In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic $$\mathsf{KD}$$ KD . Secondly, we introduce a new normal modal logic $$\mathsf{KDR}$$ KDR which is a proper extension of $$\mathsf{KD}$$ KD , and prove that there exists a Rosser provability predicate whose provability logic includes $$\mathsf{KDR}$$ KDR .

中文翻译：

---

Rosser 可证明性和正态模态逻辑

在本文中，我们研究了可证明性逻辑为正态模态逻辑的 Rosser 可证明性谓词。首先，我们证明存在一个 Rosser 可证明性谓词，其可证明性逻辑正是正常的模态逻辑 $$\mathsf{KD}$$ KD 。其次，我们引入了一个新的正规模态逻辑 $$\mathsf{KDR}$$ KDR ，它是 $$\mathsf{KD}$$ KD 的适当扩展，并证明存在一个 Rosser 可证明性谓词，其可证明性逻辑包含 $ $\mathsf{KDR}$$ KDR 。