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G. Berger, L. Beklemishev, H. Tompits:
"A many-sorted variant of Japaridze's polymodal provability logic";
Logic Journal of the IGPL, 26 (2018), 5; S. 505 - 538.

We consider a many-sorted variant of Japaridze´s polymodal provability logic (⁠GLP⁠). In this variant, which is denoted GLP∗⁠, propositional variables are assigned sorts α≤ω⁠, where variables of finite sort n<ω are interpreted as Πn+1-sentences of the arithmetical hierarchy, while those of sort ω range over arbitrary ones. We prove that GLP∗ is arithmetically complete with respect to this interpretation. Moreover, we relate GLP∗ to its one-sorted counterpart 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 GLP∗ 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 GLP∗ is arithmetically complete.

http://dx.doi.org/10.1093/jigpal/jzy012