Failure of Esakia's theorem in the monadic setting

TL;DR

This paper demonstrates that Esakia's theorem, which links certain modal and intuitionistic logics, does not hold in the monadic setting, showing a fundamental limitation in the existing logical correspondence.

Contribution

It proves that the one-variable fragment of intuitionistic predicate calculus lacks a greatest modal companion, revealing a failure of Esakia's theorem in this context.

Findings

Esakia's theorem fails in the monadic setting.

The one-variable fragment of intuitionistic predicate calculus has no greatest modal companion.

This result extends the understanding of limitations in modal-intuitionistic logic correspondence.

Abstract

Esakia's theorem states that Grzegorczyk's logic is the greatest modal companion of intuitionistic propositional calculus. We prove that already the one-variable fragment of intuitionistic predicate calculus does not have a greatest modal companion, yielding that Esakia's theorem fails in the monadic setting.