Arithmetical completeness theorems for monotonic modal logics

Abstract

We investigate modal logical aspects of provability predicates ${\mathrm{Pr}}_T(x)$ satisfying the following condition: $\mathbf{M}$: If $T⊢\phi \to \psi$, then $T⊢{\mathrm{Pr}}_T(┌\phi ┐)\to {\mathrm{Pr}}_T(┌\psi ┐)$. We prove the arithmetical completeness theorems for monotonic modal logics $\mathsf{MN}$, $MN4$, $\mathsf{MNP}$, $MNP4$, 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 $\mathrm{P}$: $\neg \square \perp$ and $\mathrm{D}$: $\neg (\square A\wedge \square \neg A)$ are not equivalent over non-normal modal logic and correspond to two different formalizations $\neg {\mathrm{Pr}}_T(┌0=1┐)$ and $\neg ({\mathrm{Pr}}_T(┌\phi ┐)\wedge {\mathrm{Pr}}_T(┌\neg \phi ┐))$ of consistency statements, respectively. Our results separate these formalizations in terms of modal logic.

arXiv comment33 pages