It is well known that if $\phi$ is a $\Delta_{1}$-formula and $x_{1},..,x_{n}$ in $V$ and $V[G]$ is a forcing extension, then $V\models\phi(x_{1},...,x_{n})$ if and only if $V[G]\models\phi(x_{1},...,x_{n})$. In other words, we say that $\Delta_{1}$ formulas are absolute for forcing extensions. More generally, if $\phi$ is $\Sigma_{1}$, and $V\models\phi(x_{1},...,x_{n})$, then $V[G]\models\phi(x_{1},...,x_{n})$. I am wondering about what kind of absoluteness theorems hold between $V^{B}$ and $V$. Perhaps, there are absoluteness theorems where the formulas in $V^{B}$ are slightly different than the formulas in $V$. Let me be more specific about what I mean.
Let $B$ be a complete Boolean algebra and let $V^{B}$ be the corresponding $B$-valued set theoretic universe. Suppose $\dot{X}\in V^{B}$. Let $\Gamma^{*}(\dot{X})=\{\dot{x}\in V^{B}:\|\dot{x}\in\dot{X}\|=1\}$. Let $\simeq$ be the equivalence relation on $\Gamma^{*}(\dot{X})$ where $\dot{x}\simeq\dot{y}$ iff $\|\dot{x}=\dot{y}\|=1$. Let $\Gamma(\dot{X})=\Gamma^{*}(\dot{X})/\simeq$.
Suppose that $n_{1},...,n_{r}$ are natural numbers and $V^{B}\models\dot{R}_{1}\subseteq\dot{X}^{n_{1}},...,\dot{R}_{k}\subseteq\dot{X}^{n_{r}}$. Let $X=\Gamma(\dot{X})$. If $1\leq i\leq r$, then let $R_{i}$ be the $n_{i}$-ary relation on $X$ where $([x_{1}],...,[x_{r}])\in R_{i}$ if and only if $V^{B}\models (x_{1},...,x_{n_{i}})\in\dot{R}_{i}$.
Then for which n-th order formulas $\Phi$ do we have $V^{B}\models((\dot{X},\dot{R}_{1},...,\dot{R}_{r})\models\Phi(\dot{a}_{1},...,\dot{a}_{r}))$ implies $(X,R_{1},...,R_{n})\models\Phi$ whenever $V^{B}\models\dot{a}_{1},...,\dot{a}_{r}\in\dot{X}$?
If $\Phi$ is a first order formula, then the answer to this question is fairly clear:
The structure $(X,R_{1},...,R_{r})$ is a Boolean product of the structures $(\dot{X},\dot{R}_{1},...,\dot{R}_{r})/U$ where $U$ ranges over the ultrafilters on $B$. However, Horn formulas are preserved under most Boolean products [see 1] including the Boolean products that one obtains from forcing. In particular, if $\Phi$ is a Horn formula, then $V^{B}\models((\dot{X},\dot{R}_{1},...,\dot{R}_{r})\models\Phi(\dot{a}_{1},...,\dot{a}_{r}))$ implies $(X,R_{1},...,R_{n})\models\Phi([\dot{a}_{1}],...,[\dot{a}_{n}])$ whenever $V^{B}\models\dot{a}_{1},...,\dot{a}_{r}\in\dot{X}$.
Furthermore, the Feferman-Vaught theorem also holds for most Boolean products [see 1], so the truth value of any formula in the structure $(X,R_{1},...,R_{n})$ is determined by the Boolean truth value of formulas in $(\dot{X},\dot{R}_{1},...,\dot{R}_{n})$.
However, it is not clear which higher order formulas are preserved by going between $V$ and $V^{B}$. I however must mention that some higher order sentences are preserved when one goes from $V^{B}$ to $V$. For example, it is a well known fact about iterated forcing that if $V^{B}\models``\dot{C}\,\textrm{is a complete Boolean algebra}"$, then $V\models``\Gamma(\dot{C})\,\textrm{is a complete Boolean algebra}"$. Perhaps this result can be generalize to a large class of first order formulas that contains the second order formula stating that the object is a complete Boolean algebra.
I would also appreciate it if someone supplied references for such absoluteness results. I am currently just interested in such absoluteness results between $V^{B}$ and $V$ and I am not as much interested right now in the corresponding absoluteness between $V^{B}/U$ and $V$ where $U$ is an ultrafilter.
1. Sheaf constructions and their elementary properties. Stanley Burris and Heinrich Werner. Trans. Amer. Math. Soc. 248 (1979), 269-309.
