The provability logic of all provability predicates

Abstract

Abstract We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczyński’s pure logic of necessitation $\textsf{N}$. Moreover, we introduce three extensions $\textsf{N4}$, $\textsf{NR}$ and $\textsf{NR4}$ of $\textsf{N}$ and investigate the arithmetical semantics of these logics. In fact, we prove that $\textsf{N4}$, $\textsf{NR}$ and $\textsf{NR4}$ are the provability logics of all provability predicates satisfying the third condition $\textbf{D3}$ of the derivability conditions, all Rosser provability predicates and all Rosser provability predicates satisfying $\textbf{D3}$, respectively.