The provability logic of all provability predicates

We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczyński’s pure logic of necessitation $\mathsf{N}$. Moreover, we introduce three extensions $\mathsf{N}\mathsf{4}$, $\mathsf{N}\mathsf{R}$, and $\mathsf{N}\mathsf{R}\mathsf{4}$ of $\mathsf{N}$ and investigate the arithmetical semantics of these logics. In fact, we prove that $\mathsf{N}\mathsf{4}$, $\mathsf{N}\mathsf{R}$, and $\mathsf{N}\mathsf{R}\mathsf{4}$ are the provability logics of all provability predicates satisfying the third condition $\mathbf{D}\mathbf{3}$ of the derivabiity conditions, all Rosser’s provability predicates, and all Rosser’s provability predicates satisfying $\mathbf{D}\mathbf{3}$, respectively.