History of neutrosophic set and logic
F.Smarandache
Mathematics & Science Department
University of New Mexico
705 Gurley Ave., Gallup, NM 87301,
http://fs.gallup.unm.edu/FlorentinSmarandache.htm
E-mail: fsmarandache@gmail.com
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 1998 and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood). He has coined/invented the English words “neutrosophy” and “neutrosophic”, included now in American dictionaries.
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.
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.
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 2013 Smarandache refined the neutrosophic set to n components: t1, t2, ...; i1, i2, ...; f1, f2, ... . See http://fs.gallup.unm.edu/n-ValuedNeutrosophicLogic-PiP.pdf .
Neutrosophy
<philosophy> (From Latin "neuter" - neutral, Greek "sophia" - skill/wisdom) A branch of philosophy, introduced by Florentin 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, "Anti-A" and that which is not A, "Non-A", and that which is neither "A" nor "Anti-A", denoted by "Neut-A".
Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics.
{From: The Free Online Dictionary of Computing, is edited by Denis Howe from England.
Neutrosophy is an extension of the Dialectics.}
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-neutrosophics4.pdf (4th 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
2009 – introduction of N-norm and N-conorm
http://fs.gallup.unm.edu/N-normN-conorm.pdf
2013 - development of neutrosophic probability
(chance that an event occurs, indeterminate chance of occurrence,
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