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: In this paper, we present the lattice structures of neutrosophic theories. We prove that Zhang-Zhang’s YinYang bipolar fuzzy set is a subclass of the Single-Valued bipolar neutrosophic set. Then we show that the pair structure is a particular case of reﬁned neutrosophy, and the number of types of neutralities (sub-indeterminacies) may be any ﬁnite or inﬁnite number.


Introduction
First, we prove that Klement Dand Mesiar's lattices [1] do not fit the general definition of neutrosophic set, and we construct the appropriate nonstandard neutrosophic lattices of the first type (as neutrosophically ordered set) [2], and of the second type (as neutrosophic algebraic structure, endowed with two binary neutrosophic laws, inf N and sup N ) [2].
We also present the novelties that neutrosophy, neutrosophic logic, set, and probability and statistics, with respect to the previous classical and multi-valued logics and sets, and with the classical and imprecise probability and statistics, respectively.
Second, we prove that Zhang-Zhang's YinYang bipolar fuzzy set [3,4] is not equivalent with but a subclass of the Single-Valued bipolar neutrosophic set.
Third, we show that Montero, Bustince, Franco, Rodríguez, Gómez, Pagola, Fernández, and Barrenechea's paired structure of the knowledge representation model [5] is a particular case of Refined Neutrosophy (a branch of philosophy that generalized dialectics) and of the Refined Neutrosophic Set [6].We disprove again the claim that the bipolar fuzzy set (renamed as YinYang bipolar fuzzy set) is the same of neutrosophic set as asserted by Montero et al [5].
About the three types of neutralities presented by Montero et al., we show, by examples and formally, that there may be any finite number or an infinite number of types of neutralities n, or that indeterminacy (I), as neutrosophic component, can be refined (split) into 1 ≤ n ≤ ∞ number of sub-indeterminacies (not only 3 as Montero et al. said) as needed to each application to solve.Also, we show, besides numerous neutrosophic applications, many innovatory contributions to science were brought on by the neutrosophic theories, such as: generalization of Yin Yang Chinese philosophy and dialectics to neutrosophy [7], a new branch of philosophy that is based on the dynamics of opposites and their neutralities, the sum of the neutrosophic components T, I, F up to 3, the degrees of dependence/independence between the neutrosophic components [8,9]; the distinction between absolute truth and relative truth in the neutrosophic logic [10], the introduction of nonstandard neutrosophic logic, set, and probability after we have extended the nonstandard analysis [11,12], the refinement of neutrosophic components into subcomponents [6]; the ability to express incomplete information, complete information, paraconsistent (conflicting) information [13,14]; and the extension of the middle principle to the multiple-included middle principle [15], introduction of neutrosophic crisp set and topology [16], and so on.

Oversimplification of the Neutrosophic Set
At [1], page 10 (Section 3.3) in their paper, related to neutrosophic sets, they wrote: "As a straightforward generalization of the product lattice I × I, ≤ comp , for each n ∈ N, the n-dimensional unit cube I n , ≤ comp , i.e., the n-dimensional product of the lattice (I, ≤ comp ), can be defined by means of (1) and (2).
The so-called "neutrosophic" sets introduced by F. Smarandache [93] (see also [94][95][96][97], which are based on the bounded lattices I 3 , ≤ I 3 and I 3 , ≤ I 3 , where the orders ≤ I 3 and ≤ I 3 on the unit cube I 3 are defined by the Equations below. (x 1 , x 2 , x 3 ) ≤ I 3 (y 1 , y 2 , y 3 ) ⇔ x 1 ≤ y 1 AND x 2 ≤ y 2 AND x 3 ≥ y 3 (-13) The authors have defined Equations (1) and (2) as follows: The authors did not specify what type of lattices they employ: of the first type (lattice, as a partially ordered set), or the second type (lattice, as an algebraic structure).Since their lattices are endowed with some inequality (referring to the neutrosophic case), we assume it is as the first type.
The authors have used the notations: The order relationship ≤ comp on I 3 can be defined as: The three lattices they constructed are denoted by KL 1 , KL 2 , KL 3 , respectively.
Contain only the very particular case of standard single-valued neutrosophic set, i.e., when the neutrosophic components T (truth-membership), I (indeterminacy-membership), and F (false-membership) of the generic element x(T, I, F), of a neutrosophic set N are single-valued (crisp) numbers from the unit interval [0, 1].
The authors have oversimplified the neutrosophic set.Neutrosophic is much more complex.Their lattices do not characterize the initial definition of the neutrosophic set ([10], 1998): a set whose elements have the degrees of appurtenance T, I, F, where T, I, F are standard or nonstandard subsets of the nonstandard unit interval: ] − 0, 1 + [, where ] − 0, 1 + [ overpasses the classical real unit interval [0, 1] to the left and to the right.

The Most General Neutrosophic Lattices
The authors' lattices are far from catching the most general definition of the neutrosophic set.Let  be a universe of discourse, and  ⊂  be a set.

Distinction between Absolute Truth and Relative Truth
The authors' lattices are incapable of making distinctions between absolute truth (when  = 1 > 1) and relative truth (when  = 1) in the sense of Leibniz, which is the essence of nonstandard neutrosophic logic.
While the most general refined neutrosophic lattices of the first type is: ] − Ω, Ψ + [, ≤ nonS nN , where ≤ nonS nN is the n-tuple nonstandard neutrosophic inequality dealing with nonstandard subsets, defined as:

Distinction between Absolute Truth and Relative Truth
The authors' lattices are incapable of making distinctions between absolute truth (when T = 1 + > N 1) and relative truth (when T = 1) in the sense of Leibniz, which is the essence of nonstandard neutrosophic logic.

Neutrosophic Standard Subset Lattices
Their three lattices are not even able to deal with standard subsets [including intervals [8], and hesitant (discrete finite) subsets] T, I, F ⊆ [0, 1], since they have defined the 3D-inequalities with respect to single-valued (crisp) numbers: In order to deal with standard subsets, they should use inf/sup, i.e., The Nonstandard Refined Neutrosophic Set [2,6,12], defined on ] − 0, 1 + [ n , strictly includes their n-dimensional unit cube (I n ), and we use a nonstandard neutrosophic inequality, not the classical inequalities, to deal with inequalities of monads and binads, such as ≤ nonS nN and ≤ nonS N .Not even the Standard Refined Single-Valued Neutrosophic Set [6] (2013) may be characterized with KL 1 , KL 2 , and KL 3 nor with I n , ≤ comp , since the n-D neutrosophic inequality is different from n-D ≤ comp , and from n-D extensions of ≤ I 3 or ≤ I 3 respectively, as follows: Let T be refined into T 1 , T 2 , . . ., T p ; I be refined into I 1 , I 2 , . . ., I r ; and F be refined into F 1 , F 2 , . . ., F s ; with p, r, s ≥ 1 are integers, and p + r + s = n ≥ 4, produced the following n-D neutrosophic inequality.

Sum of Neutrosophic Components up to 3
The authors do not mention the novelty of neutrosophic theories regarding the sum of single-valued neutrosophic components T + I + F ≤ 3, extended up to 3, and, similarly, the corresponding inequality when T, I, F are subsets of [0, 1]: supT + supI + supF ≤ 3, for neutrosophic set, neutrosophic logic, and neutrosophic probability never done before in the previous classic logic and multiple-valued logics and set theories, nor in the classical or imprecise probabilities.This makes a big difference, since, for a single-valued neutrosophic set S, all unit cubes [0, 1] 3 are fulfilled with points, each point P(a, b, c) into the unit cube may represent the neutrosophic coordinates (a, b, c) of an element x(a, b, c) ∈ S, which was not the case for previous logics, sets, and probabilities.This is not the case for the Picture Fuzzy Set (Cuong [21], 2013) whose domain is 1  6 of the unit cube (a cube corner): For Intuitionistic Fuzzy Set (Atanassov [22], 1986), the following is true.

Etymology of Neutrosophy and Neutrosophic
The authors [1] write ironically twice, in between quotations, "neutrosophic" because they did not read the etymology [10] of the word published into my first book (1998), etymology, which also appears into Denis Howe's 1999 The Free Online Dictionary of Computing [23], and, afterwards, repeated by many researchers from the neutrosophic community in their published papers: Neutrosophy [23]: <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.
While neutrosophic means what is derived/resulted from neutrosophy.
Unlike the "intuitionistic|" and "picture fuzzy" notions, the notion of neutrosophic was carefully and meaningfully chosen, coming from neutral (or indeterminate, denoted by <neutA>) between two opposites, A and antiA , which made the main distinction between neutrosophic logic/set/probability, and the previous fuzzy, intuitionistic fuzzy logics and sets, i.e., -For neutrosophic logic neither true nor false, but neutral (or indeterminate) in between them; -Similarly for neutrosophic set: neither membership nor non-membership, but in between (neutral, or indeterminate); -And analogously for neutrosophic probability: chance that an event E occurs, chance that the event E does not occur, and indeterminate (neutral) chance of the event E of occurring or not occuring.
Their irony is malicious and ungrounded.

Neutrosophy as Extension of Dialectics
Let A be a concept, notion, idea, or theory.Then antiA is the opposite of A , while neutA is the neutral (or indeterminate) part between them.
While in philosophy, Dialectics is the dynamics of opposites ( A and antiA ), Neutrosophy is an extension of dialectics.In other words, neutrosophy is the dynamics of opposites and their neutrals ( A , antiA , neutA ), because the neutrals play an important role in our world, interfering in one side or the other of the opposites.
The authors are uninformed so that a generalization was done in 2013 when we have published a paper [6] that introduced, for the first time, the refined neutrosophic set/logic/probability, where T, I, F were refined into n neutrosophic subcomponents: T 1 , T 2 , . . ., T p ; I 1 , I 2 , . . ., I r ; F 1 , F 2 , . . ., F s , With p, r, s ≥ 1 are integers and p + r + s = n ≥ 4. But in our lattice (I n , ≤ nN ), the neutrosophic inequality is adjusted to the categories of sub-truths, sub-indeterminacies, and sub-falsehood, respectively.

Nonstandard Refined Neutrosophic Set and Lattice
Even more, Nonstandard Refined Neutrosophic Set/Logic/Probability (which include infinitesimals, monads, and closed monads, binads and closed binads) has no connection and no isomorphism whatsoever with any of the authors' lattices or extensions of their lattices for 2D and 3D to nD.

Nonstandard Neutrosophic Mobinad Real Lattice
We have built ( [2], 2018) a more complex Nonstandard Neutrosophic Mobinad Real Lattice, on the nonstandard mobinad unit interval ] − 0, 1 + [ defined as: which is both nonstandard neutrosophic lattice of the first type (as partially ordered set, under neutrosophic inequality ≤ N ) and lattice of the second type (as algebraic structure, endowed with two binary nonstandard neutrosophic laws: inf N and sup N ).

-
The sum of the neutrosophic components is up to 3 (previously the sum was up to 1); -Degree of independence and dependence between the neutrosophic components T, I, F, making their sum T + I + F vary between 0 and 3.
For example, when T, I, and F are totally dependent with each other, then T + I + F ≤ 1.Therefore, we obtain the particular cases of intuitionistic fuzzy set (when T + I + F = 1) and picture set when -Nonstandard analysis used in order to distinguish between absolute and relative (truth, membership, chance).
-Refinement of the components into sub-components: Klement's and Mesiar's claim that the neutrosophic set (I do not talk herein about intuitionistic fuzzy set, picture fuzzy set, and Pythagorean fuzzy set that they criticized) is not a new result is far from the truth.
However, the "YinYang bipolar" is already a pleonasm, because, in Taoist Chinese philosophy, from the 6th century BC, Yin and Yang was already a bipolarity, between negative (Yin)/positive (Yang), or feminine (Yin)/masculine (Yang).
Dialectics was derived, much later in time, from Yin Yang.Neutrosophy, as the dynamicity and harmony between opposites (Yin <A> and Yang (antiA>) together with their neutralities (things which are neither Yin nor Yang, or things which are blends of both: <neutA>) is an extension of Yin Yang Chinese philosophy.Neutrosophy came naturally since, into the dynamicity, conflict, cooperation, and even ignorance between opposites, the neutrals are attracted and play an important role.

YinYang Bipolar Fuzzy Set Is the Bipolar Fuzzy Set
The authors sincerely recognize that: "In the existing papers, YinYang bipolar fuzzy set also was called bipolar fuzzy set [5] and bipolar-valued fuzzy set [13,16]." These papers are cited as References [31][32][33].
We prove that the YinYang bipolar fuzzy set is not equivalent with the neutrosophic set, but a particular case of the bipolar neutrosophic set.
The authors [3] Let U be a universe of discourse, and M ⊂ U be a set.M is a single-valued bipolar neutrosophic set, if for any element: 3.3.Dependent Indeterminacy vs. Independent Indeterminacy The authors say: "Attanassov's intuitionistic fuzzy set [4] perfectly reflects indeterminacy but not bipolarity." We disagree, since Atanassov's intuitionistic fuzzy set [22] perfectly reflects hesitancy between membership and non-membership not indeterminacy, since hesitancy is dependent on membership and non-membership: H = 1 − T − F, where H = hesitancy, T = membership, and F = non-membership.
It is the single-valued neutrosophic set that "perfectly reflects indeterminacy" since indeterminacy (I) in the neutrosophic set is independent from membership (T) and from nonmembership (F).
On the other hand, the neutrosophic set perfectly reflects the bipolarity membership/non-membership as well, since the membership (T) and nonmembership (F) are independent of each other.

Dependent Bipolarity vs. Independent Bipolarity
The bipolarity in the single-valued fuzzy set and intuitionistic fuzzy set is dependent (restrictive) in the sense that, if the truth-membership is T, then it involves the falsehood-nonmembership F ≤ 1 − T while the bipolarity in a single-valued neutrosophic set is independent (nonrestrictive): if the truth-membership T ∈ [0, 1], the falsehood-nonmebership is not influenced at all, then F ∈ [0, 1].

Equilibriums and Neutralities
Again: "While, in semantics, the YinYang bipolar fuzzy set suggests equilibrium, and neutrosophic set suggests a general neutrality.While the neutrosophic set has been successfully applied to a medical diagnosis [9,27], from the above analysis and the conclusion in [31], we see that the YinYang bipolar fuzzy set is clearly the suitable model to a bipolar disorder diagnosis and will be adopted in this paper."I'd like to add that the single-valued bipolar neutrosophic set suggests: three types of equilibrium, between: T + (x) and T − (x) , I + (x) and I − (x) , and F + (x) and F − (x) ; -and two types of neutralities (indeterminacies) between T + (x) and F + (x) , and between T − (x) and F − (x) .
Therefore, the single-valued bipolar neutrosophic set is 3 × 2 = 6 times more complex and more flexible than the YinYang bipolar fuzzy set.Due to higher complexity, flexibility, and capability of catching more details (such as falsehood-nonmembership, and indeterminacy), the single-valued bipolar neutrosophic set is more suitable than the YinYang bipolar fuzzy set to be used in a bipolar disorder diagnosis.
This sentence is false and we proved previously that what Zhang & Zhang proposed in 2004 is a subclass of the single-valued bipolar neutrosophic set.

Tripolar and Multipolar Neutrosophic Sets
Not talking about the fact that, in 2016, we have extended our bipolar neutrosophic set to tripolar and even multipolar neutrosophic sets [18], the sets have become more general than the bipolar fuzzy model.

Neutrosophic Algebraic Structures
The Montero et al. [5] continue: "Notice that none of these two equivalent models include any formal structure, as claimed in [48]".
First, we have proved that these two models (Zhang-Zhang's bipolar fuzzy set, and neutrosophic logic) are not equivalent at all.Zhang-Zhang's bipolar fuzzy set is a subclass of a particular type of neutrosophic set, called the single-valued bipolar neutrosophic set.

Neutrality (<neutA>)
Montero et al. [5] continue: " . . . the selected denominations within each model might suggest different underlying structures: while the model proposed by Zhang and Zhang suggests conflict between categories (a specific type of neutrality different from Atanassov's indeterminacy), Smarandache suggests a general neutrality that should, perhaps jointly, cover some of the specific types of neutrality considered in our paired approach." In neutrosophy and neutrosophic set/logic/probability, the neutrality <neutA> means everything in between <A> and <antiA>, everything which is neither <A> nor <antiA>, or everything which is a blending of <A> and <antiA>.
Thus, the paired structure becomes a particular case of refined neutrosophy (see next).

The Pair Structure as a Particular Case of Refined Neutrosophy
Montero et al. [5] in 2016 have defined a paired structure: "composed by a pair of opposite concepts and three types of neutrality as primary valuations: L = {concept, opposite, indeterminacy, ambivalence, conflict}."Therefore, each element x ∈ X, where X is a universe of discourse, is characterized by a degree function, with respect to each attribute value from L: where µ 1 (x) represents the degree of x with respect to the concept; µ 2 (x) represents the degree of x with respect to the opposite (of the concept); µ 3 (x) represents the degree of x with respect to 'indeterminacy'; µ 4 (x) represents the degree of x with respect to 'ambivalence'; µ 5 (x) represents the degree of x with respect to 'conflict'.However, this paired structure is a particular case of Refined Neutrosophy.

Antonym vs. Negation
First, Dialectics is the dynamics of opposites.Denote them by A and antiA , where A may be an item, a concept, attribute, idea, theory, and so on while antiA is the opposite of A .
Secondly, Neutrosophy ([10], 1998), as a generalization of Dialectics, and a new branch of philosophy, is the dynamics of opposites and their neutralities (denoted by neutA ).Therefore, Neutrosophy is the dynamics of A , antiA , and neutA .
neutA means everything, which is neither A nor antiA , or which is a mixture of them, or which is indeterminate, vague, or unknown.
The antonym of A is antiA .The negation of A (which we denote by nonA ) is what is not A , therefore: We preferred to use the lower index N (neutrosophic) because we deal with items, concepts, attributes, ideas, and theories such as A and, in consequence, its derivates antiA , neutA , and nonA , whose borders are ambiguous, vague, and not clearly delimited.

Multi-Subpolar Refined Neutrosophy
However, the Refined Neutrosophy, whose at least one of A or antiA is refined, is multi-subpolar.

Multidimensional Fuzzy Set as a Particular Case of the Refined Neutrosophic Set
Montero et al. [5] defined the Multidimensional Fuzzy Set A L as: A t = < x; (µ s (x)) s∈L > x ∈ X , where X is the universe of discourse, L = the previous qualitative scale, and µ s (x) ∈ S, where S is a valuation scale (in most cases S = [0, 1]), µ s (x) is the degree of x with respect to s ∈ L.
A Single-Valued Neutrosophic Set is defined as follows.Let U be a universe of discourse, and M ⊂ U a set.For each element x(T(x), I(x), F(x)) ∈ M, T(x) ∈ [0, 1] is the degree of truth-membership of element x with respect to the set M, I(x) ∈ [0, 1] is the degree of indeterminacy-membership of element x with respect to the set M, and F(x) ∈ [0, 1] is the degree of falsehood-nonmembership of element x with respect to the set M.
The Single-Valued Refined Neutrosophic Set is defined as follows.Let U be a universe of discourse, and M ⊂ U a set.For each element: x T 1 (x), T 2 (x), . . ., T p (x); I 1 (x), I 2 (x), . . ., I r (x); F 1 (x), F 2 (x), . . ., F s (x) ∈ M T j (x), 1 ≤ j ≤ p, are degrees of subtruth-submembership of element x with respect to the set M. I k (x), 1 ≤ k ≤ r, are degrees of subindeterminacy-membership of element x with respect to the set M.
Lastly, F l (x), 1 ≤ l ≤ s, are degrees of sub-falsehood-sub-non-membership of element x with respect to the set M, where integers p, r, s ≥ 1, and p + r + s = n ≥ 4.

Plithogeny and Plithogenic Set
Fourthly, in 2017 and in 2018 [24][25][26][27], the Neutrosophy was extended to Plithogeny, which is multipolar, being the dynamics and hermeneutics [methodological study and interpretation] of many opposites and/or their neutrals, together with non-opposites.
Unlike previous sets defined, whose elements were characterized by the attribute 'appurtenance' (to the set), which has only one (membership), or two (membership, nonmembership), or three (membership, nonmembership, indeterminacy) attribute values, respectively.For the Plithogenic Set, each element may be characterized by a multi-attribute, with any number of attribute values.

Refined Neutrosophic Set as a Unifying View of Opposite Concepts
Montero et al.'s statement [5] from their paper Abstract: "we propose a consistent and unifying view to all those basic knowledge representation models that are based on the existence of two somehow opposite fuzzy concepts." With respect to the "unifying" claim, their statement is not true, since, as we proved before, their paired structure together with three types on neutralities (indeterminacy, ambivalence, and conflict) is a simple, particular case of the refined neutrosophic set.
The real unifying view currently is the Refined Neutrosophic Set.
{I was notified about this paired structure article [5]  As in the neutrosophic set, there are three possibilities: T = percentage of USA people voting for Mr. Trump; I = percentage of USA people not voting, or voting but giving either a blank vote (not selecting any candidate) or a black vote (cutting all candidates); F = percentage of USA people voting against Mr. Trump.The opposite concepts, using Montero et al.'s knowledge representation, are T (voting for, or truth-membership) and F (voting against, or false-membership).However, T > F, or T = F, or T < F, that the Paired Structure can catch, mean only the Popular vote, which does not count in the United States.
Actually, it happened that T < F in the US 2016 presidential election, or Mr. Trump lost the Popular vote, but he won the Presidency using the Electoral vote.
The paired structure is not capable of refining the opposite concepts (T and F), while the indeterminate (I) could be refined by the paired structure only in three parts.
Therefore, the paired structure is not a unifying view of all basic knowledge that uses opposite fuzzy concepts.However, the refined neutrosophic set/logic/probability do.
Using the refined neutrosophic set and logic, and splits (refines) T, I, and F as: T j = percentage of American state S j people voting for Mr. Trump; I j = percentage of American state S j people not voting, or casting a blank vote or a black vote; F j = percentage of American state S j people voting against Mr. Trump, with T j , I j , F j ∈ [0, 1] and T j + I j + F j = 1, for all j ∈ {1, 2, . . ., 50}.
On the other hand, due to the fact that the sub-indeterminacies I 1 , I 2 , . . ., I 50 did not count towards the winner or looser (only for indeterminate voting statistics), it is not mandatory to refine I.We could simply refine it as: (T 1 , T 2 , . . ., T 50 ; I; F 1 , F 2 , . . ., F 50 ).

Finite Number and Infinite Number of Neutralities
Montero et al. [5]: "( . . . ) we emphasize the key role of certain neutralities in our knowledge representation models, as pointed out by Atanassov [4], Smarandache [70], and others.However, we notice that our notion of neutrality should not be confused with the neutral value in a traditional sense (see [22][23][24]36,54], among others).
Instead, we will stress the existence of different kinds of neutrality that emerge (in the sense of Reference [11]) from the semantic relation between two opposite concepts (and notice that we refer to a neutral category that does not entail linearity between opposites)." In neutrosophy, and, consequently, in the neutrosophic set, logic, and probability, between the opposite items (concepts, attributes, ideas, etc.) A and antiA , there may be a large number of neutralities/indeterminacies (all together denoted by neutA even an infinite spectrum-depending on the application to solve. We agree with different kinds of neutralities and indeterminacies (vague, ambiguous, unknown, incomplete, contradictory, linear and non-linear information, and so on), but the authors display only three neutralities.
In our everyday life and in practical applications, there are more neutralities and indeterminacies.
In another example (besides the previous one about Electoral voting), there may be any number of sub indeterminacies/sub neutralities.

Conclusions
The neutrosophic community thank the authors for their criticism and interest in the neutrosophic environment, and we wait for new comments and criticism, since, as Winston Churchill had said, the eagles fly higher against the wind.

Funding:
The author received no external funding.

Conflicts of Interest:
The author declares no conflict of interest.

Figure 1 .
Figure 1.Neutrosophic cube.The unit cube  used by the authors does not equal the above neutrosophic cube.The neutrosophic cube A'B'C'D'E'F'G'H' was introduced by Dezert [17] in 2002.

Figure 1 .
Figure 1.Neutrosophic cube.The unit cube I 3 used by the authors does not equal the above neutrosophic cube.The neutrosophic cube A'B'C'D'E'F'G'H' was introduced by Dezert [17] in 2002.

∼
where a, b ∈ R, and I = indeterminacy, I 2 = I , where R is the set of real numbers.And extended on: S C = a + bI|, where a, b ∈ C, and I = indeterminacy, I 2 = I , where C is the set of complex numbers.However, until 2016 [year of Montero et al.'s published paper], I did not develop a formal structure on the neutrosophic set.Montero et al. are right.
T 2 , . . ., T p ; I 1 , I 2 , . . ., I r ; F 1 , F 2 , . . ., F s with the newly introduced Refined Neutrosophic Logic/Set/Probability.-Ability to express incomplete information (T + I + F < 1) and paraconsistent (conflicting) and subjective information (T + I + F > 1).-Law of Included Middle explicitly/independently expressed as neutA (indeterminacy, neutral).-Law of Included Middle expanded to the Law of Included Multiple-Middles within the refined neutrosophic set as well as logic and probability.-A large array of applications [28-30] in a variety of fields, after two decades from their foundation ([10], 1998), such as: Artificial Intelligence, Information Systems, Computer Science, Cybernetics, Theory Methods, Mathematical Algebraic Structures, Applied Mathematics, Automation, Control Systems, Communication, Big Data, Engineering, Electrical, Electronic, Philosophy, Social Science, Psychology, Biology, Biomedical, 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, and more [30].
[5]Dr.Said Broumi, who forwarded it to me.}4.8.Counter-Example to the Paired StructureAs a counter example to the paired structure[5], it cannot catch a simple voting scenario.The election for the United States President from 2016: Donald Trump vs. Hillary Clinton.USA has 50 states and since, in the country, there is an Electoral vote, not a Popular vote, it is required to know the winner of each state.There were two opposite candidates.The candidate that receives more votes than the other candidate in a state gets all the points of that state.