  In 1933, G\"odel introduced a provability interpretation of the propositional
intuitionistic logic to establish a formalization for the BHK interpretation.
He used the modal system, $\mathbf{S4}$, as a formalization of the intuitive
concept of provability and then translated $\mathbf{IPC}$ to $\mathbf{S4}$. His
work suggested the problem to find a concrete provability interpretation of the
modal logic $\mathbf{S4}$. In this paper, we will try to answer this problem.
In fact, we will generalize Solovay's provability interpretation of the modal
logic $\mathbf{GL}$ to capture other modal logics such as $\mathbf{K4}$,
$\mathbf{KD4}$ and $\mathbf{S4}$. Then we will use these results to find a
formalization for the BHK interpretation and we will show that with different
interpretations of the BHK interpretation, we can capture some of the
propositional logics such as Intuitionistic logic, minimal logic and
Visser-Ruitenburg's basic logic. Moreover, we will show that there is no
provability interpretation for any extension of $\mathbf{KD45}$ and also there
is no BHK interpretation for the classical propositional logic.

Logic