The provability logic of all provability predicates

TL;DR

This paper characterizes the provability logic of all provability predicates as a specific logic and introduces extensions with arithmetical semantics, covering various classes of provability predicates.

Contribution

It identifies the exact provability logic for all provability predicates and introduces three new logic extensions with their arithmetical semantics.

Findings

The provability logic of all provability predicates is exactly $ ext{N}$.

Introduces three extensions $ ext{N4}$, $ ext{NR}$, $ ext{NR4}$ with specific semantics.

These logics correspond to classes of provability predicates satisfying certain conditions.

Abstract

We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczy\'nski's pure logic of necessitation $\mathsf{N}$. Moreover, we introduce three extensions $\mathsf{N4}$, $\mathsf{NR}$, and $\mathsf{NR4}$ of $\mathsf{N}$ and investigate the arithmetical semantics of these logics. In fact, we prove that $\mathsf{N4}$, $\mathsf{NR}$, and $\mathsf{NR4}$ are the provability logics of all provability predicates satisfying the third condition $\mathbf{D3}$ of the derivabiity conditions, all Rosser's provability predicates, and all Rosser's provability predicates satisfying $\mathbf{D3}$, respectively.