  For each $n\in\mathbb{N}$, let $[n]\phi$ mean "the sentence $\phi$ is true in
all $\Sigma_{n+1}$-correct transitive sets." Assuming G\"odel's axiom $V = L$,
we prove the following graded variant of Solovay's completeness theorem: the
set of formulas valid under this interpretation is precisely the set of
theorems of the linear provability logic GLP.3. We also show that this result
is not provable in ZFC, so the hypothesis V = L cannot be removed. As part of
the proof, we derive (in ZFC) the following purely modal-logical results which
are of independent interest: the logic GLP.3 coincides with the logic of closed
substitutions of GLP, and is the maximal non-degenerate, normal extension of
GLP.
