Formalization of rough sets is a key issue in rough set theory. When rough sets are formalized by propositional logic, predicate logic, or modal propositional logic, it easily suffers from some problems. For instance, an incomplete system is obtained. The concepts of “$precise$” or “$rough$” of rough sets cannot be described. To tackle these issues, a new noncontingency axiomatic system is proposed for formalizing rough sets in this paper. First, a new concise accessibility relation is defined for the axiomatic system; then, two simpler axiom schemas of the axiomatic system are designed to replace the axiom schema $K$. This is helpful to prove the soundness and completeness theorems for the axiomatic system. Finally, rough sets can be perfectly formalized by our proposed axiomatic system. Theoretical analysis proves that a complete formal system is achieved. In addition, the concepts of “$precise$” or “$rough$” of rough sets can be described without the help of semantics functions of metalanguage.

中文翻译：

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使用新的非偶然性公理系统形式化粗糙集

粗糙集的形式化是粗糙集理论中的一个关键问题。当粗糙集被命题逻辑、谓词逻辑或模态命题逻辑形式化时，很容易出现一些问题。例如，得到一个不完整的系统。“的概念$preC\mathrm{一世}se$“ 或者 ”$r○你GH$”的粗糙集无法描述。为了解决这些问题，本文提出了一种新的非偶然公理系统来形式化粗糙集。首先，为公理系统定义了一个新的简洁的可达性关系；然后，设计了公理系统的两个更简单的公理模式来代替公理模式$ķ$. 这有助于证明公理系统的健全性和完备性定理。最后，我们提出的公理系统可以完美地形式化粗糙集。理论分析证明，实现了完整的形式系统。此外，“$preC\mathrm{一世}se$“ 或者 ”$r○你GH$”的粗糙集可以在没有元语言语义功能的帮助下进行描述。