The Ground Axiom (GA), introduced by Hamkins and Reitz, asserts that the universe is not a nontrivial set forcing extension of any inner model, and it is known that GA is consistent relative to ZFC. My question is somehow related to GA.
First a definition:
Definition. Let $\Phi$ be a statement of set theory which is independent of ZFC (or ZFC+large cardinals). Call $\Phi$ forceable, if there is a model $W$ of ZFC and a set of forcing conditions $P$ in $W$ such that $\Phi$ is true in the generic extensions of $W$ by $P$ (we allow the existence of large cardinals in $W$).
Question. 1) Is there a characterization of all forceable sentences $\Phi$ such that the following holds:
If $\Phi$ holds in $V$, then there some inner model $W$ of $V$ in which $\Phi$ fails, and such that $V$ is a set generic extension of $W$.
2) Can you give at least one natural statement $\Phi$ as in (1).
Remark (suggested by Sy Friedman). In the paper "A Large $\Pi^1_2$ Set, Absolute for Set Forcings", Sy Friedman constructs a $\Pi^1_2$ formula $\phi$ which can have many solutions in a cardinal-preserving extension of $L$. $\phi$ also has the property that set-forcing will not add new solutions. So no model in which $\phi$ has more than $\aleph_1$ solutions can be a set-generic extension of a model with only $\aleph_1$ solutions.
Also there are reals $r$ which are minimal over $L$ but which cannot be obtained by set forcing over $L$. Then in $V=L[r]$ the sentence $V=L$ fails, but there is no inner model $W$ of $V$ satisfying $V=L$ such that $V$ is a set forcing extension of $W$.