# Why Some Non-classical Logics Are More Studied?

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## Abstract

It is well known that the traditional 2-valued logic is only an approximation to how we actually reason. To provide a more adequate description of how we actually reason, researchers proposed and studied many generalizations and modifications of the traditional logic, generalizations and modifications in which some rules of the traditional logic are no longer valid. Interestingly, for some of such rules (e.g., for law of excluded middle), we have a century of research in logics that violate this rule, while for others (e.g., commutativity of “and”), practically no research has been done. In this paper, we show that fuzzy ideas can help explain why some non-classical logics are more studied and some less studied: namely, it turns out that most studied are the violations which can be implemented by the simplest expressions (specifically, by polynomials of the lowest order).

## Notes

### Acknowledgements

This work was supported in part by the National Science Foundation via grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science) and HRD-1242122 (Cyber-ShARE Center of Excellence).

## References

- 1.Belohlavek, R., Dauben, J.W., Klir, G.J.: Fuzzy Logic and Mathematics: A Historical Perspective. Oxford University Press, New York (2017)CrossRefGoogle Scholar
- 2.Bouchon-Meunier, B., Kreinovich, V., Nguyen, H.T.: Non-associative operations. In: Proceedings of the Second International Conference on Intelligent Technologies InTech 2001, Bangkok, Thailand, 27–29 November 2001, pp. 39–46 (2001)Google Scholar
- 3.Gabbay, D.M., Guenthner, F. (eds.): Handbook of Philosophical Logic. Springer, Cham (2018)zbMATHGoogle Scholar
- 4.Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, Upper Saddle River (1995)zbMATHGoogle Scholar
- 5.Kreinovich, V.: Towards more realistic (e.g., non-associative) ‘and’- and ‘or’-operations in fuzzy logic. Soft Comput.
**8**(4), 274–280 (2004)Google Scholar - 6.Martinez, J., Macias, L., Esper, A., Chaparro, J., Alvarado, V., Starks, S.A., Kreinovich, V.: Towards more realistic (e.g., non-associative) and- and or-operations in fuzzy logic. In: Proceedings of the 2001 IEEE Systems, Man, and Cybernetics Conference, Tucson, Arizona, 7–10 October 2001, pp. 2187–2192 (2001)Google Scholar
- 7.Mendel, J.M.: Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions. Springer, Cham (2017)CrossRefGoogle Scholar
- 8.Nguyen, H.T., Kreinovich, V.: Nested intervals and sets: concepts, relations to fuzzy sets, and applications. In: Kearfott, R.B., Kreinovich, V. (eds.) Applications of Interval Computations, pp. 245–290. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
- 9.Nguyen, H.T., Walker, C., Walker, E.A.: A First Course in Fuzzy Logic. Chapman and Hall/CRC, Boca Raton (2019)zbMATHGoogle Scholar
- 10.Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Boston (1999)CrossRefGoogle Scholar
- 11.Priest, G.: An Introduction to Non-Classical Logic: From If to Is. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
- 12.Trejo, R., Kreinovich, V., Goodman, I.R., Martinez, J., Gonzalez, R.: A realistic (non-associative) logic and a possible explanations of \(7\pm 2\) law. Int. J. Approximate Reasoning
**29**, 235–266 (2002)CrossRefGoogle Scholar - 13.Xiang, G., Kreinovich, V.: Towards improved trapezoidal approximation to intersection (fusion) of trapezoidal fuzzy numbers: specific procedure and general non-associativity theorem. In: Proceedings of the IEEE World Congress on Computational Intelligence WCCI 2010, Barcelona, Spain, 18–23 July 2010, pp. 3120–3125 (2010)Google Scholar
- 14.Zadeh, L.A.: Fuzzy sets. Inf. Control
**8**, 338–353 (1965)CrossRefGoogle Scholar