A Geometric Interpretation of the Neutrosophic Set - A Generalization of the Intuitionistic Fuzzy Set

In this paper we generalize the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Several examples are presented. Also, a geometric interpretation of the Neutrosophic Set is given using a Neutrosophic Cube. Many distinctions between NS and IFS are underlined.Comment: 9 pages. Presented at the 2003 BISC FLINT-CIBI International Workshop on Soft Computing for Internet and Bioinformatics, University of Berkeley, California, December 15-19, 2003, under the title "Generalization of the Intuitionistic Fuzzy Set to the Neutrosophic Set

Plithogeny, Plithogenic Set, Logic, Probability, and Statistics

In this book we introduce for the first time, as generalization of dialectics and neutrosophy, the philosophical concept called plithogeny. And as its derivatives: the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), plithogenic logic (as generalization of classical, fuzzy, intuitionistic fuzzy, and neutrosophic logics), plithogenic probability (as generalization of classical, imprecise, and neutrosophic probabilities), and plithogenic statistics (as generalization of classical, and neutrosophic statistics). Plithogeny is the genesis or origination, creation, formation, development, and evolution of new entities from dynamics and organic fusions of contradictory and/or neutrals and/or non-contradictory multiple old entities. Plithogenic Set is a set whose elements are characterized by one or more attributes, and each attribute may have many values. An attribute’s value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x, v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set/logic/probability and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement/inclusion/equality are influenced by the attribute values’ contradiction (dissimilarity) degrees. Formal definitions of plithogenic set/logic/probability/statistics are presented into the book, followed by plithogenic aggregation operators, various theorems related to them, and afterwards examples and applications of these new concepts in our everyday life

Neutrosophic Set is a Generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set, q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical Fuzzy Set, while Neutrosophication is a Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision (revisited)

In this paper we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of single-valued neutrosophic components is \u3c 1, or \u3e 1, or = 1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators one gets a different result from that of applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Neutrosophic Set (NS) is also a generalization of Inconsistent Intuitionistic Fuzzy Set (IIFS) { which is equivalent to the Picture Fuzzy Set (PFS) and Ternary Fuzzy Set (TFS) }, Pythagorean Fuzzy Set (PyFS), Spherical Fuzzy Set (SFS), n-HyperSpherical Fuzzy Set (n-HSFS), and q-Rung Orthopair Fuzzy Set (q-ROFS). And all these sets are more general than Intuitionistic Fuzzy Set. We prove that Atanassov’s Intuitionistic Fuzzy Set of second type (IFS2), and Spherical Fuzzy Sets (SFS) do not have independent components. And we show that n-HyperSphericalFuzzy Set that we now introduce for the first time, Spherical Neutrosophic Set (SNS) and n-HyperSpherical Neutrosophic Set (n-HSNS) {the last one also introduced now for the first time} are generalizations of IFS2 and SFS. The main distinction between Neutrosophic Set (NS) and all previous set theories are: a) the independence of all three neutrosophic components {truth-membership (T), indeterminacy-membership (I), falsehood-nonmembership (F)} with respect to each other in NS – while in the previous set theories their components are dependent of each other; and b) the importance of indeterminacy in NS - while in previous set theories indeterminacy is completely or partially ignored. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and of Neutrosophic Probability. We have extended the Three-Ways Decision to n-Ways Decision

NEUTROSOPHIC SET – A GENERALIZATION OF THE INTUITIONISTIC FUZZY SET

In this paper one generalizes the intuitionistic fuzzy set (IFS),paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined

Neutrosophic Crisp Set Theory

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov\u27s intuitionistic fuzzy set

Plithogenic Set, an Extension of Crisp, Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets - Revisited

In this paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes’ values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. This article offers some examples and applications of these new concepts in our everyday life

Neutrosophic Theory and its Applications : Collected Papers - vol. 1

Neutrosophic Theory means Neutrosophy applied in many fields in order to solve problems related to indeterminacy. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. This theory considers every entity together with its opposite or negation and with their spectrum of neutralities in between them (i.e. entities supporting neither nor ). The and ideas together are referred to as . Neutrosophy is a generalization of Hegel\u27s dialectics (the last one is based on and only). According to this theory every entity tends to be neutralized and balanced by and entities - as a state of equilibrium. In a classical way , , are disjoint two by two. But, since in many cases the borders between notions are vague, imprecise, Sorites, it is possible that , , (and of course) have common parts two by two, or even all three of them as well. Hence, in one hand, the Neutrosophic Theory is based on the triad , , and . In the other hand, Neutrosophic Theory studies the indeterminacy, labelled as I, with In = I for n ≥ 1, and mI + nI = (m+n)I, in neutrosophic structures developed in algebra, geometry, topology etc. The most developed fields of the Neutrosophic Theory are Neutrosophic Set, Neutrosophic Logic, Neutrosophic Probability, and Neutrosophic Statistics - that started in 1995, and recently Neutrosophic Precalculus and Neutrosophic Calculus, together with their applications in practice. Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic). In neutrosophic logic a proposition has a degree of truth (T), a degree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ]-0, 1+[. Neutrosophic Probability is a generalization of the classical probability and imprecise probability. Neutrosophic Statistics is a generalization of the classical statistics. What distinguishes the neutrosophics from other fields is the , which means neither nor . And , which of course depends on , can be indeterminacy, neutrality, tie (game), unknown, contradiction, vagueness, ignorance, incompleteness, imprecision, etc