A many-sorted variant of Japaridze’s polymodal provability logic

Abstract

We consider a many-sorted variant of Japaridze’s polymodal provability logic (⁠|$\textsf{GLP}$|⁠). In this variant, which is denoted |$\textsf{GLP}^\ast $|⁠, propositional variables are assigned sorts |$\alpha \leq \omega $|⁠, where variables of finite sort |$n < \omega $| are interpreted as |$\varPi _{n+1}$|-sentences of the arithmetical hierarchy, while those of sort |$\omega $| range over arbitrary ones. We prove that |$\textsf{GLP}^\ast $| is arithmetically complete with respect to this interpretation. Moreover, we relate |$\textsf{GLP}^\ast $| to its one-sorted counterpart |$\textsf{GLP}$| and prove that the former inherits some well-known properties of the latter, like Craig interpolation and polynomial space (PSpace) decidability. We also study a positive variant of |$\textsf{GLP}^\ast $| that allows for an even richer arithmetical interpretation—variables are permitted to range over theories rather than single sentences. This interpretation in turn allows the introduction of a modality that corresponds to the full uniform reflection principle. We show that our positive variant of |$\textsf{GLP}^\ast $| is arithmetically complete.