Abstract
We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \(\Sigma _2\) provability predicate of T whose provability logic is precisely the modal logic \(\mathsf{K}\). For this purpose, we introduce a new bimodal logic \(\mathsf{GLK}\), and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \(\mathsf{GLK}\).
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Kurahashi, T. Arithmetical Completeness Theorem for Modal Logic \(\mathsf{K}\) . Stud Logica 106, 219–235 (2018). https://doi.org/10.1007/s11225-017-9735-y
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DOI: https://doi.org/10.1007/s11225-017-9735-y