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Abstract
This paper presents the construction of an axiomatic system of Generalized Ambiguous Set Theory (AGAST)founded upon a four-valued logic, where the membership relation accepts four discrete degrees: True $(\alpha_A (x))$, Partially True $(\beta_A (x))$, Partially False $(\gamma_A (x))$, and Falsity $(\eta_A (x))$ membership degree functions, The contribution of this work is the rigorous adaptation of foundational Zermelo-Frankel set theory (ZFC) and Axiomatic Fuzzy Set Theory principles, including the Axiom of Extensionality and the Axiom Schema of Ambiguous Separation, to cohere within the four-degree semantics. Furthermore, the theory introduces the Anti-Classicality Axiom, which postulates the existence of sets exhibiting non-classical membership degrees.
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References
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- P. Singh, “Ambiguous set theory: A new approach to deal with unconsciousness and ambiguousness of human perception,” Journal of Neutrosophic and Fuzzy Systems (JNFS), vol. 5, no. 1, pp. 52–58, 2023. https://doi.org/10.54216/JNFS.050106.
- A. Aji, M. I. Bello, and B. A. Umar, “Operators and properties of generalized ambiguous four-degree membership functions set theory,” International Journal of Scientific Research in Mathematical and Statistical Sciences, vol. 12, no. 5, pp. 40–50, 2025. https://www.isroset.org/journal/IJSRMSS/full_paper_view.php?paper id=3967.
- A. Ali, M. I. Bello, Y. Haruna, and A. M. Kwami, “Formal examination of operators and properties of generalized ambiguous four-degree logic,” Dutse Journal of Pure and Applied Sciences, vol. 11, no. 4c, pp. 84–98, 2025. https://www.ajol.info/index.php/dujopas/article/view/310368.
- P. Singh, “An investigation of ambiguous sets and their application to decision-making from partial order to lattice ambiguous sets,” Decision Analytics Journal, vol. 8, p. 100286, 2023. https://doi.org/10.1016/j.dajour.2023.100286.
References
E. Zermelo, “Untersuchungen über die grundlagen der mengenlehre. i,” Mathematische Annalen, vol. 65, no. 2, pp. 261–281, 1908. https://link.springer.com/article/10.1007/bf01449999.
S. Robert and E. Herbert, Set Theory. Neumann-Bernays-Gödel set theory, Encyclopedia Britannica. 2023. https://www.britannica.com/science/set-theory/The-Neumann-Bernays-Godel-axioms.
T. Banakh, “Classical set theory: theory of sets and classes,” arXiv preprint arXiv:2006.01613, 2020. https://doi.org/10.48550/arXiv.2006.01613.
S. Gottwald, “Universes of fuzzy sets and axiomatizations of fuzzy set theory. part i: Model based and axiomatic approaches,” Studia Logica, vol. 82, no. 2, pp. 211–244, 2006. https://link.springer.com/article/10.1007/s11225-006-7197-8.
H. K. Baruah, “Theory of fuzzy sets: An axiomatic approach,” International Journal of Energy, Information and Communications, vol. 6, no. 5, pp. 15–18, 2015. http://dx.doi.org/10.14257/ijeic.2015.6.5.02.
D. E. Tamir, C. Zhi-Qiang, A. Kandel, and J. L. Mott, “An axiomatic approach to fuzzy set theory,” Information Sciences, vol. 52, no. 1, pp. 75–83, 1990. https://doi.org/10.1016/0020-0255(90)90036-A.
J. Zhang, “The theory of membership degree of [gamma]-conclusion in several n-valued logic systems,” American Journal of Operations Research, vol. 2, no. 2, p. 147, 2012. https://doi.org/10.4236/ajor.2012.22017.
Y. Khomskii and H. V. Oddsson, “Paraconsistent and paracomplete zermelo–fraenkel set theory,” The Review of Symbolic Logic, vol. 17, no. 4, pp. 965–995, 2024. https://doi.org/10.1017/S1755020323000382.
J. Bhattacharya, “Some special operations and related results on intuitionistic fuzzy sets,” International journal of scientific research in mathematical and statistical sciences, vol. 8, no. 4, pp. 10–13, 2021. https://doi.org/10.26438/ijsrmss/v8i4.1013.
F. Smarandache, “Short introduction to standard and nonstandard neutrosophic set and logic,” Neutrosophic Sets and Systems, vol. 77, no. 1, p. 20, 2025. https://digitalrepository.unm.edu/nss_journal/vol77/iss1/20/.
A. A. Salama and S. A. Alblowi, “Generalized neutrosophic set and generalized neutrosophic topological spaces,” Computer Science and Engineering, vol. 2, no. 7, pp. 129–132, 2012. https://doi.org/10.5923/j.computer.20120207.01.
A. P. Sai, A. Arulselvam, and A. BobinArulselvam, “Intuitionistic quadri-partitioned neutrosophic n-soft set to evaluate the emotional resilience of students in a dynamic learning environment,” Neutrosophic Sets and Systems, vol. 87, no. 1, p. 57, 2025. https://digitalrepository.unm.edu/nss_journal/vol87/iss1/57/.
P. Singh, “Ambiguous set theory: A new approach to deal with unconsciousness and ambiguousness of human perception,” Journal of Neutrosophic and Fuzzy Systems (JNFS), vol. 5, no. 1, pp. 52–58, 2023. https://doi.org/10.54216/JNFS.050106.
A. Aji, M. I. Bello, and B. A. Umar, “Operators and properties of generalized ambiguous four-degree membership functions set theory,” International Journal of Scientific Research in Mathematical and Statistical Sciences, vol. 12, no. 5, pp. 40–50, 2025. https://www.isroset.org/journal/IJSRMSS/full_paper_view.php?paper id=3967.
A. Ali, M. I. Bello, Y. Haruna, and A. M. Kwami, “Formal examination of operators and properties of generalized ambiguous four-degree logic,” Dutse Journal of Pure and Applied Sciences, vol. 11, no. 4c, pp. 84–98, 2025. https://www.ajol.info/index.php/dujopas/article/view/310368.
P. Singh, “An investigation of ambiguous sets and their application to decision-making from partial order to lattice ambiguous sets,” Decision Analytics Journal, vol. 8, p. 100286, 2023. https://doi.org/10.1016/j.dajour.2023.100286.