Luca Carai (Università di Milano) will give a talk on
Extending the Blok-Esakia Theorem to the monadic setting
Abstract:
Intuitionistic and modal logic are closely connected: the intuitionistic propositional calculus IPC can be faithfully translated into the propositional modal logic S4 via the Gödel translation. This allows one to determine whether a formula is a theorem of IPC by checking the validity of its translation in S4. A normal extension M of S4 is called a modal companion of an extension L of IPC if L can be faithfully translated into M via the Gödel translation. By Esakia’s Theorem, the largest modal companion of IPC is the Grzegorczyk modal logic Grz. The study of modal companions culminated into the celebrated Blok-Esakia Theorem, which states that assigning to each extension of IPC its largest modal companion yields an isomorphism between the lattices of extensions of IPC and of normal extensions of Grz.
In this talk, we will examine the challenges that arise when studying modal companions of extensions of the monadic fragment (that is, the one-variable fragment) of predicate intuitionistic logic. We will see that the natural generalizations of Esakia’s and Blok-Esakia Theorems fail in the monadic setting. The talk will conclude on a positive note, showing how Esakia’s Theorem can be recovered by adding a suitable axiom. This talk is based on a joint work with G. Bezhanishvili.