas with low complexity or to the class of forcings Ikegami and Schlicht proposed a principle, namely generic absoluteness, which is stated below using Hamkins' and Lowe's terminology:
Working in $ZF$:
(Generic Absoluteness): For all formulas in language of set theory
$$\diamondsuit\Phi\longrightarrow\Phi$$
Which means if something is forceable then is already true.
This principle is false as Woodin gave the following counterexample:
$$\psi\equiv~~~\forall A\subseteq \omega_1~~\text{there exists a random real over}~ L[A]$$
In the sense that both $\Psi$ and $\neg \Psi$ are forceable over $ZF$.
I am not sure about the complexity of $\Psi$, but it seems it is $\Pi_3$.

Now my questions are:

1) Is it consistent to have generic absoluteness restricted to formulas with complexity less than $\Pi_3$, or more precisely less than complexity of $\Psi$?


2) Is there any nontrivial reasonable class of forcing notions avoiding a counter example?


For example, as suggested by Schlicht:


Is there a model of $ZF$ such that generic absoluteness holds for adding arbitrary many Cohen reals?



Edit: The first problem is solved by Joel's answer.
 A: Your sentence $\psi$ is a local property (see my blog post Local properties in set theory), since we can check whether or not it is true by looking inside $V_{\omega+5}$ or so. So it has complexity $\Delta_2$. So one cannot have the principle for assertions at that level of complexity.
Meanwhile, if one goes down to $\Sigma_1$, then the axiom $\Diamond\varphi\to\varphi$ for sentences $\varphi$ is consistent with ZFC, since it follows from the maximality principle, which asserts $\Diamond\square\varphi\to\varphi$, since every forceable $\Sigma_1$ statement is automatically forceably necessary, since once it becomerue it remains true in all further extensions. My article 


*

*J. D. Hamkins, A simple maximality principle, J. Symbolic Logic, vol. 68, iss. 2, pp. 527-550, 2003.


explains why MP is relatively consistent with ZFC, and with a little strength, you can also allow real parameters.
In the case of $\Pi_1$ assertions $\varphi$, the implication $\Diamond\varphi\to\varphi$ is simply provable in ZF, since if a universal statement holds in some extension, then it is already true. 
