History of Neutrosophic Theory and its Applications by Florentin Smarandache
History of neutrosophic set and logic, generalization of intuitionistic fuzzy set and logic
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 defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood). The words “neutrosophy” and “neutrosophic” were coined/invented by F. Smarandache in his 1998 book. Etymologically, “neutrosophy” (noun) [French neutre <Latin neuter, neutral, and Greek sophia, skill/wisdom] means knowledge of neutralthought.
While “neutrosophic” (adjective), means having the nature of, or having the characteristic of Neutrosophy.
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 3DNeutrosophic 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 nonstandard 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 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 2013 Smarandache refined the neutrosophic set to n components:
(T_{1}, T_{2}, ...; I_{1}, I_{2}, ...; F_{1}, F_{2}, ...); See http://fs.gallup.unm.edu/nValuedNeutrosophicLogicPiP.pdf
Neutrosophy (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, "AntiA" and that which is not A, "NonA", and that which is neither "A" nor "AntiA", denoted by "NeutA".
Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics. {From: The Free Online Dictionary of Computing, edited by Denis Howe from England. Neutrosophy is an extension of the Dialectics.}
The Most Important Books and Papers in the Development of Neutrosophics
 19951998  introduction of neutrosophic set/logic/probability/statistics; generalization of dialectics to neutrosophy; http://fs.gallup.unm.edu/ebookneutrosophics6.pdf (last edition)
 2003 – introduction of neutrosophic numbers (a+bI,where I = indeterminacy); introduction of Ineutrosophic algebraic structures; 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 degree of dependence and degree of independence; between the neutrosophic components T, I, F, http://fs.gallup.unm.edu/ebookneutrosophics6.pdf (p. 92), http://fs.gallup.unm.edu/NSS/DegreeOfDependenceAndIndependence.pdf

2007 – The Neutrosophic Set was extended
[Smarandache, 2007] to Neutrosophic Overset (when some neutrosophic component is > 1), and to Neutrosophic Underset (when some neutrosophic component is < 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., http://fs.gallup.unm.edu/ebookneutrosophics6.pdf (pp. 9293), http://fs.gallup.unm.edu/IFSgeneralized.pdf  2007 – Smarandache introduced the Neutrosophic Tripolar Set and Neutrosophic Multipolar Set; the Neutrosophic Tripolar Graph and Neutrosophic Multipolar Graph, http://fs.gallup.unm.edu/ebookneutrosophics6.pdfm(p. 93), http://fs.gallup.unm.edu/IFSgeneralized.pdf
 2009 – introduction of Nnorm and Nconorm, http://fs.gallup.unm.edu/NnormNconorm.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 (T_{1}, T_{2}, ...; I_{1}, I_{2}, ...; F_{1}, F_{2}, ...),
http://fs.gallup.unm.edu/nValuedNeutrosophicLogic.pdf 
2014 – introduction of the law of included multiple middle (<A>; <neut1A>, <neut2A>, …;
<antiA>), http://fs.gallup.unm.edu/LawIncludedMultipleMiddle.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+ b_{1}I_{1} + b_{2}I_{2} + … + b_{n}I_{n}), where I_{1}, I_{2}, …, I_{n} are subindeterminacies of indeterminacy I;
 2015 – (t,i,f)neutrosophic graphs;
 2015 ThesisAntithesisNeutrothesis, and Neutrosynthesis, Neutrosophic Axiomatic System, neutrosophic dynamic systems, symbolic neutrosophic logic, (t, i, f)Neutrosophic Structures, INeutrosophic Structures, Refined Literal Indeterminacy, Multiplication Law of Subindeterminacies: http://fs.gallup.unm.edu/SymbolicNeutrosophicTheory.pdf
 2015 – Introduction of the subindeterminacies of the form , for k ∈ {0, 1, 2, …, n1}, into the ring of modulo integers Z_{n}  called natural neutrosophic indeterminacies [VasanthaSmarandache], http://fs.gallup.unm.edu/MODNeutrosophicNumbers.pdf
 2015 – Introduction of neutrosophic triplet structures and mvalued refined neutrosophic triplet structures [Smarandache  Ali]