Abstract

The Gödel 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 provability 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 Gödel 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 a provability interpretation of both$\mathsf {M^{+}IPC}$and$\mathsf {M^{+}Grz}$.