  Given a graph $G$ and a vertex set $X$, the annotated treewidth tw$(G,X)$ of
$X$ in $G$ is the maximum treewidth of an $X$-rooted minor of $G$, i.e., a
minor $H$ where the model of each vertex of $H$ contains some vertex of $X$.
That way, tw$(G,X)$ can be seen as a measure of the contribution of $X$ to the
tree-decomposability of $G$. We introduce the logic CMSO/tw as the fragment of
monadic second-order logic on graphs obtained by restricting set quantification
to sets of bounded annotated treewidth. We prove the following Algorithmic
Meta-Theorem (AMT): for every non-trivial minor-closed graph class, model
checking for CMSO/tw formulas can be done in quadratic time. Our proof works
for the more general CMSO/tw+dp logic, that is CMSO/tw enhanced by
disjoint-path predicates. Our AMT can be seen as an extension of Courcelle's
theorem to minor-closed graph classes where the bounded-treewidth condition in
the input graph is replaced by the bounded-treewidth quantification in the
formulas. Our results yield, as special cases, all known AMTs whose
combinatorial restriction is non-trivial minor-closedness.
