Basic proof theory
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The paper introduces Hilbert--- and Gentzen-style calculi which correspond to systems $${\mathsf{C}_{n}}$$ from Gupta and Belnap [3]. Systems $${\mathsf{C}_{n}}$$ were shown to be sound and complete with respect to the semantics of finite revision. Here, it is shown that Gentzen-style systems $${\mathsf{GC}_{n}}$$ admit a syntactic proof of cut elimination. As a consequence, it follows that they are consistent.