Indeterminacy in Neutrosophic Theories and their Applications

Indeterminacy makes the main distinction between fuzzy / intuitionistic fuzzy (and other extensions of fuzzy) set / logic vs. neutrosophic set / logic, and between classical probability and neutrosophic probability. Also, between classical statistics vs. neutrosophic and plithogenic statistics, between classical algebraic structures vs. neutrosophic algebrais structures, between crisp numbers vs. neutrosophic numbers. We present a broad definition of indeterminacy, various types of indeterminacies, and many practical applications.

Firstly, let's define the neutrosophic triplets. Let <A> be an item (concept, notion, idea, sentence, theory etc.) and <antiA> its opposite. In between the opposites <A> and <antiA>, there is a neutral (or indeterminacy) part, denoted by <neutA>.
The <neutA> is neither <A> not <antiA>, or sometimes the <neutA> is a mixture of partial <A> and partial <antiA>.
Of course, we consider the neutrosophic triplets (<A>, <neutA>, <antiA>) that make sense in the world, and there are plenty of such triplets in our every day life [1].

Neutrosophic Definition of Indeterminacy
In neutrosophy, which is a new branch of philosophy, we interpret Indeterminacy in the broadest possible sense, i.e. Indeterminacy, denoted by <neutA>, is everything that is in between the opposites <A> and <antiA>.
The word "Indeterminacy" is a generic name for <neutA> (or the letter "I"). It should not be taken literally (in a narrow sense) as in a lexical dictionary (such as Webster, Larousse, etc.).
Indeterminacy depends on each application, or problem to solve, and on the experts. That's why there are many types of Indeterminacies.
In general, Indeterminacy I is not the complement of T and F, since the neutrosophic components T, I, F are independent from each other.
As a middle side, <neutA> is neither <A> nor <antiA>, but in between them, or sometimes, a combination of them.
For the neutrosophic triplet (Matter, Unmatter, Antimatter), the Indeterminacy = Unmatter (Unmatter is formed by combinations of matter and antimatter that bound together, or by long-range mixture of matter and antimatter forming a weakly-coupled phase) [12].
In Fuzzy Set and Logic, T = the truth (or membership), while F = 1 -T = the falsehood (or nonmembership), while I = 0 is the indeterminacy.
In Intuitionistic Fuzzy Set and Logic, T = the truth (or membership), F = the falsehood (or nonmembership), and the indeterminacy is called hesitancy H = 1 -T -F.
In Picture Fuzzy Set and Logic, T = the truth (or membership), F = the falsehood (or nonmembership), and the indeterminacy (I) was split/refined into N = neutrality (or the first subindeterminacy I1), and the hesitancy H = 1 -T -F -N (or the second subindeterminacy I2). Therefore: T, I1 = N, I2 = H, F. Picture Fuzzy Set and Logic (also called Inconsistent Intuitionistic Fuzzy Set and Logic, or Ternary Fuzzy Set and Logic)) are particular cases of Refined Neutrosophic Set and respectively Logic (where T is split/refined into T1, T2, …, Tp; I is split/refined into I1, I2, …, Ir; and F is split/refined into F1, F2, …, Fs; with integers p, r, s ≥ 0 and at least one of p, r, or s is ≥ 2; if some T0, I0, F0 occur, it is discarded) [3].
Similarly for other fuzzy extension sets and logics {such as: Pythagorean Fuzzy Set and Logic (also called Atanassov's Intuitionistic Fuzzy Set and Logic of second type), q-Rung Orthopair Fuzzy Set and Logic, Fermatean Fuzzy Set and Logic, also Spherical Fuzzy Set and Logic, n-HyperSpherical Fuzzy Set and Logic, etc.} [13]. They have either two components (T and F) or three (T, I, and F), but with the restrictions that 0
Therefore, the (total) Indeterminacy is the union ( U )of all Subindeterminacies:

Example of Refined Indeterminacy
For the I-refined neutrosophic triplet (White; Yellow, Pink, Red, Blue, Violet; Black), the Indeterminacy = Yellow U Pink U Red U Blue U Violet.
There also is possible to have an infinite I-refined neutrosophic triplet by considering the infinite color spectrum between White and Black.

The Neutrosophic Logic Triplet [1]
The Neutrosophic Logic (NL) truth-value of a proposition P is: or T = truth, I = indeterminacy, F = falsehood. We prefer to use these descriptive notations T, I, F all over for the neutrosophic components.

The Neutrosophic Set Triplet
The Neutrosophic Set (NS) membership-value of an element x with respect to a give set M is: In this case, the Indeterminacy = ch(neutA). [5,6] While the Classical Statistics deals with determinate data, determinate probability distributions, and determinate inference methods only, the Neutrosophic Statistics may deals with indeterminate data {i.e. data that has some degree of indeterminacy (unclear, vague, partially unknown, contradictory, incomplete, etc.)}, indeterminate probability distributions, and indeterminate inference methods {i.e. distributions and inferences that contain some degrees of indeterminacy as well (for example, instead of crisp arguments and values for the probability distributions and inference methods, charts, diagrams, algorithms, functions etc. one may deal with inexact or ambiguous arguments and values)}.

Indeterminacy in Neutrosophic Statistics
For example: -The sample's size or population's size are not exactly known (for example, the size may be between 200 -250 individuals). -Not all individuals may belong 100% to the sample or populations, some may only partially belong (their degree of belongingness T < 1), others may over-belonging (their degree of belongingness T > 1).
An application: Upon their work for a factory, John belongs 100%, George 50%, and Mary 110% (because she works overtime). John is 40 years old, George 60, and Mary 20. What is the age average of this company's workers?

When Indeterminacy = 0
Let T, I, F belonging to the interval [0, 1] be the neutrosophic components. If Indeterminacy I = 0, the neutrosophic components (T, 0, F) are still more flexible and more general than fuzzy components and intuitionistic fuzzy components. Because, we get: -for the fuzzy set and the intuitionistic fuzzy set (they coincide): T + F = 1.
-while for the neutrosophic set: 0 ≤ T + F ≤ 2, whence we may have any of these situations: T + F < 2 (for incomplete information); T + F = 2 (for complete information); T + F > 2 (for paraconsistent / conflicting information, coming from independent sources).
Therefore, the neutrosophic set is more flexible and more general than the other sets, no matter the value of indeterminacy.

Classification of Indeterminacies
Since there are many types of indeterminacies, it is possible to define many types of neutrosophic measures in any field of knowledge. And, in general, because of dealing with lots of types of indeterminacies, we can extend any classical scientific or cultural concept from various indeterminate/neutrosophic viewpoints.
(i) There is the Numerical Indeterminacy, as part of the numerical neutrosophic triplet (T, I, F), when "I" is a numerical subset (interval, hesitant subset, single-valued number, etc.) of [0, 1], and it is used in neutrosophic set, neutrosophic logic, and neutrosophic probability.
(ii) And the Literal Indeterminacy, where I^2 = I, with "I" just a letter [7], used in neutrosophic algebraic structures (such as: neutrosophic group, neutrosophic ring, neutrosophic vector space, etc.) that are built on the sets of the form: S = {a + bI, with I^2 = I, and a, b in M}, where M is a given real or complex set.
The Literal Indeterminacy (I) is also used in neutrosophic calculus and in some neutrosophic graphs and neutrosophic cognitive maps, when the edge between two vertexes is unknown and it is denoted by a dotted line (meaning indeterminate edge).
(iii) TransIndeterminacies, inspired from the transreal numbers [11], some of which are: (iv) Also, the Neutrosophic Number, N = d + e I, where a and b are real or complex numbers introduced in [7], and they were interpreted as N = d + e I, where d is the determinate part of the number N, and e I is the indeterminate part of the number N in [5].
There are transcendental, irrational etc. numbers that are not well known, they are only partially known and partially unknown, and they have infinitely many decimals. Not even the most modern supercomputers can compute more than a few thousands decimals, but the infinitely many left decimals still remain unknown. Therefore, such numbers are very little known (because only a finite number of decimals are known), and infinitely unknown (because an infinite number of decimals are unknown). The neutrosophic number is used in neutrosophic statistics, and in neutrosophic precalculus [8].
(v) In the Quadruple Neutrosophic Number, which has the form QN = a + b·T + c·I + d·F, where the known part of QN is a, and the unknown part of QN is b·T + c·I + d·F, then the unknown part is split into three subparts: degree of confidence (T), degree of indeterminacy between confidence-nonconfidence (I), and degree of nonconfidence (F).
QN is a four-dimensional vector that can also be written as : QN = (a, b, c, d).
T, I, F are herein literal parameters. The multiplication amongst these literal parameters uses the absorbance (prevalence) law, i.e. one parameter absorbs (includes) another (see [9]). But in specific applications T, I, F may be numerical too (in general, subsets of [0, 1]).

(vi) The Over-/Under-/Off-Indeterminacy
For OverIndeterminacy we have I > 1 within the frame of Neutrosophic Overset; for UnderIndeterminacy we have I < 0 within the Neutrosophic Underset; and in general for OffIndeterminacy we have sometimes I > 1 and other times I < 0 within the frame of neutrosophic Offset.
For your information, there are cases when the degrees of membership, indeterminacy or nonmembership may be each of them > 1 or < 0, and these are happening in our real life applications (see [10,11]).