Prof. Florentin Smarandache, PhD The University of New Mexico Math & Science Dept. 705 Gurley Ave. Gallup, NM 87301, USA http://fs.gallup.unm.edu/ neutrosophy.htm Foundations of Neutrosophic Logic and Set and their Applications Contents Theory
Definition of Neutrosophy A Short History of the Logics Introduction to Non-Standard Analysis Operations with Classical Sets Neutrosophic Logic (NL) Refined Neutrosophic Logic and Set
Classical Mass and Neutrosophic Mass Differences between Neutrosophic Logic and Intuitionistic Fuzzy Logic Neutrosophic Logic generalizes many Logics Neutrosophic Logic Connectors Neutrosophic Set (NS) Neutrosophic Cube as Geometric Interpretation of the Neutrosophic Set Neutrosophic Set Operators Differences between Neutrosophic Set and Intuitionistic Fuzzy Set Partial Order in Neutrosophics N-Norm and N-conorm Interval Neutrosophic Operators Remarks on Neutrosophic Operators Examples of Neutrosophic Operators resulted from N-norms and N-conorms
Contents - contd. Applications Application of Fuzzy Logic to Information Fusion Application of Neutrosophic Logic to Information
Fusion How to Compute with Labels General Applications of Neutrosophic Logic General Applications of Neutrosophic Sets Neutrosophic Numbers Neutrosophic Algebraic Structures Neutrosophic Matrix Neutrosophic Graphs and Trees Neutrosophic Cognitive Maps & Neutrosophic Relational Maps Neutrosophic Quadruple Algebraic Structures Neutrosophic Probability and Statistics Applications of Neutrosophy to Extenics and Indian Philosophy Neutrosophics as a situation analysis tool
Application to Robotics The Need for a Novel Decision Paradigm in Management (F. S. & S. Bhattacharya) Application of Neutrosophics in Production Facility Layout Planning and Design (F. S. & S. Bhattacharya) Applications to Neutrosophic and Paradoxist Physics More Applications Definition of Neutrosophy A new branch of philosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra (1995). Neutrosophy opened a new field or research in metaphilosophy. Etymologically, neutro-sophy [French neutre < Latin neuter, neutral, and
Greek sophia, skill/wisdom] means knowledge of neutral thought and started in 1995. Extension of dialectics. Connected with Extenics (Prof. Cai Wen, 1983), and Paradoxism (F. Smarandache, 1980) The Fundamental Theory: Every idea tends to be neutralized, diminished, balanced by
for Neutrosophic Logic, Neutrosophic Set, Neutrosophic A Short History of Logics The fuzzy set (FS) was introduced by L. Zadeh in 1965, where each element had a degree of membership.
The intuitionistic fuzzy set (IFS) on a universe X was introduced by K. Atanassov in 1983 as a generalization of FS, where besides the degree of membership A(x) [0,1] of each element x to a set A there was considered a degree of non-membership A(x)[0,1], but such that for x X, A(x)+A(x)1. According to Cornelis et al. (2003), Gehrke et al. (1996) stated that Many people believe that assigning an exact number to an experts opinion is too restrictive, and the assignment of an interval of values is more realistic, which is somehow similar with the imprecise probability theory where instead of a crisp probability one has an interval (upper and lower) probabilities as in Walley (1991). Atanassov (1999) defined the interval-valued intuitionistic fuzzy set (IVIFS) on a universe X as an object A such that: A= {(x, MA(X), NA(x)), xX}, with MA:XInt([0,1]) and NA:XInt([0,1]) and x X, supMA(x)+ supNA(x) 1. A Short History of Logics contd. Belnap (1977) defined a four-valued logic, with truth (T), false (F), unknown (U), and contradiction (C). He used a bi-lattice where the four
components were inter-related. In 1995, starting from philosophy (when I fretted to distinguish between absolute truth and relative truth or between absolute falsehood and relative falsehood in logics, and respectively between absolute membership and relative membership or absolute non-membership and relative non-membership in set theory) I began to use the non-standard analysis. Also, inspired from the sport games (winning, defeating, or tie scores), from votes (pro, contra, null/black votes), from positive/negative/zero numbers, from yes/no/NA, from decision making and control theory (making a decision, not making, or hesitating), from accepted/rejected/ pending, etc. and guided by the fact that the law of excluded middle did not work any longer in the modern logics, I combined the non-standard analysis with a tri-component logic/set/probability theory and with philosophy (I was excited by paradoxism in science and arts and letters, as well as by paraconsistency and incomplete-ness in knowledge). How to deal with all of them at once, is it possible to unify them?
A Short History of Logics contd. I proposed the term "neutrosophic" because "neutrosophic" etymologically comes from "neutrosophy" [French neutre < Latin neuter, neutral, and Greek sophia, skill/wisdom] which means knowledge of neutral thought, and this third/neutral represents the main distinction between "fuzzy" and "intuitionistic fuzzy" logic/set, i.e. the included middle component (Lupasco-Nicolescus logic in philosophy), i.e. the neutral/ indeterminate/ unknown part (besides the "truth"/"membership" and "falsehood"/"non-membership" components that both appear in fuzzy logic/set). Introduction to Nonstandard Analysis
Abraham Robinson developed the nonstandard analysis (1960s); x is called infinitesimal if |x|<1/n for any positive n; A left monad (-a) = {a-x: x in R*, x>0 infinitesimal} = a-, and a right monad (b+) = {a+x: x in R*, x>0 infinitesimal} = b+, where >0 is infinitesimal; a, b called standard parts, called nonstandard part. Operations with Nonstandard Finite Real Numbers - a*b = -(a*b), a*b+ = (a*b)+, -a*b+ = -(a*b)+, -
a*-b = -(a*b) [the left monads absorb themselves], a+*b+ = (a*b)+ [the right monads absorb themselves], where * can be addition, subtraction, multiplication, division, power. Operations with Classical Sets S1 and S2 two real standard or nonstandard sets. Addition: Subtraction: Multiplication: Division of a set by a non-null number:
Neutrosophic Logic Consider the nonstandard unit interval ]-0, 1+[, with left and right borders vague, imprecise; Let T, I, F be standard or nonstandard subsets of ]-0, 1+[ ; Neutrosophic Logic (NL) is a logic in which each proposition is T% true, I % indeterminate, and F% false; - 0 <= inf T + inf I + inf F <= sup T + sup I + sup F <= 3+ ; T, I, F are not necessary intervals, but any sets (discrete, continuous, open or closed or half-open/half-closed interval, intersections or unions of the previous sets, etc.); Example:
proposition P is between 30-40% or 45-50% true, 20% indeterminate, and 60% or between 66-70% false (according to various analyzers or parameters); NL is a generalization of Zadehs fuzzy logic (FL), and especially of Atanassovs intuitionistic fuzzy logic (IFL), and of other logics. Refined Neutrosophic Logic and Set Component I, indeterminacy, can be split into more subcomponents in order to better catch the vague information we work with, and such, for example, one can get more accurate answers to the Question-Answering Systems initiated by Zadeh (2003). {In Belnaps four-valued logic (1977) indeterminacy was split into Uncertainty (U) and Contradiction (C), but they were interrelated.} Even more, we proposed to split "I" into Contradiction, Uncertainty, and
Unknown, and we get a five-valued logic. In a general Refined Neutrosophic Set, "T" can be split into subcomponents T1, T2, ..., Tm, and "I" into I1, I2, ..., In, and "F" into F1, F2, ..., Fp because there are more types of truths, of indeterminacies, and respectively of falsehoods. Classical Mass & Neutrosophic Mass Let be a frame of discernment, defined as: = {1, 2, , n}, n 2, and its Super-Power Set (or fusion space): S^ = ( , = ( , U, , C ) which means: the set closed under union, intersection, and respectively complement. Classical Mass. We recall that a classical mass m(.) is defined as: m: S^ -> [0,1] -> [0,1] such that m(X)=1.
X in S^ -> [0,1] . Classical Mass & Neutrosophic Mass contd. We extend the classical basic belief assignment (or classical mass) bba m(.) to a neutrosophic basic belief assignment (nbba) (or neutrosophic mass) mn( .) in the following way. mn : S^ -> [0,1] -> [0,1]^ -> [0,1] 3 with mn(A) = (T(A), I(A), F(A)) where T(A) means the (local) chance that hypothesis A occurs, F(A) means the (local) chance that hypothesis A does not occur (non-chance), while I(A) means the (local) indeterminate chance of A (i.e. knowing neither if A occurs nor if A doesnt occur), such that: [T(X)+I(X)+F(X)]=1. X in S^ -> [0,1] . Classical Mass &
Neutrosophic Mass contd. In a more general way, the summation can be less than 1 (for incomplete neutrosophic information), equal to 1 (for complete neutrosophic information), or greater than 1 (for paraconsistent/conflicting neutrosophic information). But in this paper we only present the case when summation is equal to 1. Of course 1 <= T(X), I(X), F(X) <= 1 for all X in S^ -> [0,1] . Differences between Neutrosophic Logic and Intuitionistic Fuzzy Logic In NL there is no restriction on T, I, F, while in IFL the sum of components (or their superior limits) = 1; thus NL can characterize the incomplete information (sum < 1), paraconsistent information (sum > 1). NL can distinguish, in philosophy, between absolute truth [NL(absolute
truth)=1+] and relative truth [NL(relative truth)=1], while IFL cannot; absolute truth is truth in all possible worlds (Leibniz), relative truth is truth in at least one world. In NL the components can be nonstandard, in IFL they dont. NL, like dialetheism [some contradictions are true], can deal with paradoxes, NL(paradox) = (1,I,1), while IFL cannot. Neutrosophic Logic Generalizes Many Logics Let the components reduced to scalar numbers, t,i,f, with t+i+f=n; NL generalizes: - the Boolean logic (for n = 1 and i = 0, with t, f either 0 or 1); - the multi-valued logic, which supports the existence of many values between true and false [Lukasiewicz, 3 values; Post, m values] (for n = 1, i = 0, 0 <= t, f <= 1); - the intuitionistic logic, which supports incomplete theories, where A\/nonA (Law of Excluded Middle) not always true, and There exist x such that P(x) is true needs an algorithm constructing x [Brouwer,
1907] (for 0 < n < 1 and i = 0, 0 <= t, f < 1); - the fuzzy logic, which supports degrees of truth [Zadeh, 1965] (for n = 1 and i = 0, 0 <= t, f <= 1); - the intuitionistic fuzzy logic, which supports degrees of truth and degrees of falsity while whats left is considered indeterminacy [Atanassov, 1982] (for n = 1); Neutrosophic Logic Generalizes Many Logics contd. - the paraconsistent logic, which supports conflicting information, and anything follows from contradictions fails, i.e. A/\nonA->B fails; A/\ nonA is not always false (for n > 1 and i = 0, with both 0< t, f < 1); the dialetheism, which says that some contradictions are true, A/\ nonA=true (for t = f = 1 and i = 0; some paradoxes can be denoted
this way too); the faillibilism, which says that uncertainty belongs to every proposition (for i > 0). Neutrosophic Logic Connectors A1(T1, I1, F1) and A2(T2, I2, F2) are two propositions. Neutrosophic Logic Connectors contd. Many properties of the classical logic operators do not apply in neutrosophic logic. Neutrosophic logic operators (connectors) can be defined in many ways
according to the needs of applications or of the problem solving. Neutrosophic Set (NS) Let U be a universe of discourse, M a set included in U. An element x from U is noted with respect to the neutrosophic set M as x(T, I, F) and belongs to M in the following way: it is t% true in the set (degree of membership), i% indeterminate indeterminacy), (unknown if it is in
the set) (degree of and f% false (degree of non-membership), where t varies in T, i varies in I, f varies in F. Definition analogue to NL Generalizes the fuzzy set (FS), especially the intuitionistic fuzzy set (IFS), intuitionistic set (IS), paraconsistent set (PS) Example: x(50,20,40) in A means: with a believe of 50% x is in A, with a believe of 40% x is not in A (disbelieve), and the 20% is undecidable.
Neutrosophic Cube as Geometric Interpretation of Neutrosophic Set The most important distinction between IFS and NS is showed in the below Neutrosophic Cube ABCDEFGH introduced by J. Dezert in 2002. Because in technical applications only the classical interval is used as range for the neutrosophic parameters , we call the cube the technical neutrosophic cube and its extension the neutrosophic cube (or absolute neutrosophic cube), used in the fields where we need to differentiate between absolute and relative (as in philosophy) notions. Neutrosophic Cube as Geometric Interpretation of Neutrosophic Set contd.
Neutrosophic Cube as Geometric Interpretation of Neutrosophic Set contd. Lets consider a 3D-Cartesian system of coordinates, where t is the truth axis with value range in ]-0,1+[, i is the false axis with value range in ]0,1+[, and similarly f is the indeterminate axis with value range in ]-0,1+[. We now divide the technical neutrosophic cube ABCDEFGH into three disjoint regions: 1) The equilateral triangle BDE, whose sides are equal to V2, which represents the geometrical locus of the points whose sum of the coordinates is 1. If a point Q is situated on the sides of the triangle BDE or inside of it, then tQ+iQ+fQ=1 as in Atanassov-intuitionistic fuzzy set (A-IFS). 2) The pyramid EABD {situated in the right side of the triangle EBD, including its faces triangle ABD (base), triangle EBA, and triangle EDA (lateral faces), but excluding its face: triangle BDE} is the locus of the points whose sum of coordinates is less than 1. 3) In the left side of triangle BDE in the cube there is the solid EFGCDEBD (excluding triangle BDE) which is the locus of points whose sum of their coordinates is greater than 1 as in the paraconsistent set.
Neutrosophic Cube as Geometric Interpretation of Neutrosophic Set contd. It is possible to get the sum of coordinates strictly less than 1 or strictly greater than 1. For example: We have a source which is capable to find only the degree of membership of an element; but it is unable to find the degree of nonmembership; another source which is capable to find only the degree of non- membership of an element; or a source which only computes the indeterminacy. Thus, when we put the results together of these sources, it is possible that their sum is not 1, but smaller or greater. Also, in information fusion, when dealing with indeterminate models (i.e. elements of the fusion space which are indeterminate/unknown, such as intersections we dont know if they are empty or not since we dont have enough information, similarly for complements of indeterminate elements,
etc.): if we compute the believe in that element (truth), the disbelieve in that element (falsehood), and the indeterminacy part of that element, then the sum of these three components is strictly less than 1 (the difference to 1 is the missing information). Neutrosophic Set Operators A and B two sets over the universe U. An element x(T1, I1, F1) in A and x(T2, I2, F2) in B [neutrosophic membership appurtenance to A and respectively to B]. NS operators (similar to NL connectors) can also be defined in many ways. Differences between Neutrosophic Set and Intuitionistic Fuzzy Set In NS there is no restriction on T, I, F, while in IFS the sum of components (or their superior limits) = 1; thus NL can characterize the incomplete information (sum < 1), paraconsistent information (sum > 1). NS
can distinguish, in philosophy, between absolute membership [NS(absolute membership)=1+] and relative membership [NS(relativemembership)=1], while IFS cannot; absolute membership is membership in all possible worlds, relative membership is membership in at least one world. In NS the components can be nonstandard, in IFS they dont. NS, like dialetheism [some contradictions are true], can deal with paradoxes, NS(paradox element) = (1,I,1), while IFS cannot. NS operators can be defined with respect to T,I,F while IFS operators are defined with respect to T and F only. I can be split in NS in more subcomponents (for example in Belnaps four-
valued logic (1977) indeterminacy contradiction), but in IFS it cannot. is split into uncertainty and Partial Order in Neutrosophics We define a partial order relationship on the neutrosophic set/logic in the following way: x(T1, I1, F1) y(T2, I2, F2) iff (if and only if) T1 T2, I1 I2, F1 F2 for crisp components. And, in general, for subunitary set components:
x(T1, I1, F1) y(T2, I2, F2) iff inf T1 inf T2, sup T1 sup T2, inf I1 inf I2, sup I1 sup I2, inf F1 inf F2, sup F1 sup F2. If we have mixed - crisp and subunitary - components, or only crisp components, we can transform any crisp component, say a with a in [0,1] or a in ]-0, 1+[, into a subunitary set [a, a]. So, the definitions for subunitary set components should work in any case. N-Norm and N-Conorm As a generalization of T-norm and T-conorm from the Fuzzy Logic and Set, we now introduce the N-norms and N-conorms for the Neutrosophic Logic and Set. N-norm Nn: ( ]-0,1+[ ]-0,1+[ ]-0,1+[ )2 ]-0,1+[ ]-0,1+[ ]-0,1+[ Nn (x(T1,I1,F1), y(T2,I2,F2)) = (NnT(x,y), NnI(x,y), NnF(x,y)), where NnT(.,.), NnI(.,.), NnF(.,.) are the truth/membership, indeterminacy, and respectively falsehood/nonmembership components. Nn have to satisfy, for any x, y, z in the neutrosophic logic/set M of the universe of discourse U, the following axioms:
a) Boundary Conditions: Nn(x, 0) = 0, Nn(x, 1) = x. b) Commutativity: Nn(x, y) = Nn(y, x). c) Monotonicity: If x y, then Nn(x, z) Nn(y, z). d) Associativity: Nn(Nn (x, y), z) = Nn(x, Nn(y, z)). N-Norm and N-Conorm contd. There are cases when not all these axioms are satisfied, for example the associativity when dealing with the neutrosophic normalization after each neutrosophic operation. But, since we work with approximations, we can call these N-pseudo-norms, which still give good results in practice. Nn represent the and operator in neutrosophic logic, and respectively the intersection operator in neutrosophic set theory. Let J in {T, I, F} be a component. Most known N-norms, as in fuzzy logic and set the T-norms, are: The Algebraic Product N-norm: NnalgebraicJ(x, y) = x y The Bounded N-Norm: NnboundedJ(x, y) = max{0, x + y 1} The Default (min) N-norm: NnminJ(x, y) = min{x, y}.
N-Norm and N-Conorm contd. A general example of N-norm would be this: Let x(T1, I1, F1) and y(T2, I2, F2) be in the neutrosophic set/logic M. Then: Nn(x, y) = (T1/\T2, I1\/I2, F1\/F2) where the /\ operator, acting on two (standard or non-standard) subunitary sets, is a N-norm (verifying the above N-norms axioms); while the \/ operator, also acting on two (standard or non-standard) subunitary sets, is a N-conorm (verifying the below N-conorms axioms). For example, /\ can be the Algebraic Product T-norm/N-norm, so T1/\T2 = T1T2 (herein we have a product of two subunitary sets using simplified notation); and \/ can be the Algebraic Product Tconorm/N-conorm, so T1\/T2 = T1+T2-T1T2 (herein we have a sum, then a product, and afterwards a subtraction of two subunitary sets). N-Norm and N-Conorm contd. Nc: ( ]-0,1+[ ]-0,1+[ ]-0,1+[ )2 ]-0,1+[ ]-0,1+[ ]-0,1+[ Nc (x(T1,I1,F1), y(T2,I2,F2)) = (NcT(x,y), NcI(x,y), NcF(x,y)), where NnT(.,.), NnI(.,.), NnF(.,.) are the truth/membership, indeterminacy, and respectively falsehood/nonmembership
components. Nc have to satisfy, for any x, y, z in the neutrosophic logic/set M of universe of discourse U, the following axioms: a) Boundary Conditions: Nc(x, 1) = 1, Nc(x, 0) = x. b) Commutativity: Nc (x, y) = Nc(y, x). c) Monotonicity: if x y, then Nc(x, z) Nc(y, z). d) Associativity: Nc (Nc(x, y), z) = Nc(x, Nc(y, z)). N-Norm and N-Conorm contd. There are cases when not all these axioms are satisfied, for example the associativity when dealing with the neutrosophic normalization after each neutrosophic operation. But, since we work with approximations, we can call these N-pseudo-conorms, which still give good results in practice. Nc represent the or operator in neutrosophic logic, and respectively the union operator in neutrosophic set theory.
Let J in {T, I, F} be a component. Most known N-conorms, as in fuzzy logic and set the Tconorms, are: The Algebraic Product N-conorm: NcalgebraicJ(x, y) = x + y xy The Bounded N-conorm: NcboundedJ(x, y) = min{1, x + y} The Default (max) N-conorm: NcmaxJ(x, y) = max{x, y}. N-Norm and N-Conorm contd. A general example of N-conorm would be this. Let x(T1, I1, F1) and y(T2, I2, F2) be in the neutrosophic set/logic M. Then: Nn(x, y) = (T1\/T2, I1/\I2, F1/\F2) where as above - the /\ operator, acting on two (standard or nonstandard) subunitary sets, is a N-norm (verifying the above N-norms axioms); while the \/ operator, also acting on two (standard or nonstandard) subunitary sets, is a N-conorm (verifying the above N-conorms axioms). For example, /\ can be the Algebraic Product T-norm/N-norm, so T1/\T2 = T1T2 (herein we have a product of two subunitary sets); and \/ can be the Algebraic Product Tconorm/N-conorm, so T1\/T2 = T1+T2-T1T2 (herein we have a sum, then a product, and afterwards a subtraction of two subunitary sets).
N-Norm and N-Conorm contd. Or /\ can be any T-norm/N-norm, and \/ any T-conorm/N-conorm from the above; for example the easiest way would be to consider the min for crisp components (or inf for subset components) and respectively max for crisp components (or sup for subset components). If we have crisp numbers, we can at the end neutrosophically normalize. Interval Neutrosophic Operators Interval Neutrosophic Conjunction/Intersection: x/\y=(T/\,I/\,F/\), where inf T/\ = min{inf T1, inf T2}; sup T/\ = min{sup T1, sup T2}; inf I/\ = max{inf I1, inf I2}; sup I/\ = max{sup I1, sup I2}; inf F/\ = max{inf F1, inf F2};
sup F/\ = max{sup F1, sup F2}. Interval Neutrosophic Operators contd. Interval Neutrosophic Disjunction/Union: x\/y=(T\/,I\/,F\/), where inf T\/ = max{inf T1, inf T2}; sup T\/ = max{sup T1, sup T2}; inf I\/ = min{inf I1, inf I2}; sup I\/ = min{sup I1, sup I2}; inf F\/ = min{inf F1, inf F2}; sup F\/ = min{sup F1, sup F2}. Interval Neutrosophic Operators contd. Interval Neutrosophic Containment: We say that the neutrosophic set A is included in the neutrosophic set B of the universe of discourse U, iff for
any x(TA, IA, FA) A with x(TB, IB, FB) B we have: inf TA inf TB ; sup TA sup TB; inf IA inf IB ; sup IA sup IB; inf FA inf FB ; sup FA sup FB. Remarks on Neutrosophic Operators A. The non-standard unit interval ]-0, 1+[ is merely used for philosophical applications, especially when we want to make a distinction between relative truth (truth in at least one world) and absolute truth (truth in all possible worlds), and similarly for distinction between relative or absolute falsehood, and between relative or absolute indeterminacy. But, for technical applications of neutrosophic logic and set, the domain of definition and range of the N-norm and N-conorm can be restrained to the normal standard real unit interval [0, 1], which is easier to use, therefore: Nn: ( [0,1] [0,1] [0,1] )2 [0,1] [0,1] [0,1] and Nc: ( [0,1] [0,1] [0,1] )2 [0,1] [0,1] [0,1].
Remarks on Neutrosophic Operators contd. B. Since in NL and NS the sum of the components (in the case when T, I, F are crisp numbers, not sets) is not necessary equal to 1 (so the normalization is not required), we can keep the final result unnormalized. But, if the normalization is needed for special applications, we can normalize at the end by dividing each component by the sum all components. Remarks on Neutrosophic Operators contd. C. If T, I, F are subsets of [0, 1] the problem of neutrosophic normalization is more difficult. If sup(T)+sup(I)+sup(F) < 1, we have an intuitionistic proposition/set. If inf(T)+inf(I)+inf(F) > 1, we have a paraconsistent proposition/set. If there exist the crisp numbers t in T, i in I, and f in F such that t + i + f =1, then we can say that we have a plausible normalized proposition/set. Examples of Neutrosophic
Operators resulting from N-norms or N-pseudonorms The neutrosophic conjunction (intersection) operator component truth, indeterminacy, and falsehood values result from the multiplication (T1+I1 +F1)(T2+I2+F2), since we consider in a prudent way T
m1(Tank) = 0.4, then m1(not Tank) = 0.6; m2(Tank) = 0.5, then m2(not Tank) = 0.5. Then we use the product-sum fuzzy operators: and for the truth component: a/\b = ab (T-norm); or for the false component: a\/b = a+b-ab (T-conorm): (m1/\m2)(Tank) = 0.4(0.5) = 0.2. Then, of course (m1\/m2)(not Tank) = 1-.2 = 0.8 {or through a different calculation using the above T-conorm (m1\/m2)(not Tank) = 0.6+0.5-0.6(0.5) = 0.8}. T-norm is a class of and (conjunction/intersection) fuzzy operators, while Tconorm is a class of or (disjunction/union) fuzzy operators. Application of Fuzzy Logic to Information Fusion contd. Suppose we look for a target type identification: is the target a friend, a neutral, or an enemy? Then = { F(riend), N(eutral), E(nemy) }. Two neutrosophic sources nm1 and nm2 give us information about the target type: nm1(F) = 0.2, nm1(N) = 0.3, nm1(E) = 0.5; nm2(F) = 0.6, nm2(N) = 0.1, nm2(E) = 0.3. Then we use the neutrosophic product operator and (N-norm):
(a1, a2, a3)/\(b1, b2, b3) = (a1b1, a2b2, a3b3,) and then normalize; (nm1/\nm2)( F, N, E ) = (0.2 0.6, 0.3 0.1, 0.5 0.3) = (0.12, 0.03, 0.15) and then divide by their sum 0.30 (normalize): = (0.4, 0.1, 0.5). So, it is mostly (with a believe of 50%) an Enemy target. N-norm is a class of and (conjunction/intersection) neutrosophic operators, while N-conorm is a class of or (disjunction/union) neutrosophic operators. So, there are many and/or neutrosophic operators. How to Compute with Labels Type of sources of information: 1) Numerical Source, which gives us believe estimation in numbers. Example: The likelihood that the aircraft is a Fighter is 80%. 2) Qualitative Source, which gives us words (in natural language), also called labels.
Example: The likelihood that the aircraft is a Fighter is high. How to Compute with Labels? Either try to convert the labels into approximate corresponding numbers in [0,1]; Or, directly compute with labels. Previous Example: Then = { F(riend), N(eutral), E(nemy) }. Two qualitative neutrosophic sources nm1 and nm2 give us information about the target type: qnm1(F) = very low, qnm1(N) = above low, qnm1(E) = medium;
qnm2(F) = above medium, qnm2(N) = very low, qnm2(E) = low; How to Compute with Labels contd. The set of ordered labels is: Lmin < Very Low < Low < Above Low < Medium < Above Medium < High < Very High < Lmax But we can renumber them: L0 = Lmin < L1 < L2 < < Lmax L3 < L4 < L5
< L6 < L7 (qnm1/\qnm2)( F, N, E ) = (L1, L3, L4) /\ (L5, L1, L2) = ( min{L1,L5}, min{L3,L1}, min{L4,L2} ) [amongst the neutrosophic and operators we have used the min operator] = (L1, L1, L2) and we quasi-normalize the result by increasing with the same quality each label; so we get = (L2, L2, L3) (qnm1/\qnm2)( F, N, E ) = (low, low, above low), so the largest believe [= above low] is that the target is an Enemy. General Applications of Neutrosophic Logic Voting (pro, contra, neuter). The candidate C, who runs for election in a metropolis M of p people with right to vote, will win.
This proposition is, say, 20-25% true (percentage of people voting for him), 35-45% false (percentage of people voting against him), and 40% or 50% indeterminate (percentage of people not coming to the ballot box, or giving a blank vote not selecting anyone, or giving a negative vote cutting all candidates on the list). Epistemic/subjective uncertainty (which has hidden/unknown parameters). Tomorrow it will rain. This proposition is, say, 50% true according to meteorologists who have investigated the past years' weather, between 20-30% false according to today's very sunny and droughty summer, and 40% undecided. General Applications of Neutrosophic Logic contd. Paradoxes. This is a heap (Sorites Paradox). We may now say that this proposition is 80% true, 40% false, and 25-35%
indeterminate (the neutrality comes for we don't know exactly where is the difference between a heap and a nonheap; and, if we approximate the border, our 'accuracy' is subjective). Vagueness plays here an important role. The Medieval paradox, called Buridans Ass after Jean Buridan (near 1295-1356), is a perfect example of complete indeterminacy. An ass, equidistantly from two quantitatively and qualitatively heaps of grain, starves to death because there is no ground for preferring one heap to another. The neutrosophic value of asss decision, NL = (0, 1, 0). Games (win, defeated, tied). Electrical charge, temperature, altitude, numbers, and other 3-valued systems (positive, negative, zero). Business (M. Khoshnevisan, S. Bhattacharya). Investors who are: Conservative and security-oriented (risk shy), Chance-oriented and progressive (risk happy), or Growth-oriented and dynamic (risk neutral). General Applications of Neutrosophic
Sets Philosophical Applications: Or, how to calculate the truth-value of Zen (in Japanese) / Chan (in Chinese) doctrine philosophical proposition: the present is eternal and comprises in itself the past and the future? In Eastern Philosophy the contradictory utterances form the core of the Taoism and Zen/Chan (which emerged from Buddhism and Taoism) doctrines. How to judge the truth-value of a metaphor, or of an ambiguous statement, or of a social phenomenon which is positive from a standpoint and negative from another standpoint? Physics Applications: How to describe a particle in the infinite micro-universe of Quantum Physics that belongs to two distinct places P1 and P2 in the same time? in P1 and is not in P1 as a true contradiction, or in P1 and in nonP1.
General Applications of Neutrosophic Sets contd. Dont we better describe, using the attribute neutrosophic than fuzzy and others, a quantum particle that neither exists nor non-exists? [high degree of indeterminacy] In Schroedingers Equation on the behavior of electromagnetic waves and matter waves in Quantum Theory, the wave function Psi which describes the superposition of possible states may be simulated by a neutrosophic function, i.e. a function whose values are not unique for each argument from the domain of definition (the vertical line test fails, intersecting the graph in more points). A cloud is a neutrosophic set, because its borders are ambiguous, and each element (water drop) belongs with a neutrosophic probability to the set (e.g. there are a kind of separated water drops, around a compact mass of water drops, that we don't know how to consider them: in or out of the cloud).
Neutrosophic Numbers The Neutrosophic Numbers have been introduced by W.B. Vasantha Kandasamy and F. Smarandache, which are numbers of the form N = a+bI, where a, b are real or complex numbers, while I is the indeterminacy part of the neutrosophic number N, such that I2 = I and I + I = (+)I. Of course, indeterminacy I is different from the imaginary number i. In general one has I n = I if n > 0, and is undefined if n 0. Neutrosophic Algebraic Structures The algebraic structures using neutrosophic numbers gave birth to the
neutrosophic algebraic structures [see for example neutrosophic groups, neutrosophic rings, neutrosophic vector space, neutrosophic matrices, bimatrices, , n-matrices, etc.], introduced by W.B. Vasantha Kandasamy, F. Smarandache, Adesina Agboola, B. Davvaz et al. Example of Neutrosophic Ring: ({a+bI, with a, b R}, +, ), where of course (a+bI)+(c+dI) = (a+c)+(b+d)I, and (a+bI) (c+dI) = (ac) + (ad+bc+bd)I. Neutrosophic Matrix A Neutrosophic Matrix is a matrix which has neutrosophic numbers. See an example: Neutrosophic Graphs and Trees Also, I led to the definition of the neutrosophic graphs
(graphs which have at least either one indeterminate edge or one indeterminate node), and neutrosophic trees (trees which have at least either one indeterminate edge or one indeterminate node), which have many applications in social sciences. An edge is said indeterminate if we dont know if it is any relationship between the nodes it connects, or for a directed graph we dont know if it is a directly or inversely proportional relationship. A node is indeterminate if we dont know what kind of node it is since we have incomplete information. Neutrosophic Graphs and Trees contd. Example of Neutrosophic Graph (edges V1V3, V1V5, V2V3 are indeterminate and they are drawn as dotted):
Neutrosophic Graphs and Trees contd. The graphs neutrosophic adjacency matrix is below. The edges mean: 0 = no connection between nodes, 1 = connection between nodes, I = indeterminate connection (not known if it is or if it is not). Such notions are not used in the fuzzy theory. Maps & Neutrosophic Relational Maps As a consequence, the Neutrosophic Cognitive Maps and Neutrosophic Relational Maps are generalizations of fuzzy cognitive maps and respectively fuzzy relational maps (W.B. Vasantha Kandasamy, F. Smarandache et al.). A Neutrosophic Cognitive Map (NCM) is a neutrosophic directed graph
with concepts like policies, events etc. as nodes and causalities or indeterminates as edges. It represents the causal relationship between concepts. Neutrosophic Cognitive Maps & Neutrosophic Relational Maps example An Example of Neutrosophic Cognitive Map (NCM), which is a generalization of the Fuzzy Cognitive Maps (FCM). Lets have the following nodes: C1 - Child Labor; C2 - Political Leaders; C3 - Good Teachers; C4 Poverty; C5 Industrialists; C6 - Public practicing/encouraging Child Labor; C7 - Good Non-Governmental Organizations (NGOs). Neutrosophic Cognitive Maps & Neutrosophic Relational Maps example
The edges mean: 0 = no connection between nodes, 1 = directly proportional connection, -1 = inversely proportionally connection, and I = indeterminate connection (not knowing what kind of relationship is between the nodes the edge connects). Neutrosophic Cognitive Maps & Neutrosophic Relational Maps example The corresponding neutrosophic adjacency matrix related to this neutrosophic cognitive map is below. Neutrosophic Quadruple Algebraic Structures Developed by F. Smarandache, A. A. A. Agboola, and B. Davvaz (20152017). A neutrosophic quadruple number is a number of the form (a, bT, cI, dF) where T, I, F have their usual neutrosophic logic meanings and a, b, c, d are real or complex numbers. The set NQ defined by NQ = {(a, bT, cI, dF): a, b, c, d R or C}
is called a neutrosophic set of quadruple numbers. A neutrosophic quadruple number (a, bT, cI, dF) represents any entity, which may be a number, an idea, an object etc., where a is called the known part and (bT, cI, dF) is called the unknown part. 61 Neutrosophic Probability & Statistics The neutrosophics introduced (in 1995) the Neutrosophic Probability (NP), which is a generalization of the classical and imprecise probabilities. Neutrosophic Probability of an event E is the chance that event E occurs, the chance that event E doesnt occur, and the chance of indeterminacy (not knowing if the event E occurs or not). In classical probability nsup 1, while in neutrosophic probability nsup 3+. In imprecise probability: the probability of an event is a subset T in [0, 1],
not a crisp number p in [0, 1], whats left is supposed to be the opposite, subset F (also from the unit interval [0, 1]); there is no indeterminate subset I in imprecise probability. Consequently the Neutrosophic Statistics (NS), which is the analysis of the neutrosophic events. Neutrosophic statistics deals with neutrosophic numbers, neutrosophic probability regression. distribution, neutrosophic estimation, neutrosophic Neutrosophic Distribution
The function that models the neutrosophic probability of a random variable x is called neutrosophic distribution: NP(x) = ( T(x), I(x), F(x) ), where T(x) represents the probability that value x occurs, F(x) represents the probability that value x does not occur, and I(x) represents the indeterminate / unknown probability of value x. Applications of Neutrosophy to Extenics and Indian Philosophy Extenics, founded by Prof. Cai Wen in 1983, means solving contradictions problems in information fusion, management, design, automation etc. using computers and Internet. In India's VIII-th - IX-th centuries one promulgated the Non-Duality
(Advaita) through the non-differentiation between Individual Being (Atman) and Supreme Being (Brahman). The philosopher Sakaracharya (782-814 A.C.) was then considered the savior of Hinduism, just in the moment when the Buddhism and the Jainism were in a severe turmoil and India was in a spiritual crisis. Non-Duality means elimination of ego, in order to blend yourself with the Supreme Being (to reach the happiness). Or, arriving to the Supreme was done by Prayer (Bhakti) or Cognition (Jnana). It is a part of Sakaracharya's huge merit (charya means teacher) the originality of interpreting and synthesizing the Source of Cognition (Vedas, IV th century B.C.), the Epic (with many stories), and the Upanishads (principles of Hindu philosophy) concluding in Non-Duality. Applications of Neutrosophy to Extenics and Indian Philosophy contd. Then Special Duality (Visishta Advaita) follows, which asserts that Individual Being and Supreme Being are different in the beginning, but end
to blend themselves (Rmnujacharya, XI-th century). And later, to see that the neutrosophic scheme perfectly functions, Duality (Dvaita) ensues, through whom the Individual Being and Supreme Being were differentiated (Madhvacharya, XIII-th - XIV-th centuries). Thus: Non-Duality converged to Duality, i.e.
challenge for the designer is to transform the raw, imprecise, conflicting and often paradoxical information received from the different sources into statements understandable by both man and machines Hence, two levels of processing coexist in SA: measuring of the world and reasoning about the world. Another great challenge in SA is the reconciliation of both aspects. As a consequence, SA applications need frameworks general enough to take into account the different types of uncertainty observed in the SA context, coupled with a semantics allowing reasoning on those situations (Jousselme and Maupin, 2004) Neutrosophics as a Situation Analysis Tool contd. A particularity of SA is that most of the time it is impossible to list every possible situation that can occur. Corresponding frames of discernment cannot,
thus, be considered as exhaustive Furthermore, in SA situations are not clear-cut elements of the frames of discernment. Considering these particular aspects of SA, a neutrosophic logic paradigm incorporating the Dezert-Smarandache Theory (DSmT) appears as an appropriate modeling tool It has been recently shown that the neutrosophic logic paradigm does have the capacity to cope with the epistemic and uncertainty-related problems of SA In particular, it has been formally demonstrated that the neutrosophic logic paradigm incorporating DSmT has the ability to process symbolic and numerical statements on belief and knowledge using the possible worlds semantics (Jousselme and Maupin, 2004) Neutrosophics as a
Situation Analysis Tool contd. A Kripke Model MK = (S, , R) is a directed labeled graph. The graphs nodes are the possible worlds s belonging to a set S of possible worlds, labeled by truth assignments . A world s is considered possible with respect to another world s whenever there is an edge linking s and s. This link is defined by an arbitrary binary relation, technically called the accessibility relation R. A proposition is known if it is TRUE in all possible worlds of S. A proposition is believed if it is TRUE in at least one possible world s of S. A Neutrosophic Kripke Model, extends Kripke structure in order to take into account triplets of truth assignments (not only Boolean assignments). The concepts of knowledge and believe are represented with hyperreal values (truth, falsity, indeterminacy) assignments on possible worlds. NL() = (1+, 0, 0) if is known (i.e. true in all possible worlds - absolute truth), and NL() = (1, 0, 0) if is believed (i.e. true in at least one world]. While in a Kripke Model a proposition can only be TRUE or FALSE, in a
Neutrosophic Kripke Model is allowed to be T% TRUE and F% FALSE, and I% INDETERMINATE, where T, I, F are hyperreal subsets of ]-0, 1+[. Application to Robotics For the fusion of information received from various sensors, information that can be conflicting in a certain degree, the robot uses the fuzzy and neutrosophic logic or set. In a real time it is used a neutrosophic dynamic fusion, so an autonomous robot can take a decision at any moment. The Need for a Novel Decision Paradigm in Management The process of scientific decision-making necessarily follows an input- output system; The primary input is in the form of raw data (quantitative, qualitative or both);
This raw data is subsequently cleaned, filtered and organized to yield information; The available information is then processed according to either (a) very well-structured, hard rules or (b) partially-structured semi-soft rules or (c) almost completely unstructured soft rules; The output is the final decision which may be a relatively simple and routine one such as deciding on an optimal inventory re-ordering level or a much more complex and involved one such as discontinuing a product line or establishing a new startegic business unit (SBU). It has been observed that most of these complex and involved decision problems are those that need to be worked out using the soft rules of information processing; The Need for a Novel Decision Paradigm in Management contd.
Besides being largely subjective, soft decision rules are often ambiguous, inconsistent and even contradictory; The main reason is that the event spaces governing complex decision problems are not completely known. However, the human mind abhors incompleteness when it comes to complex cognitive processing. The mind invariably tries to fill in the blanks whenever it encounters incompleteness; Therefore, when different people form their own opinions from a given set of incomplete information, it is only to be expected that there will be areas of inconsistency, because everybody will try to complete the set in their own individual ways, governed by their own subjective utility preferences; The Need for a Novel Decision Paradigm in Management contd.
Looking at the following temporal trajectory of the market price of a share in ABC Corp. over the past thirty days, would it be considered advisable to invest in this asset? The hard decision rule applicable in this case is that one should buy an asset when its price is going up and one should sell an asset when its price is going down The share price as shown above, is definitely trending in a particular direction. But will the observed trend over the past thirty days continue in the future? It is really very hard to say because most financial analysts will find this information rather inadequate to arrive at an informed judgment; Although this illustration is purely anecdotal, it is nevertheless a matter of fact that the world of managerial decision-making is fraught with such inadequacies and complete information is often an unaffordable luxury .
The Need for a Novel Decision Paradigm in Management contd. The more statistically minded decision-takers would try to forecast the future direction of the price trend of a share in ABC Corp. from the given (historical) information The implied logic is that the more accurate this forecast the more profitable will be the outcome resulting from the decision Let us take two financial analysts Mr. X and Ms. Y trying to forecast the price of a share in ABC Corp. To fit their respective trendlines, Mr. X considers the entire thirty days of data while Ms. Y (who knows about Markovian property of stock prices)
considers only the price movement over a single day Mr. Xs forecast trend forecast trend Ms. Ys Who do you think is more likely to make the greater profit? (Please try answering the question before moving on to the next slide!) The Need for a Novel Decision Paradigm in Management contd. Most people will have formed their opinions after having made a spontaneous assumption about the orientation of the coordinate axes i.e. the temporal order of the price data! This is an example of how our minds sub-consciously complete an incomplete set of information prior to cognitive processing!
Obviously, without a definite knowledge about the orientation of the axes it is impossible to tell who is more likely to make a greater profit. This has nothing to do with which one of Mr. X or Ms. Y has the better forecasting model. In fact it is a somewhat paradoxical situation - we may know who among Mr. X and Ms. Y has a technically better forecasting model and yet dont know who will make more profit! That will remain indeterminate as long as the exact orientation of the two coordinate axes is unknown! The Need for a Novel Decision Paradigm in Management contd. The neutrosophic probability approach makes a distinction between relative sure event, event that is true only in certain world(s) and absolute sure event, event that is true for all possible world(s) Similar relations can be drawn for relative impossible event / absolute impossible event and relative indeterminate event / absolute indeterminate event
In case where the truth- and falsity-components are complimentary i.e. they sum up to unity and there is no indeterminacy, then it is reduced to classical probability. Therefore, neutrosophic probability may be viewed as a three-way generalization of classical and imprecise probabilities The Need for a Novel Decision Paradigm in Management contd. In our little anecdotal illustration, we may visualize a world where stock prices follow a Markovian path and Ms. Y knows the correct orientation of the coordinate axes. That Ms. Y will make a greater profit thereby becomes a relative sure event and that Mr. X will make a greater profit becomes a relative impossible event. Similarly we may visualize a different world where stock prices follow a linear path and Mr. X knows the correct orientation of the coordinate axes. That Mr. X will make a greater profit thereby becomes a relative sure
event and that Ms. Y will make a greater profit thereby becomes a relative impossible event. Then there is our present world where we have no knowledge at all as to the correct orientation of the coordinate axes and hence both thereby become relative indeterminate events! Because real-life managers have to mostly settle for incomplete sets of information, the arena of managerial decision-making is replete with such instances of paradoxes and inconsistencies. This is where neutrosophy can play a very significant role as a novel addition to the managerial decision paradigm! Application of Neutrosophics in Production Facility Layout Planning and Design The original CRAFT (Computerized Relative Allocation of Facilities Technique)
model for cost-optimal relative allocation of production facilities as well as many of its later extensions tend to be quite heavy in terms of CPU engagement time due to their heuristic nature; A Modified Assignment (MASS) model (first proposed by Bhattacharya and Khoshnevisan in 2003) increases the computational efficiency by developing the facility layout problem as primarily a Hungarian assignment problem but becomes indistinguishable from the earlier CRAFT-type models beyond the initial configuration; However, some amount of introspection will reveal that the production facilities layout problem is basically one of achieving best interconnectivity by optimal fusion of spatial information. In that sense, the problem may be better modeled in terms of mathematical information theory whereby the best layout is obtainable as the one that maximizes relative entropy (or equivalently, minimizes relative negentropy) of the spatial configuration; Application of Neutrosophics in Production Facility Layout Planning and Design
contd. Going a step further, one may hypothesize a neutrosophic dimension to the problem. Given a combination rule like the Dezert-Smarandache formula, the layout optimization problem may be formulated as a normalized basic probability assignment for optimally comparing between several alternative interconnectivities; The neutrosophic argument can be justified by considering the very practical possibility of conflicting bodies of evidence for the structure of the load matrix possibly due to conflicting assessments of two or more design engineers; If for example we consider two mutually conflicting bodies of evidence 1 and 2, characterized respectively by their basic probability assignments 1 and 2 and their cores k (1) and k (2) then one has to look for the optimal combination rule which maximizes the joint entropy of the two conflicting information sources; Mathematically, it boils down to the general optimization problem of
evaluating max [H ()] min [ H ()] subject to the constraints that (a) the marginal basic probability assignments 1 (.) and 2 (.) are obtainable by the summation over each column and summation over each row respectively of the relevant information matrix and (b) the sum of all cells of the information matrix is unity. Applications to Neutrosophic and Paradoxist Physics Neutrosophic Physics. Let be a physical entity (i.e. concept, notion, object, space, field, idea, law, property, state, attribute, theorem, theory, etc.),
objects, neutrosophic states, etc. Paradoxist Physics. Neutrosophic Physics is an extension of Paradoxist Physics, since Paradoxist Physics is a combination of physical contradictories and
coexist: anions in two spatial dimensions are arbitrary spin particles that are neither bosons (integer spin) nor fermions (half integer spin); among possible Dark Matter candidates there may be exotic particles that are neither Dirac nor Majorana fermions; mercury (Hg) is a state that is neither liquid nor solid under normal conditions at room temperature; non-magnetic materials are neither ferromagnetic nor anti-ferromagnetic; quark gluon plasma (QGP) is a phase formed by quasi-free quarks and gluons that behaves neither like a conventional plasma nor as an ordinary liquid; unmatter, which is formed by matter and antimatter that bind together (Smarandache, 2004);
Several Examples of Paradoxist and Neutrosophic Entities contd. neutral kaon, which is a pion & anti-pion composite (Santilli, 1978) and thus a form of unmatter; neutrosophic methods in General Relativity (Rabounski-SmarandacheBorissova, 2005); neutrosophic cosmological model (Rabounski-Borissova, 2011); neutrosophic gravitation (Rabounski); neutrino-photon doublet (Goldfain);
semiconductors are neither conductors nor isolators; semi-transparent optical components are neither opaque nor perfectly transparent to light; quantum states are metastable (neither perfectly stable, nor unstable); in Quantum Field Theory the observables (i.e. the physical characteristics that can be measured in the laboratory) are represented by operators. For example, the Hamiltonian of a quantum electric oscillator determines the energy and it can be expressed as a function of the of the operators of creation and annihilation of oscillation quanta; Several Examples of Paradoxist and Neutrosophic Entities contd. this idea of unparticle was first considered by F. Smarandache in 2004, 2005 and 2006, when he uploaded a paper on CERN web site and published three papers about what he called 'unmatter', which is a new form of matter formed by matter and antimatter that bind together. In 2006 E. Goldfain introduced the concept of "fractional number of field quanta" and
he conjectured that these exotic phases of matter may emerge in the near or deep ultraviolet sector of quantum field theory. H. Georgi proposed the theory of unparticle physics in 2007 that conjectures matter that cannot be explained in terms of particles using the Standard Model of particle physics, because its components are scale invariant. -- etc. qubit and generally quantum superposition of states; the multiplet of elementary particles is a kind of neutrosophic field with two or more values (Ervin Goldfain, 2011); a neutrosophic field can be generalized to that of operators whose action is selective. The effect of the neutrosophic field is somehow equivalent with the tunneling from the solid physics, or with the "spontaneous symmetry breaking" (SSB) where there is an internal symmetry which is broken by a particular selection of the vacuum state (Ervin Goldfain). More Applications Neutrosophy and Neutrosophic Logic/Set/Probability/Statistics are used in: Extenics (to resolve contradictory problems); Description Logic, Relational Data Model, Semantic Web Service Agent; Image Segmentation; Remedy for Effective Cure of Diseases using Combined Neutrosophic
Relational Maps; Neutrosophic Research Method; Transdisciplinarity, Multispace & Multistructure; Qualitative Causal Reasoning on Complex Systems; Study on suicide problem using combined overlap block Neutrosophic Cognitive Maps; Neutrosophic Topologies; Discrimination of outer membrane proteins using reformulated support vector machine based on neutrosophic set; More Applications contd. Decision support tool for knowledge based institution using neutrosophic cognitive maps; Imprecise query solving; Answering queries in Relational Database using Neutrosophic Logia; Ensemble Neural Networks Using Interval Neutrosophic Sets and
Bagging; Lithofacies Classification from Well Log Data using Neural Networks, Interval Neutrosophic Sets and Quantification of Uncertainty; Redesigning Decision Matrix Method with an indeterminacy-based inference process; Neural network ensembles using interval neutrosophic sets and bagging for mineral prospectivity prediction and quantification of uncertainty; Processing Uncertainty and Indeterminacy in Information Systems success mapping; Combination of Paradoxical, Uncertain, and Imprecise Sources of Information based on DSmT and Neutro-Fuzzy Inference;
Neutrosophic Cognitive Maps in context of knowledge-based organizations, etc. Download books, articles, PhD theses on NEUTROSOPHICS from: http://fs.gallup.unm.edu/neutrosophy.htm
Heading specialisation and achieving better quality? The ...
, planning to first establish a conferral system for granting professional lawyer licenses for seven categories: real estate law, family law, labor law, construction and engineering law, tax law, financial law, and intellectual property law.