Relative interpretability of modal logics

This paper introduces the notion of modality as an operator $\nabla_\psi$, defined on the set of propositional modal formulas by the equality $\nabla_\psi(F)=\psi(F)$, where $\psi(p)$ is a formula of one variable $p$. Defining the logic $L(\nabla)$ of modality $\nabla$ over logic $L$ as the set of all provable in $L$ formulas of the propositional language extended by the operator $\nabla$, the notion of exact interpretability $(\hookrightarrow)$ of a logic $L_1$ in a logic $L_2$ can be formalized as follows: $L_1\hookrightarrow L_2$ iff $L_1=L_2(\nabla)$ for some modality $\nabla$. The question about the number of logics, which are exactly interpretable in some fixed logic, is considered in this paper. Answers to this question are obtained for the following family of known modal logics: logics of boolean modalities, normal logics $\mathrm{K}$, $\mathrm{K4}$, $\mathrm{T}$, $\mathrm{S4}$, $\mathrm{S5}$, $\mathrm{GL}$, $\mathrm{Grz}$, logics of provability. A number of results concerning the absence of exact interpretability of some logics of this family in others are offered as well.