  The branch of provability logic investigates the provability-based behavior
of the mathematical theories. In a more precise way, it studies the relation
between a mathematical theory $T$ and a modal logic $L$ via the provability
interpretation which interprets the modality as the provability predicate of
$T$. In this paper we will extend this relation to investigate the
provability-based behavior of a hierarchy of theories. More precisely, using
the modal language with infinitely many modalities,
$\{\Box_n\}_{n=0}^{\infty}$, we will define the hierarchical counterparts of
some of the classical modal theories such as $\mathbf{K4}$, $\mathbf{KD4}$,
$\mathbf{GL}$ and $\mathbf{S4}$. Then we will define their canonical
provability interpretations and their corresponding soundness-completeness
theorems.
