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

Provability Interpretation of Propositional and Modal Logics