The Notion of Weak-Contradiction: Definition and Measures

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Authors

Humberto Bustince

Nicolás Madrid

Manuel Ojeda-Aciego

Published

1 January 2015

Publication details

{IEEE} Trans. Fuzzy Syst. vol. 23 (4), pages 1057–1069.

Links

Abstract

In this paper, we present a way to represent contradiction between fuzzy sets. This representation is given in terms of the notion of f-weak-contradiction. Unlike other approaches, we do not define contradiction just by using one of the relations of f-weak-contradiction but by considering the whole set of relations. This consideration avoids the need to fix an operator beforehand in order to take into account all the information between two fuzzy sets. As a result, we characterize the contradiction between fuzzy sets and define a family of measures of contradiction satisfying four interesting properties: symmetry; antitonicity; if the intersection is empty, then the measure is one; and if there is an element in the intersection with degree of membership 1, then the measure is zero.

Citation

Please, cite this work as:

[BMO15] H. Bustince, N. Madrid, and M. Ojeda-Aciego. “The Notion of Weak-Contradiction: Definition and Measures”. In: IEEE Trans. Fuzzy Syst. 23.4 (2015), pp. 1057-1069. DOI: 10.1109/TFUZZ.2014.2337934. URL: https://doi.org/10.1109/TFUZZ.2014.2337934.

BibTeX

@Article{Bustince2015,
     author = {Humberto Bustince and Nicol{’a}s Madrid and Manuel Ojeda-Aciego},
     journal = {{IEEE} Trans. Fuzzy Syst.},
     title = {The Notion of Weak-Contradiction: Definition and Measures},
     year = {2015},
     number = {4},
     pages = {1057–1069},
     volume = {23},
     bibsource = {dblp computer science bibliography, https://dblp.org},
     biburl = {https://dblp.org/rec/journals/tfs/BustinceMO15.bib},
     doi = {10.1109/TFUZZ.2014.2337934},
     timestamp = {Mon, 03 Jan 2022 00:00:00 +0100},
     url = {https://doi.org/10.1109/TFUZZ.2014.2337934},
}

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Citations
CrossRef - Citation Indexes: 11
Scopus - Citation Indexes: 19

Captures
Mendeley - Readers: 7

Cites

The following graph plots the number of cites received by this work from its publication, on a yearly basis.

Papers citing this work

The following is a non-exhaustive list of papers that cite this work:

[1] Z. Ashraf, P. K. Muhuri, Q. M. D. Lohani, et al. “Type-2 fuzzy reliability–redundancy allocation problem and its solution using particle-swarm optimization algorithm”. In: Granular Computing 4.2 (Jul. 2018), p. 145–166. ISSN: 2364-4974. DOI: 10.1007/s41066-018-0106-5. URL: http://dx.doi.org/10.1007/s41066-018-0106-5.

[2] M. E. Cornejo, D. Lobo, and J. Medina. “Extended multi-adjoint logic programming”. In: Fuzzy Sets and Systems 388 (Jun. 2020), p. 124–145. ISSN: 0165-0114. DOI: 10.1016/j.fss.2019.03.016. URL: http://dx.doi.org/10.1016/j.fss.2019.03.016.

[3] M. E. Cornejo, D. Lobo, and J. Medina. “Relating Multi-Adjoint Normal Logic Programs to Core Fuzzy Answer Set Programs from a Semantical Approach”. In: Mathematics 8.6 (Jun. 2020), p. 881. ISSN: 2227-7390. DOI: 10.3390/math8060881. URL: http://dx.doi.org/10.3390/math8060881.

[4] M. E. Cornejo, D. Lobo, and J. Medina. “Selecting the Coherence Notion in Multi-adjoint Normal Logic Programming”. In: Advances in Computational Intelligence. Springer International Publishing, 2017, p. 447–457. ISBN: 9783319591537. DOI: 10.1007/978-3-319-59153-7_39. URL: http://dx.doi.org/10.1007/978-3-319-59153-7_39.

[5] R. Halas, R. Mesiar, and J. Pocs. “Generators of Aggregation Functions and Fuzzy Connectives”. In: IEEE Transactions on Fuzzy Systems 24.6 (Dec. 2016), p. 1690–1694. ISSN: 1941-0034. DOI: 10.1109/tfuzz.2016.2544353. URL: http://dx.doi.org/10.1109/tfuzz.2016.2544353.

[6] R. Halaš, R. Mesiar, and J. Pócs. “Description of sup- and inf-preserving aggregation functions via families of clusters in data tables”. In: Information Sciences 400–401 (Aug. 2017), p. 173–183. ISSN: 0020-0255. DOI: 10.1016/j.ins.2017.02.060. URL: http://dx.doi.org/10.1016/j.ins.2017.02.060.

[7] N. Madrid, J. Medina, and E. Ramírez-Poussa. “Rough sets based on Galois connections”. In: International Journal of Applied Mathematics and Computer Science 30.2 (2020). ISSN: 2083-8492. DOI: 10.34768/amcs-2020-0023. URL: http://dx.doi.org/10.34768/amcs-2020-0023.

[8] N. Madrid and M. Ojeda-Aciego. “A View of f-indexes of Inclusion Under Different Axiomatic Definitions of Fuzzy Inclusion”. In: Scalable Uncertainty Management. Springer International Publishing, 2017, p. 307–318. ISBN: 9783319675824. DOI: 10.1007/978-3-319-67582-4_22. URL: http://dx.doi.org/10.1007/978-3-319-67582-4_22.

[9] N. Madrid and M. Ojeda-Aciego. “Approaching the square of opposition in terms of the f-indexes of inclusion and contradiction”. In: Fuzzy Sets and Systems 476 (Jan. 2024), p. 108769. ISSN: 0165-0114. DOI: 10.1016/j.fss.2023.108769. URL: http://dx.doi.org/10.1016/j.fss.2023.108769.

[10] N. Madrid and M. Ojeda-Aciego. “Approaching the Square of Oppositions in Terms of the f -indexes of Inclusion and Contradiction”. In: Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP). ifsa-eusflat-agop-21. Atlantis Press, 2021. DOI: 10.2991/asum.k.210827.055. URL: http://dx.doi.org/10.2991/asum.k.210827.055.

[11] N. Madrid and M. Ojeda-Aciego. “Functional degrees of inclusion and similarity between L-fuzzy sets”. In: Fuzzy Sets and Systems 390 (Jul. 2020), p. 1–22. ISSN: 0165-0114. DOI: 10.1016/j.fss.2019.03.018. URL: http://dx.doi.org/10.1016/j.fss.2019.03.018.

[12] N. Madrid and M. Ojeda-Aciego. “On Contradiction and Inclusion Using Functional Degrees”. In: International Journal of Computational Intelligence Systems 13.1 (2020), p. 464. ISSN: 1875-6883. DOI: 10.2991/ijcis.d.200409.001. URL: http://dx.doi.org/10.2991/ijcis.d.200409.001.

[13] N. Madrid and M. Ojeda-Aciego. “Some Relationships Between the Notions of f-Inclusion and f-Contradiction”. In: Computational Intelligence and Mathematics for Tackling Complex Problems 2. Springer International Publishing, 2022, p. 175–181. ISBN: 9783030888176. DOI: 10.1007/978-3-030-88817-6_20. URL: http://dx.doi.org/10.1007/978-3-030-88817-6_20.

[14] N. Madrid and M. Ojeda-Aciego. “The f-index of inclusion as optimal adjoint pair for fuzzy modus ponens”. In: Fuzzy Sets and Systems 466 (Aug. 2023), p. 108474. ISSN: 0165-0114. DOI: 10.1016/j.fss.2023.01.009. URL: http://dx.doi.org/10.1016/j.fss.2023.01.009.

[15] N. Madrid, M. Ojeda‐Aciego, and E. Ramírez‐Poussa. “On the φ $\varphi$ ‐Index of Inclusion: Studying the Structure Generated by a Subset of Indexes”. In: Mathematical Methods in the Applied Sciences (Feb. 2025). ISSN: 1099-1476. DOI: 10.1002/mma.10833. URL: http://dx.doi.org/10.1002/mma.10833.

[16] N. Madrid and E. Ramírez-Poussa. “Analysis of the $\varphi$ -Index of Inclusion Restricted to a Set of Indexes”. In: Information Processing and Management of Uncertainty in Knowledge-Based Systems. Springer Nature Switzerland, 2024, p. 3–11. ISBN: 9783031740039. DOI: 10.1007/978-3-031-74003-9_1. URL: http://dx.doi.org/10.1007/978-3-031-74003-9_1.

[17] J. M. Mendel. “A comparison of three approaches for estimating (synthesizing) an interval type-2 fuzzy set model of a linguistic term for computing with words”. In: Granular Computing 1.1 (Dec. 2015), p. 59–69. ISSN: 2364-4974. DOI: 10.1007/s41066-015-0009-7. URL: http://dx.doi.org/10.1007/s41066-015-0009-7.

[18] J. M. Mendel. “Type-2 Fuzzy Sets”. In: Uncertain Rule-Based Fuzzy Systems. Springer International Publishing, 2017, p. 259–306. ISBN: 9783319513706. DOI: 10.1007/978-3-319-51370-6_6. URL: http://dx.doi.org/10.1007/978-3-319-51370-6_6.

[19] L. Yang and S. Zhang. “An Improved Fuzzy TOPSIS Based on Fuzzy Sets with Three Kinds of Negations”. In: 2022 34th Chinese Control and Decision Conference (CCDC). IEEE, Aug. 2022, p. 4847–4851. DOI: 10.1109/ccdc55256.2022.10034271. URL: http://dx.doi.org/10.1109/ccdc55256.2022.10034271.

[20] S. Zhang and Y. Li. “A novel table look-up scheme based on GFScom and its application”. In: Soft Computing 21.22 (Jun. 2016), p. 6767–6781. ISSN: 1433-7479. DOI: 10.1007/s00500-016-2226-7. URL: http://dx.doi.org/10.1007/s00500-016-2226-7.