Three-Mode Upside-Down Logic within Plithogenic Double-valued and Triple-valued Neutrosophic Sets

Takaaki Fujita; Ajoy Kanti Das; Sankar Prasad Mondal; Arif Mehmood; Suman Das; Volkan Duran; Arkan A. Ghaib

Authors

Takaaki Fujita Independent Researcher, Tokyo, Japan Author https://orcid.org/0009-0007-1509-2728
Ajoy Kanti Das Department of Mathematics, Tripura University, Agartala-799022, Tripura, India Author https://orcid.org/0000-0002-9326-1677
Sankar Prasad Mondal Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, India Author https://orcid.org/0000-0003-4690-2598
Arif Mehmood Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan Author https://orcid.org/0000-0002-6230-9236
Suman Das Department of Education (ITEP), NIT Agartala, Jirania, 799046, Tripura, India Author https://orcid.org/0000-0001-5682-9334
Volkan Duran Department of Computer Engineering, Igdır University, Turkey Author https://orcid.org/0000-0003-0692-0265
Arkan A. Ghaib Department of Information Technology, Management Technical College, Southern Technical University, Basrah, 61004, Iraq Author https://orcid.org/0009-0001-3355-0880

Keywords:

Upside-down-logic, Plithogenic Set, Three-Mode Upside-Down Logic, Double–Valued Neutrosophic Logic, Triple–Valued Neutrosophic Logic

Abstract

In the real world, reversal phenomena are common—for example, assertions once judged false may later be recognized as true. Upside–Down Logic formalizes such reversals by contextually transforming truth and falsity (and their attendant uncertainties), thereby capturing ambiguity and temporal change in reasoning. A Plithogenic Set represents elements via attribute–driven membership and contradiction functions, extending fuzzy, intuitionistic, and neutrosophic paradigms. A Plithogenic Neutrosophic Set further models truth, indeterminacy, and falsity under explicit contradictions, enriching neutrosophic semantics with contextual sensitivity. Double–Valued Neutrosophic Logic assigns truth, falsity, and two indeterminacy components (one leaning toward truth, one toward falsity) to each proposition, whereas Triple–Valued Neutrosophic Logic adds a neutral indeterminacy component. Despite recent interest, extended forms of the Plithogenic Neutrosophic Set remain underexplored in terms of properties and concrete use–cases. To bridge this gap, this paper introduces and formalizes Plithogenic Double–Valued and Plithogenic Triple–Valued Neutrosophic Sets, establishes their fundamental properties and reduction relations, and presents application scenarios demonstrating the practical deployment of Upside–Down Logic.

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References

Jech, T. (2003). Set theory: The third millennium edition, revised and expanded. Springer. https://doi.org/10.1007/3-540-44761-X

Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

Atanassov, K. T. (2020). Circular intuitionistic fuzzy sets. Journal of Intelligent & Fuzzy Systems, 39(5), 5981–5986. https://doi.org/10.3233/JIFS-189072

Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. https://doi.org/10.1002/int.20418

Cuong, B. C. (2015). Picture fuzzy sets. Journal of Computer Science and Cybernetics, 30, 409.

Broumi, S., Talea, M., Bakali, A., & Smarandache, F. (2016). Single valued neutrosophic graphs. Journal of New Theory, 10, 86–101. https://doi.org/10.5281/zenodo.50940

Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Infinite Study.

Smarandache, F. (2013). N-valued refined neutrosophic logic and its applications to physics. Infinite Study, 4, 143–146.

Smarandache, F. (2018). Plithogenic set, an extension of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets-revisited. Infinite Study.

Smarandache, F. (2020). Extension of hypergraph to n-superhypergraph and to plithogenic n-superhypergraph, and extension of hyperalgebra to n-ary (classical-/neutro-/anti-) hyperalgebra. Infinite Study. https://fs.unm.edu/nss8/index.php/111/article/view/198

Fujita, T., & Smarandache, F. (2025a). A unified framework for U-structures and functorial structure: Managing super, hyper, superhyper, tree, and forest uncertain over/under/off models. Neutrosophic Sets and Systems, 91, 337–380. https://fs.unm.edu/nss8/index.php/111/article/view/6984

Fujita, T., & Smarandache, F. (2025b). A dynamic survey of fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and extensional sets. Neutrosophic Science International Association (NSIA). https://philarchive.org/rec/FUJADS

Molodtsov, D. (1999). Soft set theory-first results. Computers & Mathematics with Applications, 37(4-5), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5

Abdel-Basset, M., Mohamed, M., Elhoseny, M., Son, L. H., Chiclana, F., & Zaied, A. N. H. (2019). Cosine similarity measures of bipolar neutrosophic set for diagnosis of bipolar disorder diseases. Artificial Intelligence in Medicine, 101, 101735.

Pang, Y., & Yang, W. (2018). Hesitant neutrosophic linguistic sets and their application in multiple attribute decision making. Information, 9, 88.

Radha, R., Mary, A. S. A., & Smarandache, F. (2021). Quadripartitioned neutrosophic pythagorean soft set. International Journal of Neutrosophic Science, 14(1). https://doi.org/10.54216/IJNS.140101

Das, S., Poojary, P., GR, V. B., & Sharma, S. K. (2024). Single-valued pentapartitioned neutrosophic bi-topological spaces. International Journal of Neutrosophic Science (IJNS), 23(4). https://doi.org/10.54216/IJNS.230415

bin Mohammad Kamari, M. S., Rodzi, Z. B. M., Al-Obaidi, R., Al-Sharq, F., Al-Quran, A., et al. (2025). Deciphering the geometric bonferroni mean operator in pythagorean neutrosophic sets framework. Neutrosophic Sets and Systems, 75, 139–161. https://doi.org/10.5281/zenodo.13932052

Priyadharshini, S., Nirmala Irudayam, F., & Ramkumar, J. (2025). An unique overture of plithogenic cubic overset, underset and offset. In Neutrosophic paradigms: Advancements in decision making and statistical analysis: Neutrosophic principles for handling uncertainty (pp. 139–156). Springer. https://doi.org/10.1007/978-3-031-78505-4_7

Sultana, F., Gulistan, M., Ali, M., Yaqoob, N., Khan, M., Rashid, T., & Ahmed, T. (2023). A study of plithogenic graphs: Applications in spreading coronavirus disease (covid-19) globally. Journal of Ambient Intelligence and Humanized Computing, 14(10), 13139–13159. https://doi.org/10.1007/s12652-022-03772-6

Smarandache, F. (2023). Introduction to the symbolic plithogenic algebraic structures (revisited). Infinite Study. https://fs.unm.edu/nss8/index.php/111/article/view/3250

Fujita, T., Das, A. K., Das, S., Duran, V., Ghaib, A. A., Mehmood, A., Mondal, S. P., & Datta, M. (2025). Plithogenic directed offset and plithogenic multidirected offsets. Neutrosophic Knowledge, 8, 109–134. https://doi.org/10.5281/zenodo.17915302

Azeem, M., Rashid, H., Jamil, M. K., Gütmen, S., & Tirkolaee, E. B. (2024). Plithogenic fuzzy graph: A study of fundamental properties and potential applications. Journal of Dynamics and Games.

Bharathi, T., & Sampath, A. P. (2025). Plithogenic product intuitionistic fuzzy graph. Afrika Matematika, 36(4), 1–18. https://doi.org/10.1007/s13370-025-01378-7

He, H. (2025). A novel approach to assessing art education teaching quality in vocational colleges based on double-valued neutrosophic numbers and multi-attribute decision-making with tree soft sets. Neutrosophic Sets and Systems, 78, 206–218. https://doi.org/10.5281/zenodo.14248915

Zhao, Q., & Li, W. (2025). Incorporating intelligence in multiple-attribute decision-making using algorithmic framework and double-valued neutrosophic sets: Varied applications to employment quality evaluation for university graduates. Neutrosophic Sets and Systems, 76, 59–78. https://doi.org/10.5281/zenodo.13988798

Lin, L. (2025). Double-valued neutrosophic offset for enhancing humanistic competence: An effectiveness study of humanities instruction in vocational college students. Neutrosophic Sets and Systems, 88, 680–689. https://doi.org/10.5281/zenodo.15843644

Fujita, T. (2025a). Triple-valued neutrosophic set, quadruple-valued neutrosophic set, quintuple-valued neutrosophic set, and double-valued indetermsoft set. Neutrosophic Systems with Applications, 25(5), 3. https://doi.org/10.61356/2993-7159.1276

Wang, H. (2025). Professional identity formation in traditional chinese medicine students: An educational perspective using triple-valued neutrosophic set. Neutrosophic Sets and Systems, 88, 83–92. https://doi.org/10.5281/zenodo.15773799

Kandasamy, I. (2018). Double-valued neutrosophic sets, their minimum spanning trees, and clustering algorithm. Journal of Intelligent Systems, 27(2), 163–182. https://doi.org/10.1515/jisys-2016-0088

Du, K., & Du, Y. (2023). Research on performance evaluation of intangible assets operation and management in sports events with double-valued neutrosophic sets. Journal of Intelligent & Fuzzy Systems, 45, 2813–2822.

Guo, Y. (2025). Enhancing dvnn-wcsm technique for double-valued neutrosophic multiple-attribute decision-making in digital economy: A case study on enhancing the quality of development of henan’s cultural and tourism industry. Neutrosophic Sets and Systems, 75, 391–407. https://doi.org/10.5281/zenodo.13955811

Smarandache, F. (1999). A unifying field in logics: Neutrosophic logic. In Philosophy (pp. 1–141). American Research Press.

Fujita, T., & Smarandache, F. (2024). Introduction to upside-down logic: Its deep relation to neutrosophic logic and applications. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Third Volume).

Smarandache, F. (2024). Upside-down logics: Falsification of the truth and truthification of the false. Systems Assessment and Engineering Management, 1, 1. https://doi.org/10.61356/j.saem.2024.1248

Bu, Y. (2025). An upside-down logic-based neutrosophic methodology: Assessment of online marketing effectiveness in e-commerce enterprises. Neutrosophic Sets and Systems, 85, 801–813. https://doi.org/10.5281/zenodo.15380951

Fujita, T., & Mehmood, A. (2026). Formalizing contextual truth flips: Upside-down logic and its variants. Neutrosophic Computing and Machine Learning, 41, 191–212. https://fs.unm.edu/NCML2/index.php/112/article/view/907

Dreman, D. (2008). Contrarian investment strategies: The next generation. Simon & Schuster.

Applebaum, A., Miller, D., Strom, B., Korban, C., & Wolf, R. (2016). Intelligent, automated red team emulation. Proceedings of the 32nd Annual Conference on Computer Security Applications, 363–373.

Smarandache, F. (2022). Plithogeny, plithogenic set, logic, probability and statistics: A short review. Journal of Computational and Cognitive Engineering, 1(2), 47–50. https://doi.org/10.47852/bonviewJCCE2202191

Fujita, T. (2025b). Three-mode upside-down logic within plithogenic neutrosophic set [Accepted]. Journal of Data Science and Intelligent Systems (JDSIS).