In Intuitionistic fuzzy logic (IF), a certain amount incomplete information or indeterminacy is allowed. Technically,  v (p) = (t, f) with t + f ≤ 1. In neutrosophic logic, inconsistent beliefs are allowed as well. Hence, it do not even assume that the “indeterminacy degree” can often be attained by 1 − (t + f). 

The neutrosophic logic has already found its  practical applications in many fields, such as semantic web services and relational DB systems. its applications in multi-criteria decision-making problems are also clear. There are many publication that  use this logic, you can see the next papers, if you seek some real-life applications:

Book Multispace & Multistructure. Neutrosophic Transdisciplinarit...

Although I answered this question, somebody keeps deleting my answer. Please, Dr. Mohanty, for the sake of the truth let my answer below.

RK Mohanty

1) In neutrosophic sets all three measures (Truth, Falsehood, indeterminacy) are independent, how one effects another in decision making. (Ex: In case of intuitionistic fuzzy, if membership increases, then, certainly the sum of other two measure will decrease.)

FS:

In intuitionistic fuzzy set, the components membership and nonmembership are dependent, so their sum should be 1.

We have in this case contradictory information (when the sum > 1, but if the

sum < 1 we have incomplete information (we do not know all information), while if sum = 1 we have complete information as in intuitionistic fuzzy set.

Please see below a paper on degrees of dependence and independence between the components (T, I, and F):

http://fs.gallup.unm.edu/nss/DegreeOfDependenceAndIndependence.pdf .

See also the UNM website of the neutrosophics:

http://fs.gallup.unm.edu/neutrosophy.htm .

RK Mohanty

2) When the following conditions occur in a real life situation:

                                        (T, I, F) : (1,0,1), (1,1,0), (1,1,1), (0,0,0), (0,1,1)

FS:

I change the order to (T, I, F) since this is the classical in neutrosophics, without altering the result.

A paradox is a proposition which is true and false in the same time, i.e.

T = 1, I = 0 (well defined), F = 1.

See for example the last U.S. Presidential election:

the pools predicted that Hilary Clinton will win100% (or T = 1 for Hilary), while the outcome was the opposite (Donald Trump won, so F = 1 for Hilary).

Depending on the type of paradox, we may also have:

T = 1, I = 1 (not well defined, i.e. totally indeterminate proposition/paradox), F = 1.

For (0, 0, 0) again: in a soccer game India - Pakistan a source may say that tha chance of winning for India is 0%, other independent source may say that the chance for India to loose if 0%, and a third independent source may say that the chance of tie game is 0%.

RK Mohanty

3) In decision making applications how to sort in best to worse (higher truth value is better) order for following:

                            (T, I, F) : (1,0,0) (1,0,1), (1,1,0), (1,1,1), (0,0,0), (0,1,1), (0,1,0)

FS:

Sorting depends on each specific application.

On the triplets (T, I, F) we have partial orders, but total orders can be defined as well.

T has a positive quality, while I and F have negative quality. We may say that:

(T1, I1, F1) > (T2, I2, F2) if T1 > T2 and I1 ≤ I2, F1 ≤ F2.

As I said above, other orders can be defined too on the triplets, depending on the optimistic or pessimistic point of view.

Dear Mohanty,

Each of the set theory has some advantages and disadvantages.  It is just an expert has to decide in which context the calculus of available set can be applied. As Prof. Samarandache given you the answer. Of course your doubt is obvious that very rare case exists when a given event is mutually exclusive and their sum is more then universe. If those case are sparse then why we need analysis. Due to that first we need a real life data set to validate it.   

If the components T, I, F are dependent of each other, then T+I+F = 1, as in intuitionistic fuzzy logic. But applying the neutrosophic operators: union, intersection etc. the result is different from the IFL, since in IFL the indeterminacy "I" is ignored, while in NL the indeterminacy is taken into consideration.

If the components are independent in some degree (please see the attached paper), then T+I+F > 1.

@Florentin Smarandache

Dear Sir, Thank you so much for your  effort to give such a long and comprehensive comments. I appreciate your time taking for this.

Sir, I have never deleted any of your comments. Your comments are of the utmost important to me as you are the founder of the concept. I'm very fortunate to learn about basics of NL from the founder himself. 

I'm not a doctorate but pursuing my doctoral degree now.

Thank you again for your valuable comments. 

It is very good to keep in touch, Mr. Mohanty.

If you have questions, you can email me at [email protected].

If you use neutrosophic set or logic in your Ph D thesis, I can help you by answering your questions and by sending you adequate materials (papers or books).

See the site of neutrosophics at UNM for some Ph D theses on neutrosophics:

http://fs.gallup.unm.edu/neutrosophy.htm

Fuzzy logic is very useful method for representing the contents in logic with rule style.

@Florentin Smarandache

Thank you Sir.

See many papers on neutrosophic MCDM in these UNM websites:

http://fs.gallup.unm.edu/neutrosophy.htm

and

http://fs.gallup.unm.edu/NSS/NSSArticles.htm .

@ Florentin Smarandache well done. am impressed by your response. i may need your help in this aspect, even though am still learning the fuzzy theory. kindly assist me with some elementary book or papers on this topics (fuzzy set, hesitant fuzzy set, and Neutrosophic set. am actually a statistician but interested in this fuzzy logic. perharp it may help me broading my statistical research. thank you.

Revolution of Neutrosophic Set from Classical Set as follows:

The classical set, I = ∅, inf T = sup T = 0 or 1, inf F = sup F = 0 or 1.

The fuzzy set, I = ∅, inf T = sup T ∈ [0, 1], inf F = supF ∈ [0, 1] and sup T + sup F = 1.

The interval valued fuzzy set, I = ∅, inf T, sup T, inf F, sup F ∈ [0, 1], sup T + inf F = 1 and inf T + sup F = 1.

The intuitionistic fuzzy set, I = ∅, inf T = sup T ∈ [0, 1], inf F = sup F ∈[0, 1] and sup T + sup F ≤ 1.

The interval-valued intuitionistic fuzzy set, I = ∅, inf T, sup T, inf F, sup F ∈ [0, 1] and sup T + sup F ≤ 1.

An interval neutrosophic set (INS), An INS 𝐴 in 𝑋 is characterized by a truth-membership function 𝑇𝐴(𝑥), an indeterminacy-membership function 𝐼𝐴(𝑥), and a falsity-membership function 𝐹𝐴(𝑥). For each point 𝑥 in 𝑋, we have that

𝑇𝐴(𝑥) = [inf 𝑇𝐴(𝑥), sup 𝑇𝐴(𝑥)]

𝐼𝐴(𝑥) = [inf 𝐼𝐴(𝑥), sup 𝐼𝐴(𝑥)]

𝐹𝐴(𝑥) = [inf 𝐹𝐴(𝑥), sup 𝐹𝐴(𝑥)]

𝑇𝐴(𝑥), 𝐼𝐴(𝑥), 𝐹𝐴(𝑥) ⊆ [0, 1] and

0 ≤ sup 𝑇𝐴(𝑥) + sup 𝐼𝐴(𝑥) + sup 𝐹𝐴(𝑥) ≤ 3, 𝑥 ∈ 𝑋.

We only consider the sub unitary interval of [0, 1]. It is the subclass of an NS.

Therefore, all INSs are clearly NSs.

A neutrosophic set, A NS 𝐴 in 𝑋 is characterized by a truth-membership function 𝑇𝐴(𝑥), an indeterminacy-membership function 𝐼𝐴(𝑥), and a falsity membership function 𝐹𝐴(𝑥).

𝑇𝐴(𝑥), 𝐼𝐴(𝑥), and 𝐹𝐴(𝑥) are real standard or nonstandard subsets of ]0−, 1+[

That is,

𝑇𝐴(𝑥):𝑋 →]0−, 1+[

𝐼𝐴(𝑥): 𝑋 →]0−, 1+[

𝐹𝐴(𝑥): 𝑋 →]0−, 1+[

There is no restriction on the sum of 𝑇𝐴(𝑥), 𝐼𝐴(𝑥), and 𝐹𝐴(𝑥)

So 0−≤ sup 𝑇𝐴(𝑥) + sup 𝐼𝐴(𝑥) + sup 𝐹𝐴(𝑥) ≤ 3+.

Reference paper for details:

https://arxiv.org/abs/math/0409113

http://dx.doi.org/10.1155/2014/645953

Logically, indeterminacy function (I) is a complement of the member and non-member functions (T, F). That is, indeterminacy function implicitly exists in intuitionistic fuzzy sets and explicitly exists in neutrosophic sets. It should noted that indeterminacy function takes a value (>0) when available information of phenomenon or problem are incomplete. With respect to the independent and dependent of each other, they must dependent of each other, how do they are independent?! For example, it is not logic that someone expects that A will win with a chance of 40% and will fail with a chance of 70%. But the chances of winning and loss can be respectively: 40% and 50% or 30% and 70% or...(

In neutrosophic set, we may have any of these situations:

T + F < 1 (for incomplete information);

T + F = 1 (for complete information; this coincides with fuzzy set and intuitionistic fuzzy set;

T + F > 1 (for paraconsistent / conflicting information coming from independent sources).

Then neutrosophic set is more general and more complex than the other sets, no matter the value of indeterminacy.

The neutrosophic approach outperform the fuzzy approach for the following reasons:

1- The neutrosophic approach uses permutation of dependent / independent three factors ( Truth, Indeterminacy, Falsity).

On the other hand,

2- The fuzzy approach uses permutation of dependent / independent two factors ( Truth, Falsity). so the neutrosophic approach is generalized more than the fuzzy approach.

There many real life applications indicate this truth. see the following references:

References.

[1] Florentin Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information &Learning, Ann Arbor, Michigan, USA, 105 p.,

[2] 1998, 2000, 2003, 2005, 2006;

http://fs.gallup.unm.edu/eBook-neutrosophics5.pdf (fifth edition).

[3] Florentin Smarandache, Degree of Dependence and Independence of Neutrosophic Logic Components Applied in Physics, 2016 Annual Spring Meeting of the American Physical Society (APS) Ohio-Region Section, Dayton, Ohio, USA, 08-09 April 08 2016.

Neutrosophic theory is more flexible and useful as compared to all fuzzy-like and intui fuzzy-like theories. Respected Prof. Dr. Florentin Smarandache has provided a vivid details to prove it with strong arguments in his above qouted papers. No doubt, Excellent work done by him

Neutrosophic logic offers significant advantages over Intuitionistic fuzzy logic in real-life decision-making due to its ability to explicitly handle indeterminacy along with truth and falsity. This makes neutrosophic logic more flexible and adaptable for modeling complex, uncertain, and contradictory data, which are common in real-world scenarios. It is particularly useful in applications where precise data is unavailable or inherently vague, such as in medical diagnostics, economic forecasting, and social sciences.