  We investigate modal logical aspects of provability predicates
$\mathrm{Pr}_T(x)$ satisfying the following condition:
  $\mathbf{M}$: If $T \vdash \varphi \to \psi$, then $T \vdash
\mathrm{Pr}_T(\ulcorner \varphi \urcorner) \to \mathrm{Pr}_T(\ulcorner \psi
\urcorner)$.
  We prove the arithmetical completeness theorems for monotonic modal logics
$\mathsf{MN}$, $\mathsf{MN4}$, $\mathsf{MNP}$, $\mathsf{MNP4}$, and
$\mathsf{MND}$ with respect to provability predicates satisfying the condition
$\mathbf{M}$. That is, we prove that for each logic $L$ of them, there exists a
$\Sigma_1$ provability predicate $\mathrm{Pr}_T(x)$ satisfying $\mathbf{M}$
such that the provability logic of $\mathrm{Pr}_T(x)$ is exactly $L$. In
particular, the modal formulas $\mathrm{P}$: $\neg \Box \bot$ and $\mathrm{D}$:
$\neg (\Box A \land \Box \neg A)$ are not equivalent over non-normal modal
logic and correspond to two different formalizations $\neg
\mathrm{Pr}_T(\ulcorner 0=1 \urcorner)$ and $\neg \big(\mathrm{Pr}_T(\ulcorner
\varphi \urcorner) \land \mathrm{Pr}_T(\ulcorner \neg \varphi \urcorner)
\bigr)$ of consistency statements, respectively. Our results separate these
formalizations in terms of modal logic.
