# An Approximate Reasoning Method and Its Application to Fuzzy Information Systems

## Abstract

The present paper focus on studying the approximate reasoning models based on fuzzy information systems. Firstly, we introduce a concept of \( \lambda \)-truth degree in Lukasiewicz propositional logic by using the probability measure on the set of all valuations, which shows that all kinds of truth degree of formulas in quantitative logic can all be brought as special cases into the unified framework of the \( \lambda \)-truth degree. It is proved that the \( \lambda \)-truth degree satisfies the Kolmogorov axioms and hence the quantitative logic and the probability logic are integrated by means of \( \lambda \)-truth degree. Secondly, by using the \( \lambda \)-truth degree we define the \( \lambda \)-similarity degree and the logic \( \lambda \)-metric among two logic formulas, and prove that there exists not isolated point under some conditions and various logic operations are continuous in the logic \( \lambda \)-metric space. Thirdly, we analyze some applications of \( \lambda \)-truth degree to approximate reasoning in fuzzy information systems, propose two diverse approximate reasoning models in the logic \( \lambda \)-metric space, and give some examples to illustrate the application of approximate reasoning models.

## Keywords

\( \lambda \)-truth degree \( \lambda \)-similarity degree \( \lambda \)-logic metric Approximate reasoning Information system## Notes

### Acknowledgement

This work has been supported by the Hunan Provincial Social Science Achievement Evaluation Committee (No. XSP19YBZ111), the Natural Science Foundation of Hunan province (No. 2016JJ6138, No. 2017JJ2241).

## References

- 1.Lu, R.Q., Ying, M.S.: A model of reasoning about knowledge. Sci. China E
**41**, 527–534 (1998)MathSciNetCrossRefGoogle Scholar - 2.Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
- 3.Wang, G.J.: Classical Mathematical Logic and Approximate Reasoning. Science Press, Beijing (2006)Google Scholar
- 4.Wang, G.J., Zhou, H.J.: Introduction to Mathematical Logic and Resolution Principle. Science Press, Beijing (2009)zbMATHGoogle Scholar
- 5.Wang, G.J., Leung, Y.: Integrated semantics and logic metric spaces. Fuzzy Sets Syst.
**136**, 71–91 (2003)MathSciNetCrossRefGoogle Scholar - 6.Wang, G.J., Li, B.J.: Theory of truth degree of formulas in
*n*valued propositional logic and a limit theorem. Sci. China F**8**, 727–736 (2005)MathSciNetzbMATHGoogle Scholar - 7.Wang, G.J., Zhou, H.J.: Quantitative logic. Inf. Sci.
**179**, 226–247 (2009)MathSciNetCrossRefGoogle Scholar - 8.Hui, X.J., Wang, G.J.: Randomization of classical inference patterns and its application. Sci. China F
**50**, 867–877 (2007)MathSciNetzbMATHGoogle Scholar - 9.Zhou, H.J.: A probabilistically quantitative reasoning system based on
*n*valued Lukasiewicz propositional logic. Pattern Recogn. Artif. Intell.**26**(6), 521–528 (2013)Google Scholar - 10.Zhang, J.L., Chen, X.G.: Theory of probability semantics of classical propositional logic and its application. Chin. J. Comput.
**37**(8), 1775–1785 (2014)Google Scholar - 11.Zhang, J.L., Chen, X.G.: Approximate reasoning model based on probability valuation of propositional logic. Pattern Recogn. Artif. Intell.
**28**(9), 769–780 (2015)Google Scholar - 12.Yan, S.J., Wang, S.J., Liu, X.F.: Probability Theory. Science Press, Beijing (1986)Google Scholar
- 13.Klement, E.P., Lowen, R., Schwyhla, W.: Fuzzy probability measures. Fuzzy Sets Syst.
**1**, 21–30 (1981)MathSciNetCrossRefGoogle Scholar