Logic and Foundations Commons^™

Fifteenth International Photovideoanthology On Paradoxism, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Paradoxism is an international movement in science and culture, founded by Florentin Smarandache in 1980s, based on excessive use of antitheses, oxymoron, contradictions, and paradoxes. During three decades (1980-2020) hundreds of authors from tenth of countries around the globe contributed papers to 15 international paradoxist anthologies.

In 1995, the author extended the paradoxism to a new branch of philosophy called neutrosophy, that gave birth to many scientific branches, such as: neutrosophic logic, neutrosophic set, neutrosophic probability and statistics, neutrosophic algebraic structures and so on with multiple applications in engineering, computer science, administrative work, medical research etc.

“May your imagination blossom …

New Challenges In Neutrosophic Theory And Applications, Florentin Smarandache, Stefan Vladutescu, Miihaela Colhon, Wadei Al-Omeri, Saeid Jafari, Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, Abdur Razzaque Mughal

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of …

(Φ, Ψ)-Weak Contractions In Neutrosophic Cone Metric Spaces Via Fixed Point Theorems, Florentin Smarandache, Wadei F. Al-Omeri

Branch Mathematics and Statistics Faculty and Staff Publications

In this manuscript, we obtain common fixed point theorems in the neutrosophic cone metric space. Also, notion of (Φ, Ψ)-weak contraction is defined in the neutrosophic cone metric space by using the idea of altering distance function. Finally, we review many examples of cone metric spaces to verify some properties.

Special Subset Vertex Multisubgraphs For Multi Networks, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors study special type of subset vertex multi subgraphs; these multi subgraphs can be directed or otherwise. Another special feature of these subset vertex multigraphs is that we are aware of the elements in each vertex set and how it affects the structure of both subset vertex multisubgraphs and edge multisubgraphs. It is pertinent to record at this juncture that certain ego centric directed multistar graphs become empty on the removal of one edge, there by theorising the importance, and giving certain postulates how to safely form ego centric multi networks. Given any subset vertex multigraph we …

Fundamentals Of Neutrosophic Logic And Sets And Their Role In Artificial Intelligence (Fundamentos De La Lógica Y Los Conjuntos Neutrosóficos Y Su Papel En La Inteligencia Artificial ), Florentin Smarandache, Maykel Leyva-Vazquez

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophy is a new branch of philosophy which studies the origin, nature and scope of neutralities. This has formed the basis for a series of mathematical theories that generalize the classical and fuzzy theories such as the neutrosophic sets and the neutrosophic logic. In the paper, the fundamental concepts related to neutrosophy and its antecedents are presented. Additionally, fundamental concepts of artificial intelligence will be defined and how neutrosophy has come to strengthen this discipline.

Neutrosophic Operational Research - Vol. 3., Florentin Smarandache, Mohamed Abdel Basset, Victor Chang

Branch Mathematics and Statistics Faculty and Staff Publications

Foreword John R. Edwards This book is an excellent exposition of the use of Data Envelopment Analysis (DEA) to generate data analytic insights to make evidence-based decisions, to improve productivity, and to manage cost-risk and benefitopportunity in public and private sectors. The design and the content of the book make it an up-to-date and timely reference for professionals, academics, students, and employees, in particular those involved in strategic and operational decisionmaking processes to evaluate and prioritize alternatives to boost productivity growth, to optimize the efficiency of resource utilization, and to maximize the effectiveness of outputs and impacts to stakeholders. It …

Neutrosophic Logic: The Revolutionary Logic In Science And Philosophy -- Proceedings Of The National Symposium, Florentin Smarandache, Huda E. Khalid, Ahmed K. Essa

Branch Mathematics and Statistics Faculty and Staff Publications

The first part of this book is an introduction to the activities of the National Symposium, as well as a presentation of Neutrosophic Scientific International Association (NSIA), based in New Mexico, USA, also explaining the role and scope of NSIA - Iraqi branch. The NSIA Iraqi branch presents a suggestion for the international instructions in attempting to organize NSIA's work. In the second chapter, the pivots of the Symposium are presented, including a history of neutrosophic theory and its applications, the most important books and papers in the advancement of neutrosophics, a biographical note of Prof. Florentin Smarandache in Arabic …

Neutrosophic Multi-Criteria Decision Making, Florentin Smarandache, Jun Ye, Yanhui Guo

Branch Mathematics and Statistics Faculty and Staff Publications

The notion of a neutrosophic quadruple BCK/BCI-number is considered in the first article (“Neutrosophic Quadruple BCK/BCI-Algebras”, by Young Bae Jun, Seok-Zun Song, Florentin Smarandache, and Hashem Bordbar), and a neutrosophic quadruple BCK/BCI-algebra, which consists of neutrosophic quadruple BCK/BCI-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple BCK-algebra and a closed ideal in a neutrosophic quadruple BCI-algebra are studied. Given subsets A and B of a BCK/BCI-algebra, the set NQ(A,B), which consists of neutrosophic quadruple BCK/BCInumbers with a condition, is established. Conditions for the set NQ(A,B) to be a (positive implicative) ideal of a …

Neutrosophic N -Structures And Their Applications In Semigroups, Florentin Smarandache, Madad Khan, Saima Anis, Young Bae Jun

Branch Mathematics and Statistics Faculty and Staff Publications

The notion of neutrosophic N -structure is introduced, and applied it to semigroup. The notions of neutrosophic N -subsemigroup, neutrosophic N -product and ε-neutrosophic N -subsemigroup are introduced, and several properties are investigated. Conditions for neutrosophic N -structure to be neutrosophic N -subsemigroup are provided. Using neutrosophic N -product, characterization of neutrosophic N -subsemigroup is discussed. Relations between neutrosophic N -subsemigroup and εneutrosophic N -subsemigroup are discussed. We show that the homomorphic preimage of neutrosophic N -subsemigroup is a neutrosophic N - subsemigroup, and the onto homomorphic image of neutrosophic N - subsemigroup is a neutrosophic N -subsemigroup.

Neutrosophic Operational Research - Vol. 2, Florentin Smarandache, Mohamed Abdel Basset, Victor Chang

Branch Mathematics and Statistics Faculty and Staff Publications

Foreword John R. Edwards This book is an excellent exposition of the use of Data Envelopment Analysis (DEA) to generate data analytic insights to make evidence-based decisions, to improve productivity, and to manage cost-risk and benefitopportunity in public and private sectors. The design and the content of the book make it an up-to-date and timely reference for professionals, academics, students, and employees, in particular those involved in strategic and operational decisionmaking processes to evaluate and prioritize alternatives to boost productivity growth, to optimize the efficiency of resource utilization, and to maximize the effectiveness of outputs and impacts to stakeholders. It …

Complex Valued Graphs For Soft Computing, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce in a systematic way the notion of complex valued graphs, strong complex valued graphs and complex neutrosophic valued graphs. Several interesting properties are defined, described and developed. Most of the conjectures which are open in case of usual graphs continue to be open problems in case of both complex valued graphs and strong complex valued graphs. We also give some applications of them in soft computing and social networks. At this juncture it is pertinent to keep on record that Dr. Tohru Nitta was the pioneer to use complex valued graphs …

Neutrosophic Triplet Groups And Their Applications To Mathematical Modelling, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K.

Branch Mathematics and Statistics Faculty and Staff Publications

The innovative notion of neutrosophic triplet groups, introduced by Smarandache and Ali in 2014-2016, happens to yield the anti-element and neutral element once the element is given. It is established that the neutrosophic triplet group collection forms the classical group under product for Zn, for some specific n. However the collection is not even closed under sum. These neutrosophic triplet groups are built using only modulo integers or Cayley tables. Several interesting properties related with them are defined. It is pertinent to record that in Zn, when n is a prime number, we cannot get a neutral element which can …

New Trends In Neutrosophic Theory And Applications - Vol. 1, Florentin Smarandache, Surapati Pramanik

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, …

Nidus Idearum. Scilogs, I: De Neutrosophia, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Welcome into my scientific lab! My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). In this first books of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others …

Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, And Neutrosophic Offsets, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

We have proposed since 1995 the existence of degrees of membership of an element with respect to a neutrosophic set to also be partially or totally above 1 (over-membership), and partially or totally below 0 (under-membership) in order to better describe our world problems [published in 2007].

Strong Neutrosophic Graphs And Subgraph Topological Subspaces, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilantheral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of strong neutrosophic graphs. They are very different from the usual graphs and neutrosophic graphs. Using these new structures special subgraph topological spaces are defined. Further special lattice graph of subgraphs of these graphs are defined and described. Several interesting properties using subgraphs of a strong neutrosophic graph are obtained. Several open conjectures are proposed. These new class of strong neutrosophic graphs will certainly find applications in NCMs, NRMs and NREs with appropriate modifications.

Neutrosophic Set Approach To Algebraic Structures, Florentin Smarandache, Madad Khan, Fazal Tahir

Branch Mathematics and Statistics Faculty and Staff Publications

Real world is featured with complex phenomenons. As uncertainty is inevitably involved in problems arise in various elds of life and classical methods failed to handle these type of problems. Dealing with imprecise, uncertain or imperfect information was a big task for many years. Many modelswerepresentedinordertoproperlyincorporateuncertaintyintosystem description, LotA.Zadeh in 1965 introduced the idea of a fuzzy set. Zadeh replaced conventional characteristic function of classical crisp sets which takes on its values in f0;1g by membership function which takes on its values in closed interval [0;1]. Fuzzy set theory is conceptually a very powerful technique to deal with another aspect or …

Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time have made a through study of neutrosophic graphs. This study reveals that these neutrosophic graphs give a new dimension to graph theory. The important feature of this book is it contains over 200 neutrosophic graphs to provide better understanding of this concepts. Further these graphs happen to behave in a unique way inmost cases, for even the edge colouring problem is different from the classical one. Several directions and dimensions in graph theory are obtained from this study. Finally certainly these new notions of neutrosophic graphs in general and in particular the …

Neutrosophic Crisp Set Theory, Florentin Smarandache, A.A. Salama

Branch Mathematics and Statistics Faculty and Staff Publications

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 …

Α-Discounting Method For Multi-Criteria Decision Making (Α-D Mcdm), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce a new procedure called αDiscounting Method for Multi-Criteria Decision Making (α-D MCDM), which is as an alternative and extension of Saaty’s Analytical Hierarchy Process (AHP). It works for any number of preferences that can be transformed into a system of homogeneous linear equations. A degree of consistency (and implicitly a degree of inconsistency) of a decision-making problem are defined. α-D MCDM is afterwards generalized to a set of preferences that can be transformed into a system of linear and/or non-linear homogeneous and/or nonhomogeneous equations and/or inequalities. Many consistent, weak inconsistent, and strong inconsistent examples are …

Quaestiones Neutrosophicae, Florentin Smarandache, Yale Landsberg

Branch Mathematics and Statistics Faculty and Staff Publications

The following dialogue contains cuts from different non-protocolar conversations, initially not intended for publication, held by the authors by email during the beginning of 2015 – on Neutrosophy and related topics.

Many thanks to all friends and dialogue partners who payed attention to Neutrosophy and connected areas, in emails, yahoo groups, social media, letters, private discussions.

Neutrosophic Axiomatic System, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we introduce for the first time the notions of Neutrosophic Axiom, Neutrosophic Axiomatic System, Neutrosophic Deducibility and Neutrosophic Inference, Neutrosophic Proof, Neutrosophic Tautologies, Neutrosophic Quantifiers, Neutrosophic Propositional Logic, Neutrosophic Axiomatic Space, Degree of Contradiction (Dissimilarity) of Two Neutrosophic Axioms, and Neutrosophic Model. A class of neutrosophic implications is also introduced. A comparison between these innovatory neutrosophic notions and their corresponding classical notions is made. Then, three concrete examples of neutrosophic axiomatic systems, describing the same neutrosophic geometrical model, are presented at the end of the paper.

Fuzzy Abel Grassmann Groupoids, Florentin Smarandache, Madad Khan, Tariq Aziz

Branch Mathematics and Statistics Faculty and Staff Publications

Usually the models of real world problems in almost all disciplines like engineering, medical sciences, mathematics, physics, computer science, management sciences, operations research and articial intelligence are mostly full of complexities and consist of several types of uncertainties while dealing them in several occasion. To overcome these di¢ culties of uncertainties, many theories have been developed such as rough sets theory, probability theory, fuzzy sets theory, theory of vague sets, theory of soft ideals and the theory of intuitionistic fuzzy sets, theory of neutrosophic sets, Dezert-Smarandache Theory (DSmT), etc. Zadeh introduced the degree of membership/truth (t) in 1965 and dened …

Theory Of Abel Grassmann's Groupoids, Florentin Smarandache, Madad Khan

Branch Mathematics and Statistics Faculty and Staff Publications

It is common knowledge that common models with their limited boundaries of truth and falsehood are not su¢ cient to detect the reality so there is a need to discover other systems which are able to address the daily life problems. In every branch of science problems arise which abound with uncertainties and impaction. Some of these problems are related to human life, some others are subjective while others are objective and classical methods are not su¢ cient to solve such problems because they can not handle various ambiguities involved. To overcome this problem, Zadeh [67] introduced the concept of …

Euclid Squares On Infinite Planes, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time the authors study the new type of Euclid squares in various planes like real plane, complex plane, dual number plane, special dual like number plane and special quasi dual number plane. There are six such planes and they behave distinctly. From the study it is revealed that each type of squares behave in a different way depending on the plane. We define several types of algebraic structures on them. Such study is new, innovative and interesting. However for some types of squares; one is not in a position to define product. Further under …

Algebraic Structures On Fuzzy Unit Square And Neutrosophic Unit Square, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors build algebraic structures on fuzzy unit semi open square UF = {(a, b) | a, b  [0, 1)} and on the fuzzy neutrosophic unit semi open square UN = {a + bI | a, b  [0, 1)}. This study is new and we define, develop and describe several interesting and innovative theories about them. We cannot build ring on UN or UF. We have only pseudo rings of infinite order. We also build pseudo semirings using these semi open unit squares. We construct vector spaces, S-vector spaces and strong pseudo special vector space using …

Algebraic Structures On Real And Neutrosophic Semi Open Squares, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Here for the first time we introduce the semi open square using modulo integers. Authors introduce several algebraic structures on them. These squares under addition modulo ‘n’ is a group and however under product  this semi open square is only a semigroup as under  the square has infinite number of zero divisors. Apart from + and  we define min and max operation on this square. Under min and max operation this semi real open square is a semiring. It is interesting to note that this semi open square is not a ring under + and  since …

Special Pseudo Linear Algebras Using [0,N), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce some special type of linear algebras called pseudo special linear algebras using the interval [0, n). These new types of special pseudo interval linear algebras has several interesting properties. Special pseudo interval linear algebras are built over the subfields in Zn where Zn is a S-ring. We study the substructures of them. The notion of Smarandache special interval pseudo linear algebras and Smarandache strong special pseudo interval linear algebras are introduced. The former Sspecial interval pseudo linear algebras are built over the Sring itself. Study in this direction has yielded several interesting results. S-strong special …

Soft Neutrosophic Algebraic Structures And Their Generalization - Vol. 1, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and …

New Research On Neutrosophic Algebraic Structures, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures in a broader sense. The indeterminacy element “ I “ gives rise to a more bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three categories: such as neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical …