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.

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