NEUTROSOPHY, NEUTROSOPHIC LOGIC, NEUTROSOPHIC SET,
NEUTROSOPHIC PROBABILITY AND STATISTICS
(philosophy, math, science)
Short Definitions of Neutrosophics:
1. 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] is used.
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.
In a general Refined Neutrosophic Logic, T can be split into subcomponents T_{1}, T_{2}, ..., T_{p}, and I into I_{1}, I_{2}, ..., I_{r}, and F into F_{1}, F_{2}, ...,F_{s}, where p+r+s = n ≥ 1. Even more: T, I, and/or F (or any of their subcomponents T_{j ,}I_{k}, and/or F_{l}) can be countable or uncountable infinite sets.
As a particular case, one can split the Indeterminate I into Contradiction (true and false), and Uncertainty (true or false), and we get an extension of Belnap's four-valued logic.
Even more, one can split I into Contradiction, Uncertainty, and Unknown, and we get a five-valued logic.
See this most general published case here.
As an example: a statement can be between [0.4, 0.6] true, {0.1} or between (0.15,0.25) indeterminate, and either 0.4 or 0.6 false.
The distinctions between Neutrosophic Logic/Set and Intuitionistic Fuzzy Logic/Set are here.
2. Neutrosophic Set. Let U be a universe of discourse, and M a set included in U. An element x from U is noted with respect to the set M as x(T, I, F) and belongs to M in the following way: it is t% true in the set, i% indeterminate (unknown if it is) in the set, and f% false, where t varies in T, i varies in I, f varies in F.
Statically T, I, F are subsets, but dynamically T, I, F are functions/operators depending on many known or unknown parameters.
Neutrosophic Set generalizes the fuzzy set (especially intuitionistic fuzzy set), paraconsistent set, intuitionistic set, etc.
3. Neutrosophic Probability is a generalization of the classical probability and imprecise probability in which the chance that an event A occurs is t% true - where t varies in the subset T, i% indeterminate - where i varies in the subset I, and f% false - where f varies in the subset F.
In classical probability n_sup <= 1, while in neutrosophic probability n_sup <= 3^{+}.
In imprecise probability: the probability of an event is a subset T in [0, 1], not a number p in [0, 1], what’s left is supposed to be the opposite, subset F (also from the unit interval [0, 1]); there is no indeterminate subset I in imprecise probability.
A book on Introduction to Neutrosophic Probability is here.
4. Neutrosophic Statistics is the analysis of events described by the neutrosophic probability.
The function that models the neutrosophic probability of a random variable x is called neutrosophic distribution: NP(x) = ( T(x), I(x), F(x) ), where T(x) represents the probability that value x occurs, F(x) represents the probability that value x does not occur, and I(x) represents the indeterminate / unknown probability of value x.
A book on Introduction to Neutrosophic Statistics is here.
5. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.
This theory considers every notion or idea <A> together with its opposite or negation <antiA> and with their spectrum of neutralities <neutA> in between them (i.e. notions or ideas supporting neither <A> nor <antiA>). The <neutA> and <antiA> ideas together are referred to as <nonA>. Neutrosophy is a generalization of Hegel's dialectics (the last one is based on <A> and <antiA> only).
According to this theory every idea <A> tends to be neutralized and balanced by <antiA> and <nonA> ideas - as a state of equilibrium.
In a classical way <A>, <neutA>, <antiA> are disjoint two by two.
But, since in many cases the borders between notions are vague, imprecise, Sorites, it is possible that <A>, <neutA>, <antiA> (and <nonA> of course) have common parts two by two, or even all three of them as well.
The Lupasco-Nicolescu’s Law of Included Middle [<A>, <nonA>, and a third value <T> which resolves their contradiction at another level of reality] is generalized to the Law of Included Multiple-Middle [<A>, <antiA>, and <neutA>, where <neutA> is split into a multitude of neutralities between <A> and <antiA>, such as <neut_{1}A>, <neut_{2}A>, etc.]. The <neutA> value (i.e. neutrality or indeterminacy related to <A>) actually comprises the included middle value. Also the Principle of Dynamic Opposition [opposition between <A> and <antiA>] is extended to the Principle of Dynamic Neutrosophic Opposition [which means oppositions among <A>, <antiA>, and <neutA>]. This book is here.
Neutrosophy is the base of neutrosophic logic, neutrosophic set, neutrosophic probability, and neutrosophic statistics that are used in engineering applications (especially for software and information fusion), medicine, military, airspace, cybernetics, physics.
The neutrosophics were introduced by Florentin Smarandache in 1995.
An article on Neutrosophy, A New Branch of Phylosophy is here.
The most important books and papers in the development of neutrosophcis |
||||
1995-1998 - introduction of neutrosophic set/logic/probability/statistics; generalization of dialectics to neutrosophy; http://fs.gallup.unm.edu/eBook-neutrosophics5.pdf (5th edition)
2003 – introduction of neutrosophic numbers (a+bI, where I = indeterminacy) 2003 – introduction of I-neutrosophic algebraic structures 2003 – introduction to neutrosophic cognitive maps http://fs.gallup.unm.edu/NCMs.pdf
2005 - introduction of interval neutrosophic set/logic http://fs.gallup.unm.edu/INSL.pdf
2006 - introduction of the degree of dependence and degree of independence between the neutrosophic components http://fs.gallup.unm.edu/eBook-neutrosophics5.pdf (5th edition)
2009 – introduction of N-norm and N-conorm http://fs.gallup.unm.edu/N-normN-conorm.pdf
2013 - development of neutrosophic probability chance that the event does not occur) http://fs.gallup.unm.edu/NeutrosophicMeasureIntegralProbability.pdf
2013 - refinement of components (T1, T2, ...; I1, I2, ...; F1, F2, ...) http://fs.gallup.unm.edu/n-ValuedNeutrosophicLogic.pdf
2014 – introduction of the law of included multiple middle (<A>; <neut1A>, <neut2A>, …; <antiA>) http://fs.gallup.unm.edu/LawIncludedMultiple-Middle.pdf
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) http://fs.gallup.unm.edu/NeutrosophicStatistics.pdf
2015 - introduction of neutrosophic precalculus and neutrosophic calculus http://fs.gallup.unm.edu/NeutrosophicPrecalculusCalculus.pdf
2015 – refined neutrosophic numbers (a+ b1I1 + b2I2 + … + bnIn) 2015 – 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, Multiplication Law of Subindeterminacies: http://fs.gallup.unm.edu/SymbolicNeutrosophicTheory.pdf |
BOOKS ON NEUTROSOPHICS
Ph. D. Dissertations on Neutrosophic Logic/Set/Probability/Statistics and their Applications:
International Conferences on Neutrosophics:
International Conference on Information Fusion, Tutorial on Foundations of Neutrosophic Set and Logic and Their Applications to Information Fusion, by F. Smarandache, 7th July 2014, Salamanca, Spain.
International Conference on Applications of Plausible, Paradoxical, and Neutrosophic Reasoning for Information Fusion, Cairns, Queensland, Australia, 8-11 July 2003.
Seminars
Foundations of Neutrosophic Logic and Set Theory and their Applications in Science. Neutrosophic Statistics and Neutrosophic Probability. n-Valued Refined Neutrosophic Logic, by F. Smarandache, Universidad Complutense de Madrid, Facultad de Ciencia Matemáticas, Departamento de Geometría y Topología, Instituto Matemático Interdisciplinar (IMI), Madrid, Spain, 9th July 2014.
Tutorial
Tutorial on the Foundations of Neutrosophic Logic and Set and their Applications in Science
Articles on Neutrosophics:
Applications of Neutrosophic Logic to Robotics, by Florentin Smarandache, Luige
Vlădăreanu, 6 p.
Connections between Extension Logic and Refined Neutrosophic Logic, by Florentin Smarandache, 9 p.
Neutrosofia, o nouă ramură a filosofiei, by Florentin Smarandache, 10 p.
Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Set, by Florentin Smarandache, 15 p.
New Operations on Interval Neutrosophic Sets, by Said Broumi, Florentin Smarandache, 11 p.
n-Valued Refined Neutrosophic Logic and Its Applications to Physics, by Florentin Smarandache, 9 p.
Reliability and Importance Discounting of Neutrosophic Masses, by Florentin Smarandache, 14 p.
Several Similarity Measures of Neutrosophic Sets, by Said Broumi, Florentin Smarandache, 10 p.
Deployment of neutrosophic technology to retrieve answer for queries posed in natural language, by Arora, M. ; Biswas, R.; Computer Science and Information Technology (ICCSIT), 2010 3rd IEEE International Conference on, Vol. 3, DOI: 10.1109/ICCSIT.2010.5564125, 2010, 435 – 439.
Neutrosophic modeling and control,
Aggarwal, S.
;
Biswas, R.
;
Ansari, A.Q.
Computer and Communication Technology (ICCCT), 2010 International Conference
on,
DOI:
10.1109/ICCCT.2010.5640435,
2010,
718 – 723.
Truth-value based interval neutrosophic sets,
Wang, H.
;
Yan-Qing Zhang
;
Sunderraman, R.,
Granular Computing, 2005 IEEE International Conference on,
Vol. 1, DOI:
10.1109/GRC.2005.1547284,
2005,
274 – 277;
Cited by 5.
A geometric interpretation of the neutrosophic set — A generalization of the intuitionistic fuzzy set, Smarandache, F., Granular Computing (GrC), 2011 IEEE International Conference on, DOI: 10.1109/GRC.2011.6122665, 2011, 602 – 606.
MRI denoising based on neutrosophic wiener filtering, Mohan, J. ; Yanhui Guo ; Krishnaveni, V.; Jeganathan, K., Imaging Systems and Techniques (IST), 2012 IEEE International Conference on, DOI: 10.1109/IST.2012.6295518, 2012, 327 – 331.
Applications of neutrosophic logic to robotics: An introduction, Smarandache, F. ; Vladareanu, L., Granular Computing (GrC), 2011 IEEE International Conference on, DOI: 10.1109/GRC.2011.6122666, 2011, 607 – 612; Cited by 1.
A Neutrosophic approach of MRI denoising,
Mohan, J.
;
Krishnaveni, V.
;
Guo, Yanhui
Image Information Processing (ICIIP), 2011 International Conference on,
DOI:
10.1109/ICIIP.2011.6108880,
2011, 1
– 6.
Neural Network Ensembles using Interval Neutrosophic Sets and Bagging for Mineral Prospectivity Prediction and Quantification of Uncertainty, Kraipeerapun, P. ; Chun Che Fung ; Brown, W. ; Kok-Wai Wong, Cybernetics and Intelligent Systems, 2006 IEEE Conference on, DOI: 10.1109/ICCIS.2006.252249, 2006, 1 – 6; Cited by 2.
Neutrosophic masses & indeterminate models: Applications to information fusion, Smarandache, F., Information Fusion (FUSION), 2012 15th International Conference on, 2012, 1051 – 1057.
Red Teaming military intelligence - a new approach based on Neutrosophic Cognitive Mapping, Rao, S.; Intelligent Systems and Knowledge Engineering (ISKE), 2010 International Conference on, DOI: 10.1109/ISKE.2010.5680765, 2010, 622 – 627.
Neutrosophic masses & indeterminate models. Applications to information fusion, Smarandache, F., Advanced Mechatronic Systems (ICAMechS), 2012 International Conference on, 2012, 674 – 679.
Validating the Neutrosophic approach of MRI denoising based on structural similarity, Mohan, J. ; Krishnaveni, V. ; Guo, Yanhui; Image Processing (IPR 2012), IET Conference on, DOI: 10.1049/cp.2012.0419, 2012, 1 – 6.
Ensemble Neural Networks Using Interval Neutrosophic Sets and Bagging, Kraipeerapun, P. ; Chun Che Fung ; Kok Wai Wong; Natural Computation, 2007. ICNC 2007. Third International Conference on, Vol. 1, DOI: 10.1109/ICNC.2007.359, 2007, 386 – 390, Cited by 1.
Comparing performance of interval neutrosophic sets and neural networks with support vector machines for binary classification problems, Kraipeerapun, P. ; Chun Che Fung, Digital Ecosystems and Technologies, 2008. DEST 2008. 2nd IEEE International Conference on, DOI: 10.1109/DEST.2008.4635138, 2008, 34 – 37.
Quantification of Uncertainty in Mineral Prospectivity Prediction Using Neural Network Ensembles and Interval Neutrosophic Sets, Kraipeerapun, P. ; Kok Wai Wong ; Chun Che Fung ; Brown, W.; Neural Networks, 2006. IJCNN '06. International Joint Conference on, DOI: 10.1109/IJCNN.2006.247262, 2006, 3034 – 3039.
A neutrosophic description logic,
Haibin Wang
;
Rogatko, A.
;
Smarandache, F.
;
Sunderraman, R.;
Granular Computing, 2006 IEEE International Conference on
DOI:
10.1109/GRC.2006.1635801,
2006,
305 – 308.
Neutrosophic information fusion applied to financial market,
Khoshnevisan, M.
;
Bhattacharya, S.;
Information Fusion, 2003. Proceedings of the Sixth International Conference of
Vol. 2, DOI:
10.1109/ICIF.2003.177381,
2003,
1252 – 1257.
Neutrosophic set - a generalization of the intuitionistic fuzzy set,
Smarandache, F.
Granular Computing, 2006 IEEE International Conference on,
DOI:
10.1109/GRC.2006.1635754,
2006, 38
– 42.
From Fuzzification to Neutrosophication: A Better Interface between Logic and Human Reasoning, Aggarwal, S. ; Biswas, R. ; Ansari, A.Q. Emerging Trends in Engineering and Technology (ICETET), 2010 3rd International Conference on, DOI: 10.1109/ICETET.2010.26, 2010, 21 – 26.
A novel image enhancement approach for Phalanx and Epiphyseal/metaphyseal segmentation based on hand radiographs, Chih-Yen Chen ; Tai-Shan Liao ; Chi-Wen Hsieh ; Tzu-Chiang Liu ; Hung-Chun Chien; Instrumentation and Measurement Technology Conference (I2MTC), 2012 IEEE International, DOI: 10.1109/I2MTC.2012.6229651, 2012, 220-–224.
Quantification of Vagueness in Multiclass Classification Based on Multiple
Binary Neural Networks,
Kraipeerapun, P.
;
Chun Che Fung
;
Kok Wai Wong
Machine Learning and Cybernetics, 2007 International Conference on,
Vol.
1, DOI:
10.1109/ICMLC.2007.4370129,
2007 140
– 144,
Cited by 1.
Automatic Tuning of MST Segmentation of Mammograms for Registration and Mass Detection Algorithms, Bajger, M. ; Fei Ma ; Bottema, M.J.; Digital Image Computing: Techniques and Applications, 2009. DICTA '09. DOI: 10.1109/DICTA.2009.72, 2009. 400 – 407, Cited by 1.
Externalizing Tacit knowledge to discern unhealthy nuclear intentions of nation states, Rao, S., Intelligent System and Knowledge Engineering, 2008. ISKE 2008. 3rd International Conference on, Vol. 1, DOI: 10.1109/ISKE.2008.4730959, 2008, 378 – 383.
Vagueness, a multifacet concept - a case study on Ambrosia artemisiifolia predictive cartography, Maupin, P. ; Jousselme, A.-L. Geoscience and Remote Sensing Symposium, 2004. IGARSS '04. Proceedings. 2004 IEEE International, Vol. 1, DOI: 10.1109/IGARSS.2004.1369036, 2004, Cited by 1.
Analysis of information fusion combining rules under the dsm theory using ESM inputs, Djiknavorian, P. ; Grenier, D. ; Valin, P. ; Information Fusion, 2007 10th International Conference on, DOI: 10.1109/ICIF.2007.4408128, 2007, 1 – 8, Cited by 4.
· A Fast Massively Parallel Fuzzy C-Means Algorithm for Brain MRI Segmentation, by Mohamed Youssfi, Omar Bouattane, Mohammed Ouadi Bensalah, Bouchaib Cherradi. In Wulfenia Journal, nr. 1, Jan, 2015, 19 p.
· A Fuzzy Neutrosophic Soft Set Model in Medical Diagnosis, by I. Arockiarani, 2014, 8 p.
· Fuzzy Neutrosophic Product Space, by A.A. Salama, I.R. Sumathi and I. Arockiarani, 15 p.
· Trapezoidal neutrosophic set and its application to multiple attribute decision making, Jun Ye, Neural Comput & Applic (2015) 26:1157–1166; DOI 10.1007/s00521-014-1787-6.
Seminars on Neutrosophics:
An Introduction to Fusion Level 1 and to Neutrosophic Logic/Set with Applications, presented by F. Smarandache at Air Force Research Laboratory, in Rome, NY, USA, July 29, 2009.
A Neutrosophic Description Logic, by Haibin Wang, Florentin Smarandache, Andre Rogatko, Rajshekhar Sunderraman, 2006 IEEE International Conference on Granular Computing, Georgia State University, Atlanta, USA, May 11, 2006. | |||
An Introduction to Neutrosophic Logic and Set, by F. Smarandache, Invited Speaker at and sponsored by University Kristen Satya Wacana, Salatiga, Indonesia, May 24, 2006. | |||
An Introduction to Neutrosophic Logic and Set, by F. Smarandache, Invited Speaker at and sponsored by University Sekolah Tinggi Informatika & Komputer Indonesia, Malang, Indonesia, May 19, 2006. | |||
Neutrosophic Set - A Generalization of the Intuitionistic Fuzzy Set, by F. Smarandache (Chair of the Session on Soft Computing), 2006 IEEE International Conference on Granular Computing, Georgia State University, Atlanta, USA, May 11, 2006. |
Introduction to Neutrosophics and their Applications, by F. Smarandache, Invited speaker at Pushchino Institute of Theoretical and Experimental Biophysics, Pushchino (Moscow region), Russia, August 9, 2005.
To be and Not to be - An Introduction to Neutrosophy: A Novel Decision Paradigm, by F. Smarandache & S. Bhattacharya, Invited speakers at and sponsored by Jadavpur University, Seminar at the Institute of Business Management, National Council of Education, Kolkata, India, December 23, 2004. |
Generalization of the Intuitionistic Fuzzy Set to the Neutrosophic Set, by F. Smarandache & M. Khoshnevisan, 2003 BISC FLINT-CIBI International Workshop on Soft Computing for Internet and Bioinformatics, University of Berkeley, December 15-19, 2003.
Neutrosophic Logic Operators, by F. Smarandache, International Congress of Mathematicians, Beijing, China, 20-28 August 2002.
A Unifying Field in Logics: Neutrosophic Logic. / Neutrosophic Probability, Neutrosophic Set, by F. Smarandache, American Mathematical Society Meeting, University of California at Santa Barbara, USA, March 11, 2000. |
Neutrosophic Probability, Set, and Logic, by F. Smarandache, Second Conference of the Romanian Academy of Scientists, American Branch, New York City, USA, February 2, 1999.
Neutrosophic Subjects for Future Research:
Neutrosophic topologies, including neutrosophic metric spaces and smooth topological spaces. | |
Neutrosophic numbers (a+bI, where I = indeterminate and I^2 = I, mI+nI=(m+n)I, 0I = 0, and a, b are real or complex numbers) and arithmetical operations, including ranking procedures for neutrosophic numbers. | |
Neutrosophic rough sets. | |
Neutrosophic relational structures, including neutrosophic relational equations, neutrosophic similarity relations, and neutrosophic ordering. | |
Neutrosophic geometry (Smarandache geometries). | |
Neutrosophic probability. | |
Neutrosophic logical operations, including n-norms, n-conorms, neutrosophic implicators, neutrosophic quantifiers. | |
Measures of neutrosophication. | |
Deneutrosophication techniques. | |
Neutrosophic measures, and neutrosophic integrals. | |
Neutrosophic multivalued mappings. | |
Neutrosophic differential calculus. | |
Neutrosophic mathematical morphology. | |
Neutrosophic algebraic structures. |
Neutrosophic triplet structures.
Neutrosophic models.
Neutrosophic matrix, bimatrix, ..., n-matrix.Neutrosophic graph, which is a graph that has at least one indeterminate edge or one indeterminate node.
Neutrosophic tree, which is a tree that has at least one indeterminate edge or one indeterminate node.
Applications: neutrosophic relational databases, neutrosophic image (thresholding, denoising, segmentation) processing, neutrosophic linguistic variables, neutrosophic decision making and preference structures, neutrosophic expert systems, neutrosophic reliability theory, neutrosophic soft computing techniques in e-commerce and e-learning, image segmentation, etc. |
Webinar called A Perspective Shift from Fuzzy Logic to Neutrosophic Logic, by Dr. Swati Aggarwal, winner of the 2015 IEEE CIS Webinar Competition for Students and Professionals - Real World Applications and Emerging Topics in Computational Intelligenceat https://youtu.be/WryVUv5Bq98
Submit papers for the special session FUZZ-IEEE-35 Moving Towards Neutrosophic Logic, organized by Swati Aggarwal ( swati1178@gmail.com ), at the IEEE World Congress on Computational Intelligence, Vancouver, Canada, 25-29 July 2016, http://www.wcci2016.org/spsessions.php
Neutrosophic Sets and Systems (international journal): 2013 (Vol. 1); 2014 (Vol. 2; Vol. 3; Vol. 4; Vol. 5; Vol. 6); 2015 (Vol. 7; Vol. 8; Vol. 9; Vol. 10); 2016 (Vol. 11) new journal
Neutrosophic Science International Association