International Conference on Data Intelligence and Neutrosophic Sets with Applications (DI-NSA 2019),

Chairman: Prof. Dr. Xiaohong Zhang, Shaanxi University of Science and Technology,

Xi’an, P. R. China, December 20 - 22, 2019

Program of the Conference


Neutrosophic Conference Audience, 20-22 December 2019, Xi'an, P. R. China


 On Overview of Neutrosophic Theories and Applications

Prof. Dr. Florentin Smarandache, Postdoc
University of New Mexico
Mathematics, Physics, and Natural Sciences Division
705 Gurley Ave., Gallup, NM 87301, USA


This presentation is an Overview on the Foundation and Development of Neutrosophic Theories and their Applications for a period of more than two decades (1995-2019) since they were defined and studied, together with their applications, and links to many open-source articles and books that the attendees can download.
Neutrosophic Set is a Generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set (Atanassov’s Intuitionistic Fuzzy Set of second type), q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical Fuzzy Set, while Neutrosophication is a Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision.


    Zadeh introduced the degree of membership/truth (T) in 1965 and defined the fuzzy set.
    Atanassov introduced the degree of nonmembership/falsehood (F) in 1986 and defined the intuitionistic fuzzy set.
    Smarandache introduced the degree of indeterminacy/neutrality (I) as independent component in 1995 (published in 1998) and he defined the neutrosophic set on three components:
(T, I, F) = (Truth, Indeterminacy, Falsehood), where in general T, I, F are subsets of the interval [0, 1]; in particular T, I, F may be intervals, hesitant sets, single-values, etc.; see
F. Smarandache, Neutrosophy / Neutrosophic probability, set, and logic", Proquest, Michigan, USA, 1998,;
reviewed in Zentralblatt fuer Mathematik (Berlin, Germany):
and cited by Denis Howe in The Free Online Dictionary of Computing, England, 1999.
Neutrosophic Set and Logic are generalizations of classical, fuzzy, and intuitionistic fuzzy set and logic:

The words “neutrosophy” and “neutrosophic” were coined/invented by F. Smarandache in his 1998 book.
Neutrosophy: A branch of philosophy, introduced by F. Smarandache in 1980, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophy considers a proposition, theory, event, concept, or entity <A> in relation to its opposite <antiA>, and with their neutral <neutA>.
Neutrosophy (as dynamic of opposites and their neutrals) is an extension of the Dialectics and Yin Yang (which are the dynamic of opposites only).
Neutrosophy is the basis of neutrosophic set, neutrosophic logic, neutrosophic measure, neutrosophic probability, neutrosophic statistics etc.

Neutrosophic Logic is a general framework for unification of many existing logics, such as fuzzy logic (especially intuitionistic fuzzy logic), paraconsistent logic, intuitionistic logic, etc. The main idea of NL is to characterize each logical statement in a 3D-Neutrosophic Space, where each dimension of the space represents respectively the truth (T), the falsehood (F), and the indeterminacy (I) of the statement under consideration, where T, I, F are standard or non-standard real subsets of ]-0, 1+[ with not necessarily any connection between them.
For software engineering proposals the classical unit interval [0, 1] may be used.
While Neutrosophic Probability and Statistics are generalizations of classical and imprecise probability and classical statistics respectively.

The Most Important Books and Papers on the Advancement of Neutrosophics


        1980s - Foundation of Paradoxism that is an international movement in science and culture based on excessive use of contradictions, antitheses, oxymoron, and paradoxes [Smarandache]. During three decades (1980-2020) hundreds of authors from tens of countries around the globe contributed papers to 15 international paradoxist anthologies:

        1995-1998 – Smarandache extended the dialectics (which is a particular case of Chinese philosophy Yin Yan) to neutrosophy;
introduced the neutrosophic set/logic/probability/statistics;
introduces the single-valued neutrosophic set (pp. 7-8); (fourth edition) (online sixth edition)

            1998, 20019 - Introduction of Nonstandard Neutrosophic Logic, Set, Probability

        2002 – Introduction of corner cases of sets / probabilities / statistics / logics, such as:
- Neutrosophic intuitionistic set (different from intuitionistic fuzzy set), neutrosophic paraconsistent set, neutrosophic faillibilist set, neutrosophic paradoxist set, neutrosophic pseudo-paradoxist set, neutrosophic tautological set, neutrosophic nihilist set, neutrosophic dialetheist set, neutrosophic trivialist set;
- Neutrosophic intuitionistic probability and statistics, neutrosophic paraconsistent probability and statistics, neutrosophic faillibilist probability and statistics, neutrosophic paradoxist probability and statistics, neutrosophic pseudo-paradoxist probability and statistics, neutrosophic tautological probability and statistics, neutrosophic nihilist probability and statistics, neutrosophic dialetheist probability and statistics,neutrosophic trivialist probability and statistics;
- Neutrosophic paradoxist logic (or paradoxism), neutrosophic pseudo-paradoxist logic (or neutrosophic pseudo-paradoxism), neutrosophic tautological logic (or neutrosophic tautologism):

        2003 – Introduction by Kandasamy and Smarandache of Neutrosophic Numbers (a+bI, where I = indeterminacy, I^2 = I), I-Neutrosophic Algebraic Structures and Neutrosophic Cognitive Maps

        2005 - Introduction of Interval Neutrosophic Set/Logic

        2006 – Introduction of Degree of Dependence and Degree of Independence between the Neutrosophic Components T, I, F.
For single valued neutrosophic logic, the sum of the components is:
0 ≤ t+i+f ≤ 3 when all three components are independent;
0 ≤ t+i+f ≤ 2 when two components are dependent, while the third one is independent from them;
0 ≤ t+i+f ≤ 1 when all three components are dependent.
When three or two of the components T, I, F are independent, one leaves room for incomplete information (sum < 1), paraconsistent and contradictory information (sum > 1), or complete information (sum = 1).
If all three components T, I, F are dependent, then similarly one leaves room for incomplete information (sum < 1), or complete information (sum = 1).
In general, the sum of two components x and y that vary in the unitary interval [0, 1] is:
0 ≤ x + y ≤ 2 - d°(x, y), where d°(x, y) is the degree of dependence between x and y, while
d°(x, y) is the degree of independence between x and y.
Degrees of Dependence and Independence between Neutrosophic Components T, I, F are independent components, leaving room for incomplete information (when their superior sum < 1), paraconsistent and contradictory information (when the superior sum > 1), or complete information (sum of components = 1).
For software engineering proposals the classical unit interval [0, 1] is used. (p. 92)

        2007 – The Neutrosophic Set was extended [Smarandache, 2007] to Neutrosophic Overset (when some neutrosophic component is > 1), since he observed that, for example, an employee working overtime deserves a degree of membership > 1, with respect to an employee that only works regular full-time and whose degree of membership = 1;
and to Neutrosophic Underset (when some neutrosophic component is < 0), since, for example, an employee making more damage than benefit to his company deserves a degree of membership < 0, with respect to an employee that produces benefit to the company and has the degree of membership > 0;
and to and to Neutrosophic Offset (when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and some neutrosophic component < 0).
Then, similarly, the Neutrosophic Logic/Measure/Probability/Statistics etc. were extended to respectively Neutrosophic Over-/Under-/Off- Logic, Measure, Probability, Statistics etc.

        2007 – Smarandache introduced the Neutrosophic Tripolar Set and Neutrosophic Multipolar Set and consequently
the Neutrosophic Tripolar Graph and Neutrosophic Multipolar Graph (p. 93)

        2009 – Introduction of N-norm and N-conorm

        2013 - Development of Neutrosophic Measure and Neutrosophic Probability
(chance that an event occurs, indeterminate chance of occurrence,
chance that the event does not occur)

        2013 – Smarandache Refined/Split the Neutrosophic Components (T, I, F) into Neutrosophic Subcomponents (T1, T2, ...; I1, I2, ...; F1, F2, ...;):

        2014 – Introduction of the Law of Included Multiple Middle
(<A>; <neut1A>, <neut2A>, …; <antiA>)

        2014 - Development of Neutrosophic Statistics (indeterminacy is introduced into classical statistics with respect to the sample/population, or with respect to the individuals that only partially belong to a sample/population)

        2015 - Introduction of Neutrosophic Precalculus and Neutrosophic Calculus

        2015 – Refined Neutrosophic Numbers (a+ b1I1 + b2I2 + … + bnIn), where I1, I2, …, In are subindeterminacies of indeterminacy I;
        2015 – (t,i,f)-neutrosophic graphs;
        2015 - Thesis-Antithesis-Neutrothesis, and Neutrosynthesis, Neutrosophic Axiomatic System, neutrosophic dynamic systems, symbolic neutrosophic logic, (t, i, f)-Neutrosophic Structures, I-Neutrosophic Structures, Refined Literal Indeterminacy, Quadruple Neutrosophic Algebraic Structures, Multiplication Law of Subindeterminacies:

        2015 – Introduction of the Subindeterminacies of the form (I0)n = k/0, for k ∈  {0, 1, 2, …, n-1}, into the ring of modulo integers Zn - called natural neutrosophic indeterminacies (Vasantha-Smarandache)

        2015 – Introduction of Neutrosophic Crisp Set and Topology (Salama & Smarandache)

        2016 – Introduction of Neutrosophic Multisets (as generalization of classical multisets)

        2016 – Introduction of Neutrosophic Triplet Structures and m-valued refined neutrosophic triplet structures [Smarandache - Ali]

        2016 – Introduction of Neutrosophic Duplet Structures

        2017 - In biology Smarandache introduced the Theory of Neutrosophic Evolution: Degrees of Evolution, Indeterminacy or Neutrality, and Involution

        2017 - Introduction by F. Smarandache of Plithogeny (as generalization of Dialectics and Neutrosophy), and Plithogenic Set/Logic/Probability/Statistics (as generalization of fuzzy, intuitionistic fuzzy, neutrosophic set/logic/probability/statistics)

        2017 - Enunciation of the Law that It Is Easier to Break from Inside than from Outside a Neutrosophic Dynamic System (Smarandache - Vatuiu):

        2018 – Introduction to Neutrosophic Psychology (Neutropsyche, Refined Neutrosophic Memory: conscious, aconscious, unconscious, Neutropsychic Personality, Eros / Aoristos / Thanatos, Neutropsychic Crisp Personality):

        2019 - Theory of Spiral Neutrosophic Human Evolution (Smarandache - Vatuiu):

        2019 - Introduction to Neutrosophic Sociology (Neutrosociology) [neutrosophic concept, or (T, I, F)-concept, is a concept that is T% true,I% indeterminate, and F% false]:

        2019 - Refined Neutrosophic Crisp Set


        2019 - Generalized of the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras)  {whose

 operations and axioms are partially true, partially indeterminate, and partially false} as extensions of Partial Algebra, and to

 AntiAlgebraic Structures (or AntiAlgebras) {whose operations and axioms are totally false}. And, in general, he extended any

 classical Structure, in no matter what field of knowledge, to a NeutroStructure and an AntiStructure:



        Applications in:
Artificial Intelligence, Information Systems, Computer Science, Cybernetics, Theory Methods, Mathematical Algebraic Structures, Applied Mathematics, Automation, Control Systems, Big Data, Engineering, Electrical, Electronic, Philosophy, Social Science, Psychology, Biology, Biomedical, Genetics, Engineering, Medical Informatics, Operational Research, Management Science, Imaging Science, Photographic Technology, Instruments, Instrumentation, Physics, Optics, Economics, Mechanics, Neurosciences, Radiology Nuclear, Medicine, Medical Imaging, Interdisciplinary Applications, Multidisciplinary Sciences etc.
[ Xindong Peng and Jingguo Dai, A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017, Artificial Intelligence Review, Springer, 18 August 2018; ]

        Important Neutrosophic Researchers:

There are about 1,200 neutrosophic researchers, within 74 countries around the globe, that have produced about 2,000 publications within more than two decades. Many neutrosophic researchers got specialized into various fields of neutrosophics:

Xiaohong Zhang & Yingcang  Ma (neutrosophic triplet and quadruple algebraic structures), Yanhui Guo (neutrosophic image processing), Jun Ye & Peide Liu & Jianqiang Wang (neutrosophic optimization), Xindong Peng & Jingguo Dai (neutrosophic bibliometrics), Jianqiang Wang, Guiwu Wei, Donghai Liu, Xiaohong Chen, Dan Peng, Jiongmei Mo, Han-Liang Huang, Victor Chang, Hongjun Guan, Shuang Guan, Aiwu Zhao, Wen-Hua Cui, Xiaofei Yang, Xin Zhou, G.L. Tang, W.L. Liu, Wen Jiang, Zihan Zhang, Xinyang Deng, Changxing Fan, Sheng Feng, En Fan, Keli Hu, Xingsen Li, Xin Zhou, Ping Li;
Rajab Ali Borzooei & Young Bae Jun (neutrosophic BCK/BCI-algebras), Arsham Borumand Saeid (neutrosophic structures), Saied Jafari (neutrosophic topology), Prem Kumar Singh (Neutrosophic MCDM), Maikel Leyva-Vazquez (neutrosophic cognitive maps);

Amira S. Ashour, Muhammad Aslam (neutrosophic statistics), Nguyen Xuan Thao (neutrosophic similarity measures), Le Hoang Son, Vakkas Ulucay & Memet Sahin (neutrosophic quadruple structures), Irfan Deli,
Madad Khan (neutrosophic algebraic structures), Said Broumi & Muhammad Akram (neutrosophic graphs), Mohamed Abdel-Baset (neutrosophic linear and non-linear programming), Ahmed Mostafa Khalil, Ahmed Salama (neutrosophic crisp topology), etc.


        Neutrosophic Journals:

            Neutrosophic Sets and Systems (NSS) international journal started in 2013 and it is indexed by Scopus, Web of Science (ESCI), DOAJ, Index Copernicus, Redalyc - Universidad Autonoma del Estado de Mexico (IberoAmerica), Publons, CNKI (Beijing, China), Chinese Baidu Scholar, etc. ( ).
Submit papers on neutrosophic set/logic/probability/statistics etc. and their applications through our OJS system:

            International Journal of Neutrosophic Science (IJNS, in SCOPUS):
            Neutrosophic Computing and Machine Learning (NCML), in Spanish:

            Neutrosophic Knowledge (NK),
in English and Arabic:

        Encyclopedia of Neutrosophic Researchers
The authors who have published or presented papers on neutrosophics and are not included in the Encyclopedia of Neutrosophic Researchers (ENR), vols. 1, 2, 3, and 4,

are pleased to send their CV, photo, and List of Neutrosophic Publications to in order to be included into the next volume of ENR.


University of New Mexico (USA) web sites: