Arithmetical completeness theorems for monotonic modal logics

TL;DR

This paper establishes arithmetical completeness theorems for certain monotonic modal logics related to provability predicates, clarifying the formalization of consistency statements and their modal counterparts.

Contribution

It proves the existence of specific provability predicates satisfying condition M that exactly match given monotonic modal logics, and distinguishes different formalizations of consistency.

Findings

Existence of $ ext{Pr}_T(x)$ matching each logic $L$

Separation of formalizations $ eg ext{Pr}_T(ot)$ and $ eg ( ext{Pr}_T( ext{A}) ext{and} ext{Pr}_T( eg ext{A}))$

Clarification of modal formalizations of consistency statements

Abstract

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.