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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