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A New Contingency Axiomatic System for Rough Sets

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Data Mining and Big Data (DMBD 2021)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1454))

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Abstract

Contingency logic is well-known to study the principles of reasoning involving necessity, possibility, contingency and non-contingency. However, there are some defects in existing contingency axiomatic systems. For instances, (NCR)\(_i\) is an infinite inference rule. The definition of accessibility relations and the corresponding axiom schema are very complex. To tackle these issues, a new contingency axiomatic system is proposed in this paper. Firstly, 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.

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Notes

  1. 1.

    “\(\triangle \)” is a logical symbol.

  2. 2.

    “\(\Box \)” is a logical symbol.

  3. 3.

    \(\varLambda \) is S-inconsistent iff there are \(\alpha _1\),...,\(\alpha _n\in \varLambda \) such that \(\vdash _S\lnot (\alpha _1\wedge ...\wedge \alpha _n)\). The idea is that in S you can prove that a contradiction arises from the members of \(\varLambda \), where S may be the systems K,D,T,S4 and S5, and so on [1].

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Acknowledgment

This work is partially supported by the Science and Technology Project of Jiangxi Provincial Education Department (Nos. GJJ161109, GJJ201917 and GJJ190941), and the National Science Foundation of China (Nos. 61763032 and 61562061).

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Authors and Affiliations

  1. School of Information Engineering, NanChang Institute of Technology, Nanchang, 330099, China

    Sujie Guan, Shaobo Deng, Hui Wang & Min Li

Authors
  1. Sujie Guan
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  2. Shaobo Deng
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  3. Hui Wang
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  4. Min Li
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Corresponding author

Correspondence to Hui Wang .

Editor information

Editors and Affiliations

  1. Peking University, Beijing, China

    Ying Tan

  2. Southern University of Science and Technology, Shenzhen, China

    Yuhui Shi

  3. The University of Sydney, Sydney, Australia

    Albert Zomaya

  4. Guangzhou University, Guangzhou, China

    Hongyang Yan

  5. Guangdong Polytechnic Normal University, Guangzhou, China

    Jun Cai

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Guan, S., Deng, S., Wang, H., Li, M. (2021). A New Contingency Axiomatic System for Rough Sets. In: Tan, Y., Shi, Y., Zomaya, A., Yan, H., Cai, J. (eds) Data Mining and Big Data. DMBD 2021. Communications in Computer and Information Science, vol 1454. Springer, Singapore. https://doi.org/10.1007/978-981-16-7502-7_36

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  • DOI: https://doi.org/10.1007/978-981-16-7502-7_36

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  • Online ISBN: 978-981-16-7502-7

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