Monadic Intuitionistic and Modal Logics Admitting Provability Interpretations

TL;DR

This paper extends provability interpretations from classical intuitionistic and modal logics to their monadic extensions, establishing embeddings and arithmetical interpretations through added formulas and translations.

Contribution

It introduces monadic extensions of IPC, Grz, and GL, and proves embeddings and arithmetical interpretations for these extended systems.

Findings

Monadic extensions admit similar embeddings as their base logics.

Adding Casari's formula enables embeddings of monadic IPC into monadic Grz.

Solovay's theorem extends to monadic GL, allowing arithmetical interpretations.

Abstract

The G\"odel translation provides an embedding of the intuitionistic logic $\mathsf{IPC}$ into the modal logic $\mathsf{Grz}$, which then embeds into the modal logic $\mathsf{GL}$ via the splitting translation. Combined with Solovay's theorem that $\mathsf{GL}$ is the modal logic of the provability predicate of Peano Arithmetic $\mathsf{PA}$, both $\mathsf{IPC}$ and $\mathsf{Grz}$ admit arithmetical interpretations. When attempting to 'lift' these results to the monadic extensions $\mathsf{MIPC}$, $\mathsf{MGrz}$, and $\mathsf{MGL}$ of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version of Casari's formula to these monadic extensions (denoted by a '+'), obtaining that the G\"odel translation embeds $\mathsf{M^{+}IPC}$ into $\mathsf{M^{+}Grz}$ and the splitting translation embeds $\mathsf{M^{+}Grz}$ into $\mathsf{MGL}$. As proven by Japaridze, Solovay's result extends to the monadic system $\mathsf{MGL}$, which leads us to an arithmetical interpretation of both $\mathsf{M^{+}IPC}$ and $\mathsf{M^{+}Grz}$.