1 research outputs found
A Many-Sorted Variant of Japaridze's Polymodal Provability Logic
We consider a many-sorted variant of Japaridze's polymodal provability logic
. In this variant, which is denoted ,
propositional variables are assigned sorts , where
variables of finite sort are interpreted as -sentences
of the arithmetical hierarchy, while those of sort range over
arbitrary ones. We prove that is arithmetically complete
with respect to this interpretation. Moreover, we relate to
its one-sorted counterpart and prove that the former inherits
some well-known properties of the latter, like Craig interpolation and PSpace
decidability. We also study a positive variant of which
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
is arithmetically complete.Comment: {A version of this article has been published in the Logic Journal of
the IGPL, 26(5): 505--538 (2018