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A Brief Introduction to Probabilistically Quantitative Logic with Its Applications

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Quantitative Logic and Soft Computing 2016

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 510))

Abstract

Uncertainty is a fundamental and unavoidable feature of our real life, which has many distinct representation forms such as randomness, fuzziness, ambiguity, inaccuracy, incompleteness and roughness. Accordingly, many different mathematical models for dealing with these uncertainties, like probability, fuzzy set theory, Dempster-Shafer theory of evidence and rough set theory, have been introduced and also applied with great success in many fields. In order to construct uncertainty models with more powerful abilities of linguistic expression and logical inference, researchers have been devoting to intersections of different branches of mathematics of uncertainty, among which the resulting interdiscipline by combining probability and many-valued propositional logics is called probabilistically quantitative logic. This paper presents a brief introduction to probabilistically quantitative logic from four viewpoints of its research approaches and applications. Main contents include probabilistic truth degrees of propositions, logic systems for reasoning about probabilities of many-valued events, generalized state theory on residuated lattices, consistency degrees of formal theories, characterizations of maximally consistent theories, Stone representations of \(R_{0}\)-algebras, and generalized state based similarity convergence in residuated lattices with its Cauchy completion.

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Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61473336), the Youth Science and Technology Program of Shaanxi Province (2016KJXX-24) and the Fundamental Research Funds for the Central Universities (201403001).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, China

    Hong-Jun Zhou

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  1. Hong-Jun Zhou
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Correspondence to Hong-Jun Zhou .

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Editors and Affiliations

  1. Zhejiang, Zhejiang Sci-Tech University Zhejiang, Hangzhou, China

    Tai-He Fan

  2. Fujian, Jimei University Fujian, Xiamen, China

    Shui-Li Chen

  3. Zhejiang, Zhejiang Sci-Tech University Zhejiang, Hangzhou, China

    San-Min Wang

  4. Yanta, Shaanxi Normal University Yanta, Xi’an, China

    Yong-Ming Li

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Zhou, HJ. (2017). A Brief Introduction to Probabilistically Quantitative Logic with Its Applications. In: Fan, TH., Chen, SL., Wang, SM., Li, YM. (eds) Quantitative Logic and Soft Computing 2016. Advances in Intelligent Systems and Computing, vol 510. Springer, Cham. https://doi.org/10.1007/978-3-319-46206-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-46206-6_5

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