Foundations of Neutrosophic Logic and Setand their Applications in ScienceProf. Florentin Smarandache, Ph DThe University of New MexicoMath & Science DeptDept.705 Gurley Ave.Gallup,p, NM 87301,, USAfs.galllup.unm.edu/neutrosophy.htm1
ContentTHEORY Definition of Neutrosophy A Short Historyy of the Logicsg 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 FuzzyLogic Neutrosophic Logic generalizes many Logics Neutrosophic Logic Connectors Neutrosophiceut osop c Set ((NS)S) Neutrosophic Cube as Geometric Interpretation of theNeutrosophic Set Neutrosophic Set Operators DifferencesDiffbetweenbNNeutrosophichi SetS andd IntuitionisticIi i i i FuzzyFSetS Partial Order in Neutrosophics N-Norm and N-conorm2
Content (2) Interval Neutrosophic OperatorsRemarks on Neutrosophic OperatorsExamples of Neutrosophic Operators resulted from N-norms andN-conormsAPPLICATIONS Application of FuzzyFLogic to Information FusionF sion 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 Probability and Statistics Applications of Neutrosophy to Extenics and Indian Philosophy3
Content (3) Neutrosophics as a situation analysis toolApplication to RoboticsThe Need for a Novel Decision Paradigm in Management (F. S. & S.Bhattacharya)Application of Neutrosophics in Production Facility Layout Planningand Design (F. S. & S. Bhattacharya)Applicationsppto Neutrosophicpand Paradoxist PhysicsyMore Applications4
Definition of Neutrosophy--A new branch of philosophy which studies the origin, nature, and scopeof 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,Etymologically,neuter neutral,neutral andGreek sophia, skill/wisdom] means knowledge of neutral thought andstarted in 1995.Extension of dialectics.Connected with Extenics (Prof. Cai Wen, 1983), and Paradoxism (F.Smarandache, 1980)The Fundamental Theory: Every idea A tends to be neutralized,diminished,ds ed, babalanceda ced by nonA oideasdeas ((notot oonlyy antiA a tas Hegelege asseasserted)ted)- as a state of equilibrium. nonA what is not A , antiA the opposite of A , and neutA what is neither A nor antiA In a classical way A , neutA , antiA are disjoint two by two.But, since in many cases the borders between notions are vague, imprecise,Sorites, it is possible that A , neutA , antiA (and nonA of course)have common parts two by two as wellwell.Basement for Neutrosophic Logic, Neutrosophic Set, Neutrosophic5Probability, and Neutrosophic Statistics
A Short Historyy of the LogicsgThe fuzzyy set ((FS)) was introduced byy L. Zadehin 1965, where each element had a degree ofmembership.Th intuitionisticThei t iti i ti fuzzyfsett (IFS) on a universeiXwas introduced by K. Atanassov in 1983 as ageneralization of FS,FS where besides the degreeof membership μA(x) ϵ [0,1] of each elementx to a set A there was considered a degree ofnon-membershipb hi νA(x)ϵ[0,1],( ) [0 1] butb t suchh ththattfor x ϵX, μA(x) νA(x) 1.6
A Short History of the Logics (2)According to Cornelis et al. (2003), Gehrke et al. (1996) stated that“Many people believe that assigning an exact number to an expert’sopinion is too restrictive, and the assignment of an interval of valuesis more realistic”, which is somehow similar with the impreciseprobabilityb bilit theorythwherehiinsteadt d off a crispi probabilityb bilit one hhas aninterval (upper and lower) probabilities as in Walley (1991).Atanassov (1999) defined the interval-valued intuitionistic fuzzy set(IVIFS) on a universeiX as an objectbj t A suchh that:th tA {(x, MA(X), NA(x)), xϵX},with MA:X Int([0,1]) and NA:X Int([0,1])and x ϵ X, supMA(x) supNA(x) 1 .7
A Short Historyy of the Logicsg((3)) Belnapp ((1977)) defined a four-valued logic,g withtruth (T), false (F), unknown (U), and contradiction(C). He used a bi-lattice where the fourcomponentspwere inter-related. In 1995, starting from philosophy (when I fretted todistinguish between absolute truth and relativetruth or between absolute falsehood and relativefalsehood in logics, and respectively betweenabsolute membership and relative membership orabsolute non-membershipnon membership and relative nonnonmembership in set theory) I began to use the nonstandard analysis.8
A Short History of the Logics (4) Also, inspired from the sport games (winning,d f idefeating,or tiei scores),) ffrom votes ((pro, contra,null/black votes), from positive/negative/zero numbers,from yes/no/NA, from decision making and controltheory (making a decisiondecision, not makingmaking, or hesitating)hesitating),from accepted/rejected/ pending, etc. and guided bythe fact that the law of excluded middle did not workany longer in the modern logicslogics, I combined the nonstandard analysis with a tri-componentlogic/set/probability theory and with philosophy (I wasexcited by paradoxism in science and arts and letters,as well as by paraconsistency and incomplete-ness inknowledge). How to deal with all of them at once, is itpossible to unitypy them?9
A Short History of the Logics (5) Ipproposedpthe term "neutrosophic"pbecause"neutrosophic" etymologically comes from"neutrosophy" [French neutre Latin neuter,neutral, and Greek sophia,pskill/wisdom]] whichmeans knowledge of neutral thought, and thisthird/neutral represents the main distinctionbetween "fuzzy"y and "intuitionistic fuzzy"y logic/set,g,i.e. the included middle component (LupascoNicolescu’s logic in philosophy), i.e. thepart ((besides theneutral/indeterminate/unknown p"truth"/"membership" and "falsehood"/"nonmembership" components that both appear ing)fuzzyy logic/set).10
Introduction to NonstandardA l iAnalysis AbrahamRobinson developedp the nonstandard analysisy(1960s) x is called infinitesimal if x 1/n for any positive n A left monad (-a) {a{a-x:x: x in RR*, x 0 infinitesimal} aa-εεand a right monad (b ) {a x: x in R*, x 0 infinitesimal} b εwhere ε 0 is infinitesimal;a, b called standard parts, ε 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,addition subtractionsubtraction, multiplication,multiplication divisiondivision,power.11
Operations with Classical SetsS1 and S2 two real standard or nonstandard sets. Addition: Subtraction: Multiplication:M ltiplication Division of a set by a non-null number:12
Neutrosophic Logic Consider the nonstandard unit interval ]-0, 1 [, with left and rightborders vague, impreciseT I,I F be standard or nonstandard subsets of ]-0,0 1 [ Let T, Neutrosophic Logic (NL) is a logic in which each proposition isT% 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 3030-40%40% or 4545-50%50% truetrue,20% indeterminate, and 60% or between 66-70% false(according to various analyzers or parameters) NL is a generalization of ZadehZadeh’ss fuzzy logic (FL), andespecially of Atanassov’s intuitionistic fuzzy logic (IFL), and ofother logics13
Refined Neutrosophic Logic and Set Component “I”, indeterminacy, can be split into moresubcomponents in order to better catch the vague information wework with, and such, for example, one can get more accurateanswers to the Question-Answering Systems initiated by Zadeh(2003).{In Belnap’sBelnap s fourfour-valuedvalued logic (1977) indeterminacy was split intoUncertainty (U) and Contradiction (C), but they were inter-related.} Even more, we proposed to split "I" into Contradiction,Uncertainty and Unknown,Uncertainty,Unknown and we get a fivefive-valuedvaluedlogic. In a ggeneral Refined NeutrosophicpSet, "T" can be splitpinto subcomponents T1, T2, ., Tm, and "I" into I1, I2, ., In,and "F" into F1, F2, ., Fp because there are more typesof truths,truths of indeterminacies,indeterminacies and respectively offalsehoods.14
Classical Mass & NeutrosophicMassLet Ω be a frame of discernment, defined as:Ω {θ1, θ2, , θn}, n 2,and its Super-Power Set (or fusion space):S Ω ( Ω, U, , C )which means: the set Ω closed under unionunion, intersection,intersection andrespectively complement.Classical MassMass.We recall that a classical mass m(.) is defined as:m: S Ω - [0,1]such that m(X) 1.X in S Ω15
Classical Mass & NeutrosophicMMass(2)We extend the classical basic belief assignment (or classical mass) bbam(.) to a neutrosophic basic belief assignment (nbba) (orneutrosophic mass) mn( .) in the following way.mn : S Ω - [0,1] 3withmn(A) (T(A), I(A), F(A))where T(A) means the (local) chance that hypothesis A occurs,occurs F(A)means the (local) chance that hypothesis A does not occur(nonchance), while I(A) means the (local) indeterminate chance of A((i.e. knowingg neither if A occurs nor if A doesn’t occur),),such that: [T(X) I(X) F(X)] 1.X in S Ω16
Classical Mass & NeutrosophicMMass(3)In a more general way, the summation can be less than 1 (forincomplete neutrosophic information), equal to 1 (for completeneutrosophic information), or greater than 1 (forparaconsistent/conflicting neutrosophic information). But in thispaper we onlyl presentt ththe case whenh summationti iis equall tto 1.1Of course1 T(X), I(X), F(X) 1for all X in S Ω.17
Differences between NeutrosophicLogic and Intuitionistic Fuzzy Logic In NL there is no restriction on T, I, F, while in IFL the sum ofcomponents (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 nonstandardnonstandard, in IFL they don’tdon t. NL, like dialetheism [some contradictions are true], can dealwith paradoxes, NL(paradox) (1,I,1), while IFL cannot.18
Neutrosophic Logic generalizesmany LogicsLet the components reduced to scalar numbers, t,i,f, with t i f n;NL generalizes:g- 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 manyvalues between true and false [Lukasiewicz, 3 values; Post, mvalues] (for n 1, i 0, 0 t, f 1); - the intuitionistic logic, which supportsincomplete theories, whereA\/nonA ((Law of Excluded Middle)) not alwaysy true,, and “Thereexist 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 anddegrees of falsity while what’s left is considered indeterminacy19[Atanassov, 1982] (for n 1);
Neutrosophic Logic generalizesmany Logics (cont(cont’d)d)- the paraconsistent logic, which supports conflicting information,and ‘anythingyg follows from contradictions’ fails,, i.e. A/\nonA- Bfails; 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 bedenoted this way too);- the faillibilism,, which saysy that uncertaintyy belongsg to everyyproposition (for i 0);20
Neutrosophic Logic ConnectorsA1(T1, I1, F1) andd A2(T2, I2, F2) are ttwo propositions.iti21
Neutrosophic Logic Connectors(cont d)(cont’d)Many properties of the classical logic operators do not applyin neutrosophic logic.Neutrosophic logic operators (connectors) can be defined inmany ways accordingdi tot theth needsd off applicationsli tior off ththeproblem solving.22
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 theneutrosophicthi sett M as x(T,(T I,I F) andd belongsb lt M intoi thethfollowing way:it is t% true in the set (degree of membership),i% indeterminate (unknown if it is in the set) (degree ofindeterminacy),and f% false (degree of non-membership)non membership),where t varies in T, i varies in I, f varies in F.- Definition analogue to NL- GeneralizesGliththe ffuzzy sett (FS),(FS) especiallyi ll ththe iintuitionistict iti i tifuzzy set (IFS), intuitionistic set (IS), paraconsistent set (PS)- Example: x(50,20,40) in A means: with a believe of 50% xis in A, with a believe of 40% x is not in A (disbelieve), and23the 20% is undecidable
Neutrosophic Cube as geometrici tinterpretationt ti off theth NeutrosophicN thi SetS t The most importantpdistinction between IFS andNS is showed in the below Neutrosophic CubeA’B’C’D’E’F’G’H’ introduced by J. Dezert in 2002. BecauseBini ttechnicalh i l applicationsli tionlyl ththeclassical interval is used as range for thepparameters , we call the cube thepneutrosophictechnical neutrosophic cube and its extensionthe neutrosophic cube (or absoluteneutrosophic cube),cube) used in the fields where weneed to differentiate between absolute and relative(as in philosophy) notions.24
Neutrosophic Cube as geometricinterpretation of the NeutrosophicSet (2)25
Neutrosophic Cube as geometricinterpretation of the NeutrosophicSet (3)Let’s consider a 3D-Cartesian system of coordinates,where t is the truth axis with value range in ]-0,1 [, i isthe false axis with value range in ]-0,1 [, and similarly fis the indeterminate axis with value range in ]-0,1 [.We now divide the technical neutrosophic cubeABCDEFGH into three disjointjregions:g1) The equilateral triangle BDE, whose sides are equal toV,2 which represents the geometrical locus of the pointswhose sum of the coordinates is 1.If a point Q is situated on the sides of the triangle BDEor inside of it, then tQ iQ fQ 1 as in Atanassovintuitionistic fuzzy set (A(A-IFS)IFS).26
Neutrosophic Cube as geometricinterpretation of the NeutrosophicSet (4) 2) The pyramid EABD {situated in the right sideof the triangle EBD, including its faces triangleABD(base), triangle EBA, and triangle EDA(lateral faces), but excluding its face: triangleBDE } is the locus of the points whose sum ofcoordinatesdi t iis lless ththan 11. 3) In the left side of triangle BDE in the cubethere is the solid EFGCDEBD ( excludingtriangle BDE) which is the locus of points whosesum of their coordinates is greater than 1 as inthe paraconsistent set.27
Neutrosophic Cube as geometricinterpretation of the NeutrosophicSet (5) It is possible to get the sum of coordinates strictlyless than 1 or strictly greater than 1. For example: We have a source which is capable to find only thedegree of membership of an element; but it is unableto find the degree of non-membership; Another source which is capable to find only thedegree of non-membership of an element; Or a source which only computes the indeterminacy. Thus,Thus when we put the results together of thesesources, it is possible that their sum is not 1, butsmaller or greater.28
Neutrosophic Cube as geometricinterpretation of the NeutrosophicSet (6) Also, in information fusion, when dealing withindeterminate models (i.e. elements of the fusionspace which are indeterminate/unknown, such asintersections we don’t know if they are empty ornot since we don’t have enough information,similarly for complements of indeterminateelements, etc.): if we compute the believe in thatelement (truth), the disbelieve in that element(falsehood) and the indeterminacy part of that(falsehood),element, then the sum of these three componentsis strictly less than 1 (the difference to 1 is themissing information)information).29
Neutrosophic Set OperatorsA 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 andrespectively to B]. NS operators (similar to NLconnectors) can also be defined in many ways.30
Differences between NeutrosophicSet and Intuitionistic Fuzzy Set In NS there is no restriction on T, I, F, while in IFS the sum ofcomponentsp((or their superiorplimits)) 1;;thus NL can characterize the incomplete information (sum 1),paraconsistent information (sum 1). NS can distinguish, in philosophy, between absolutemembershipb hi [NS(absolute[NS( b l t membership) 1b hi ) 1 ] andd relativel timembership [NS(relativemembership) 1], while IFS cannot;absolute membership is membership in all possible worlds,relative membership is membership in at least one worldworld. In NS the components can be nonstandard, in IFS they don’t. NS, like dialetheism [some contradictions are true], can dealwith pparadoxes,, NS(paradox(pelement)) ((1,I,1),, , ), while IFScannot. NS operators can be defined with respect to T,I,F while IFSoperators are defined with respect to T and F only I can be split in NS in more subcomponents (for example inBelnap’s four-valued logic (1977) indeterminacy is split into31uncertainty and contradiction), but in IFS it cannot
Partial Order in NeutrosophicsWe define a partial order relationship on theneutrosophic set/logic in the following way:x(T( 1, I1, F1) y(y(T2, I2, F2) iff ((if and onlyy if))T1 T2, I1 I2, F1 F2 for crisp components.And in general,And,general for subunitary set components:x(T1, I1, F1) y(T2, I2, F2) iffinf T1 inf T2, sup T1 sup T2,inf I1 inf I2, sup I1 sup I2,inf F1 inf F2, sup F1 sup F2.32
Partial Order in Neutrosophicsp(2)( ) If we have mixed - crisp and subunitary components, or only crisp components, wecan transform any crisp componentcomponent, say“a” with a in [0,1] or a in ]-0, 1 [, into asubunitary set [a[a, a]a]. So,So the definitions forsubunitary set components should work inany case.case33
N-normNnorm and NN-ConormConormAs a generalization of T-normT norm and T-conormT conorm from theFuzzy Logic and Set, we now introduce the N-normsand N-conorms for the Neutrosophic Logic and Set.N-norm, [ ]-0,1, [ ]-0,1, [ )2 ]-0,1, [ ]-0,1, [ ]-0,1, [Nn: ( ]-0,1Nn (x(T1,I1,F1), y(T2,I2,F2)) (NnT(x,y), NnI(x,y), NnF(x,y)),e e NnT(.,.),( , ), NnI(.,.),( , ), NnF(.,.)( , ) area e tthee ttruth/membership,ut / e be s p,whereindeterminacy, and respectivelyfalsehood/nonmembership components.34
N-normNnorm (2)Nn have to satisfy, for any x, y, z in theneutrosophic logic/set M of the universe ofdiscourse U, the following axioms:a) Boundary Conditions: Nn(x, 0) 0, Nn(x, 1) x.y Nn((x,, y) Nn(y, x).)b)) Commutativity:c) Monotonicity: If x y, then Nn(x, z) Nn(y, z).d) Associativity:ssoc at ty Nn((Nn ((x,, y), z)) Nn((x,, Nn(y, z)).))35
N-normNnorm (3) There are cases when not all these axioms aresatisfied, for example the associativity whendealing with the neutrosophic normalization aftereach neutrosophic operation. But, since we workwith approximations, we can call these Npseudo-norms,dwhichhi h stilltill givei goodd resultslt iinpractice. Nn represent the and operator in neutrosophiclogic, and respectively the intersection operatorin neutrosophic set theorytheory.36
N-normNnorm (4)Let J in {T, I, F} be a component.Most known N-norms, as
Content THEORY Definition of Neutrosophy A Short Historyyg 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
Laws of Classical Logic That Do Not Hold in The Interval Neutrosophic Logic 184 Modal Contexts 186 Neutrosophic Score Function 186 Applications. 187 Neutrosophic Lattices 188 Conclusion 191 CHAPTER XII Neutrosophic Predicate Logic 196 Neutrosophic Quantifiers 199 Neutrosophic Existential Quantifier. 199 Neutrosophic Universal Quantifier. 199
Neutrosophic Modal Logic Florentin Smarandache University of New Mexico, Mathematics & Science Department, 705 Gurley Ave., Gallup, NM 87301, USA. . modalities. It is an extension of neutrosophic predicate logic and of neutrosophic propositional logic. Applications of neutrosophic modal logic are to neutrosophic modal metaphysics. Similarly .
integer neutrosophic complex numbers, rational neutrosophic complex numbers and real neutrosophic complex numbers and derive interesting properties related with them. Throughout this chapter Z denotes the set of integers, Q the rationals and R the reals. I denotes the indeterminacy and I 2 I. Further i is the complex number and i 2 -1 or .
Hamacher weighted averaging operator and the neutrosophic cubic Hamacher weighted geometric operator are developed to aggregate neutrosophic cubic values. Some desirable properties of these operators such as idempotency, monotonicity and boundedness are discussed. To deal with the multi-cr
Neutrosophic Sets and Systems, Vol. 48, 2022 University of New Mexico Sivaranjini J,Mahalakshmi V ,Neutrosophic Fuzzy Strong bi-idealsof Near-Subtraction Semigroups . fuzzy subnearring, fuzzy ideal and fuzzy R-subgroups. Atanassov[3] expanded the intuitionstic fuzzy set to deal with complicated version.It explained the truth and false .
Neutrosophic General Finite Automata J. Kavikumar1, D. Nagarajan2, Said Broumi3;, F . view of commutative and switching automata. In this research, the idea of a neutrosophic is incorporated in the general . In 2005, the theory of general fuzzy automata was firstly proposed by Doostfatemeh and Kermer [11] which is used to resolve the .
Analysis of and strategic plan to increase the sales level of the Company "TIENS" of Babahoyo using neutrosophic methods. 185 The Single-Valued Neutrosophic number (SVNN) is symbolized by N (t, i, f ), such that 0 t, i, f 1 and 0 t i f 3. Definition 3: ([18-19]) The single-valued trapezoidal neutrosophic number,
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