We extend Knuth's 16 Boolean binary logic operators to fuzzy logic and neutrosophic logic binary operators. Then we generalize them to n-ary fuzzy logic and neutrosophic logic operators using the smarandache codification of the Venn diagram and a defined vector neutrosophic law. In such way, new operators in neutrosophic logic/set/probability are built.

In this paper, we propose in Dezert-Smarandache Theory (DSmT) framework, a new probabilistic transformation, called DSmP, in order to build a subjective probability measure from any basic belief assignment defined on any model of the frame of discernment. Several examples are given to show how the DSmP transformation works and we compare it to main existing transformations proposed in the literature so far. We show the advantages of DSmP over classical transformations in term of Probabilistic Information Content (PIC). The direct extension of this transformation for dealing with qualitative belief assignments is also presented.

This paper deals with enriched qualitative belief functions for reasoning under uncertainty and for combining information expressed in natural language through linguistic labels. In this work, two possible enrichments (quantitative and/or qualitative) of linguistic labels are considered and operators (addition, multiplication, division, etc) for dealing with them are proposed and explained. We denote them $qe$-operators, $qe$ standing for "qualitative-enriched" operators. These operators can be seen as a direct extension of the classical qualitative operators ($q$-operators) proposed recently in the Dezert-Smarandache Theory of plausible and paradoxist reasoning (DSmT). $q$-operators are also justified in details in this paper. The quantitative enrichment of linguistic label is a numerical supporting degree in $[0,\infty)$, while the qualitative enrichment takes its values in a finite ordered set of linguistic values. Quantitative enrichment is less precise than qualitative enrichment, but it is expected more close with what human experts can easily provide when expressing linguistic labels with supporting degrees. Two simple examples are given to show how the fusion of qualitative-enriched belief assignments can be done.

In this paper we extend the new family of (quantitative) Belief Conditioning Rules (BCR) recently developed in the Dezert-Smarandache Theory (DSmT) to their qualitative counterpart for belief revision. Since the revision of quantitative as well as qualitative belief assignment given the occurrence of a new event (the conditioning constraint) can be done in many possible ways, we present here only what we consider as the most appealing Qualitative Belief Conditioning Rules (QBCR) which allow to revise the belief directly with words and linguistic labels and thus avoids the introduction of ad-hoc translations of quantitative beliefs into quantitative ones for solving the problem.

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.

This paper may look like a glossary of the fusion rules and we also introduce new ones presenting their formulas and examples: Conjunctive, Disjunctive, Exclusive Disjunctive, Mixed Conjunctive-Disjunctive rules, Conditional rule, Dempster's, Yager's, Smets' TBM rule, Dubois-Prade's, Dezert-Smarandache classical and hybrid rules, Murphy's average rule, Inagaki-Lefevre-Colot-Vannoorenberghe Unified Combination rules [and, as particular cases: Iganaki's parameterized rule, Weighting Average Operator, minC (M. Daniel), and newly Proportional Conflict Redistribution rules (Smarandache-Dezert) among which PCR5 is the most exact way of redistribution of the conflicting mass to non-empty sets following the path of the conjunctive rule], Zhang's Center Combination rule, Convolutive x-Averaging, Consensus Operator (Josang), Cautious Rule (Smets), ?-junctions rules (Smets), etc. and three new T-norm & T-conorm rules adjusted from fuzzy and neutrosophic sets to information fusion (Tchamova-Smarandache). Introducing the degree of union and degree of inclusion with respect to the cardinal of sets not with the fuzzy set point of view, besides that of intersection, many fusion rules can be improved. There are corner cases where each rule might have difficulties working or may not get an expected result.