Perhaps the solution lies in the Levy absoluteness theorem.

We define the Levy hierarchy of formulas as follows.

1. $\Sigma_0,\Pi_0,\Delta_0$ are formulas in the language of set theory which only have bounded quantifiers.

2. $\Sigma_{n+1}$ formulas equivalent to formulas of the form $\exists x\varphi$, where $\varphi$ is $\Pi_n$; and similarly $\Pi_{n+1}$ are formulas equivalent to $\forall x\varphi$ where $\varphi$ is $\Sigma_n$.

3. $\Delta_n$ are formulas which are both $\Sigma_n$ and $\Pi_n$.

Now we have that if $M$ is a transitive class (set or otherwise), then $\Delta_0$ formulas are absolute between $M$ and $V$ (as long as $M$ includes all the parameters). Next we have that $\Sigma_1$ formulas are upwards absolute and $\Pi_1$ sentences are downwards formulas, so $\Delta_1$ formulas are absolute.

And so we have that $\Sigma_2$ formulas are upwards absolute and $\Pi_2$ formulas are downwards absolute.

Perhaps the solution lies in the Levy absoluteness theorem.

We define the Levy hierarchy of formulas as follows.

1. $\Sigma_0,\Pi_0,\Delta_0$ are formulas in the language of set theory which only have bounded quantifiers.

2. $\Sigma_{n+1}$ formulas equivalent to formulas of the form $\exists x\varphi$, where $\varphi$ is $\Pi_n$; and similarly $\Pi_{n+1}$ are formulas equivalent to $\forall x\varphi$ where $\varphi$ is $\Sigma_n$.

3. $\Delta_n$ are formulas which are both $\Sigma_n$ and $\Pi_n$.

Now we have that if $M$ is a transitive class (set or otherwise), then $\Delta_0$ formulas are absolute between $M$ and $V$ (as long as $M$ includes all the parameters). Next we have that $\Sigma_1$ formulas are upwards absolute and $\Pi_1$ formulas are downwards absolute, so $\Delta_1$ formulas are absolute.