Encyclopedia > Neutrosophic set

Article Content

Neutrosophy

Redirected from Neutrosophic set

Ironically, the neutrality of this article is disputed. Please see the talk page (to access the talk page, click on "Discuss this page" in the sidebar).

Neutrosophy is a theory invented by Florentin Smarandache, which considers every notion or idea <A> together with its opposite or negation <Anti-A> and the spectrum of "neutralities" <Neut-A> (i.e. notions or ideas located between the two extremes, supporting neither <A> nor <Anti-A>). The <Neut-A> and <Anti-A> ideas together are referred to as <Non-A>. The theory claims that every idea <A> tends to be neutralized and balanced by <Anti-A> and <Non-A> ideas - as a state of equilibrium.

Expositions of neutrosophy might be difficult to understand, since Smarandache (as "leader of paradoxism") is fond of paradoxes, such as "All is possible, the impossible too!". In addition, Smarandache employs unusual grammatical constructions, leading to paragraphs such as:

"Human being is un organized chaos, endowed with abyssal reason, limited senses, and unbounded irrationalism. All is of continuous and transcendental field. Nor even phenomena are totally derivated ones from others, and there is effect without cause because the irrational has its act empire".

Smarandache, pg53 of "A Unifying Field..."

Despite the rhetorical inventiveness of Smarandache, neutrosophy has yet to make the impact he feels it deserves.

Table of contents

1 Origins

2 Extensions

3 Applications

4 External links

Origins

Neutrosophy, neutrosophic logic, neutrosophic sets, etc., were invented in the 1980s by Smarandache, after he coined the word from the Latin neuter and the Greek sophia, to mean "knowledge of neutral thought".

Smarandache promotes neutrosophy heavily. He organized the "First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics" in 2001 and published the conference's proceedings. One book about neutrosophy was published by American Research Press, a small publisher closely aligned with Smarandache. Additionally, some articles by Smarandache and Jean Dezert[?] were included in the Journal of Multiple-Valued Logic, Volume 8, Number 3, issue dedicated to neutrosophy and neutrosophic logic, and Numbers 5-6. This journal is now known as the Journal of Multiple-Valued Logic and Soft Computing. Other journals that published on neutrosophics are International Journal of Social Economics (University of California at Fresno), Libertas Mathematica (University of Texas at Arlington), Proceedings of the Second Symposium / Romanian Academy of Scientists, American Branch (City University of New York), Bulletin of the Transilvania University of Brasov (Romania), Abstracts of papers presented to the International Congress of Mathematicians (Beijing, China) and Abstracts of papers presented to the meetings of the American Mathematical Society (University of California at Santa Barbara meeting).

Extensions

Smarandache extended neutrosophy to neutrosophic logic (or Smarandache logic), neutrosophic sets, and so forth.

In bivalent logic, the truth value of a proposition is given by either one (true), or zero (false). Neutrosophic logic is a multi-valued logic, in which the truth values are given by an amount of truth, an amount of falsehood, and an amount of indeterminacy. Each of these values is between 0 and 1. In addition, the values may vary over time, space, hidden parameters, etc. Further, these values can be ranges.

In the neutrosophic logic every logical variable x is described by an ordered triple x = (T, I, F) where T is the degree of truth, F is the degree of false and I the level of indeterminacy.

(A) To maintain consistency with the classical and fuzzy logics and with probability there is the special case where T + I + F = 1.

(B) But to refer to intuitionistic logic, which means incomplete information on a variable proposition or event one has T + I + F < 1.

(C) Analogically referring to Paraconsistent logic, which means contradictory sources of information about a same logical variable, proposition or event one has T + I + F > 1.

Thus the advantage of using Neutrosophic logic is that this logic distinguishes in philosophy between relative truth that is a truth in one or a few worlds only noted by 1 and absolute truth denoted by 1+. Likewise neutrosophic logic distinguishes between relative falsehood, noted by 0 and absolute falsehood noted by -0 in non-standard analysis.

For example, a neutrosophic answer to the question "Is the pope a Catholic?" might be "80-90% true, 36-42% false, and 2-7% indeterminate". Note that these values need not sum to 100%. Smarandache claims that it can serve as a generalization of many other logics, such as: fuzzy logic, intuitionistic logic, paraconsistent logic[?], boolean logic, etc.

In neutrosophic set theory, propositions of the form "x is an element of S" are answered in terms of neutrosophic truth values. Hence, each element has a membership-degree, an indeterminacy-degree, and a non-membership degree. These are claimed to generalise paraconsistent sets[?] and intuitionistic sets[?], amongst others.

Applications

As examples of application of neutrosophy in information fusion in finance there are some papers by Dr. M. Khoshnevisan, Dr. S. Bhattacharya and Dr. F. Smarandache, where the fuzzy theory doesn't work because fuzzy theory has only two components, truth and falsehood, while the neutrosophy has three components: truth, falsehood, and indeterminacy (or <A>, <Anti-A>, and <Neut-A>), papers about investments which are: Conservative and security-oriented (risk shy), Chance-oriented and progressive (risk happy), or Growth-oriented and dynamic (risk neutral). See the paper "Fuzzy and Neutrosophic Systems and Time Allocation of Money", pp. 5-23, in their book "Artificial Intelligence and Responsive Optimization" at www.gallup.unm.edu/~smarandache/ArtificialIntelligence-book2.pdf.

Proponents of neutrosophy claim that in any field where there is indeterminacy, unknown, hidden parameters, imprecision, sorites paradoxes, high conflict between sources of information, non-exhaustive or non-exclusive elements of the frame of discernment, etc., then neutrosophy could in theory be applied.

More applications of neutrosophics:
Fuzzy Cognitive Maps (FCMs) are fuzzy structures that strongly resemble neural networks, and they have powerful and far-reaching consequences as a mathematical tool for modeling complex systems. Neutrosophic Cognitive Maps (NCMs) are generalizations of FCMs, and their unique feature is the ability to handle indeterminacy in relations between two concepts thereby bringing greater sensitivity into the results.
Some of the varied applications of FCMs and NCMs include: modeling of supervisory systems; design of hybrid models for complex systems; mobile robots and in intimate technology such as office plants; analysis of business performance assessment; formalism debate and legal rules; creating metabolic and regulatory network models; traffic and transportation problems; medical diagnostics; simulation of strategic planning process in intelligent systems; specific language impairment; web-mining inference application; child labor problem; industrial relations: between employer and employee, maximizing production and profit; decision support in intelligent intrusion detection system; hyper-knowledge representation in strategy formation; female infanticide; depression in terminally ill patients and finally, in the theory of community mobilization and women empowerment relative to the AIDS epidemic. See Dr. W. B. Vasantha Kandasamy from Indian Institute of Technology in Madras and Dr. F. Smarandache's book Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps at http://www.gallup.unm.edu/~smarandache/NCMs.pdf.

The concept of only fuzzy cognitive maps are dealt which mainly deals with the relation / non-relation between two nodes or concepts but it fails to deal the relation between two conceptual nodes when the relation is an indeterminate one. Neutrosophic logic is the tool known to us, which deals with the notions of indeterminacy. Suppose in a legal issue the jury or the judge cannot always prove the evidence in a case, in several places we may not be able to derive any conclusions from the existing facts because of which we cannot make a conclusion that no relation exists or otherwise. But existing relation is an indeterminate. So in the case when the concept of indeterminacy exists the judgment ought to be very carefully analyzed be it a civil case or a criminal case. FCMs are deployed only where the existence or non-existence is dealt with but however in the Neutrosophic Cognitive Maps we will deal with the notion of indeterminacy of the evidence also. Thus legal side has lot of Neutrosophic (NCM) applications. Also, NCMs can be used to study factors as varied as stock markets, medical diagnosis, etc.