In order to more accurately situate and fit the neutrosophic logic into the framework of nonstandard analysis, we present the neutrosophic inequalities, neutrosophic equality, neutrosophic infimum and supremum, neutrosophic standard intervals, including the cases when the neutrosophic logic standard and nonstandard components T, I, F get values outside of the classical real unit interval [0, 1], and a brief evolution of neutrosophic operators. The paper intends to answer Imamura criticism that we found benefic in better understanding the nonstandard neutrosophic logic, although the nonstandard neutrosophic logic was never used in practical applications.

In this book we introduce 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). Plithogenic Set is a set whose elements are characterized by one or more attributes, and each attribute may have many 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, 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.

In this book we introduce a new procedure called \alpha-Discounting Method for Multi-Criteria Decision Making (\alpha-D MCDM), which is as an alternative and extension of Saaty 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. \alpha-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 non-homogeneous equations and or inequalities. The general idea of \alpha-D MCDM is to assign non-null positive parameters \alpha_1, \alpha_2, and so on \alpha_p to the coefficients in the right-hand side of each preference that diminish or increase them in order to transform the above linear homogeneous system of equations which has only the null-solution, into a system having a particular non-null solution. After finding the general solution of this system, the principles used to assign particular values to all parameters \alpha is the second important part of \alpha-D, yet to be deeper investigated in the future. In the current book we propose the Fairness Principle, i.e. each coefficient should be discounted with the same percentage (we think this is fair: not making any favoritism or unfairness to any coefficient), but the reader can propose other principles. For consistent decision-making problems with pairwise comparisons, \alpha-Discounting Method together with the Fairness Principle give the same result as AHP. But for weak inconsistent decision-making problem, \alpha-Discounting together with the Fairness Principle give a different result from AHP. Many consistent, weak inconsistent, and strong inconsistent examples are given in this book.

We present in this paper some examples of how to compute by hand the PCR5 fusion rule for three sources, so the reader will better understand its mechanism. We also take into consideration the importance of sources, which is different from the classical discounting of sources.

In this paper the behavior of three combinational rules for temporal/sequential attribute data fusion for target type estimation are analyzed. The comparative analysis is based on: Dempster's fusion rule proposed in Dempster-Shafer Theory; Proportional Conflict Redistribution rule no. 5 (PCR5), proposed in Dezert-Smarandache Theory and one alternative class fusion rule, connecting the combination rules for information fusion with particular fuzzy operators, focusing on the t-norm based Conjunctive rule as an analog of the ordinary conjunctive rule and t-conorm based Disjunctive rule as an analog of the ordinary disjunctive rule. The way how different t-conorms and t-norms functions within TCN fusion rule influence over target type estimation performance is studied and estimated.

In this paper we introduce two new DSm fusion conditioning rules with example, and as a generalization of them a class of DSm fusion conditioning rules, and then extend them to a class of DSm conditioning rules.

Comments on ``A new combination of evidence based on compromise'' by K. Yamada

Martin and Osswald \cite{Martin07} have recently proposed many generalizations of combination rules on quantitative beliefs in order to manage the conflict and to consider the specificity of the responses of the experts. Since the experts express themselves usually in natural language with linguistic labels, Smarandache and Dezert \cite{Li07} have introduced a mathematical framework for dealing directly also with qualitative beliefs. In this paper we recall some element of our previous works and propose the new combination rules, developed for the fusion of both qualitative or quantitative beliefs.

The management and combination of uncertain, imprecise, fuzzy and even paradoxical or high conflicting sources of information has always been, and still remains today, of primal importance for the development of reliable modern information systems involving artificial reasoning. In this introduction, we present a survey of our recent theory of plausible and paradoxical reasoning, known as Dezert-Smarandache Theory (DSmT), developed for dealing with imprecise, uncertain and conflicting sources of information. We focus our presentation on the foundations of DSmT and on its most important rules of combination, rather than on browsing specific applications of DSmT available in literature. Several simple examples are given throughout this presentation to show the efficiency and the generality of this new approach.

In this paper we present the N-norms/N-conorms in neutrosophic logic and set as extensions of T-norms/T-conorms in fuzzy logic and set. Also, as an extension of the Intuitionistic Fuzzy Topology we present the Neutrosophic Topologies.