In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm.

在这项研究中，我们分别介绍了Gentzen型继发性结石BDm和BDi，以经典否定的形式对De和Omori的扩展Belnap-Dunn逻辑BD +进行了模态扩展和直觉修改。我们证明了将BDm和BDi分别在语法和语义上分别嵌入到普通模态逻辑和直觉逻辑的Gentzen型后续演算S4和LJ中的定理。使用这些嵌入定理可以获得BDm和BDi的割除定理，可判定性和完整性定理。此外，我们证明了将BD +嵌入BDi的Glivenko定理和将BDi嵌入BDm的McKinsey-Tarski定理。