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Neutrosophy (English) | Logic | Fuzzy Logic

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[[de: Neutrosophie]] '''Neutrosophy''' is a theory developed by [[Florentin Smarandache]] as a generalization of [[dialectics]]. This theory considers every notion or idea together with its opposite or negation and the spectrum of neutralities (i.e. notions or ideas located between the two extremes, supporting neither nor ). The and ideas together are referred to as . The theory claims that every idea tends to be neutralized and ba
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  [[de: Neutrosophie]]'''Neutrosophy''' is a theory developed by [[Florentin Smarandache]] as a generalization of [[dialectics]]. This theory considers every notion or idea <A> together with its opposite or negation <Anti-A> and the spectrum of neutralities <Neut-A> (i.e. notions or ideaslocated between the two extremes, supporting neither <A> nor <Anti-A>). The <Neut-A>and <Anti-A> ideas together are referred to as <Non-A>. The theory claims that every idea<A> tends to be neutralized and balanced by <Anti-A> and <Non-A> ideas - as a state of equilibrium.== Origins == Neutrosophy, neutrosophic logic, neutrosophic sets, etc., were invented in the [[1980s]] bySmarandache, after he [[neologism|coined]] the word from the [[Latin]] ''neuter'' and the[[Greek language|Greek]] ''sophia'', to mean knowledge of neutral thought .Smarandache promotes neutrosophy heavily. He organized the First InternationalConference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics in [[2001]]and published the conference's proceedings. One book about neutrosophy was published by''American Research Press'', a small publisher closely aligned with Smarandache.Additionally, some articles by Smarandache and [[Jean Dezert]] were included in the''Journal of Multiple-Valued Logic'', Volume 8, Number 3, issue dedicated to neutrosophyand neutrosophic logic, and Numbers 5-6. This journal is now known as the ''Journal of Multiple-Valued Logic and Soft Computing''. Other journals that published on neutrosophicsare ''International Journal of Social Economics'' (University of California at Fresno),''Libertas Mathematica'' (University of Texas at Arlington), ''Proceedings of the SecondSymposium / Romanian Academy of Scientists, American Branch'' (City University of NewYork), ''Bulletin of the Transilvania University of Brasov'' (Romania), ''Abstracts of papers presented to the International Congress of Mathematicians'' (Beijing, China) and ''Abstractsof papers presented to the meetings of the American Mathematical Society'' (University of California at Santa Barbara meeting).== Extensions ==Smarandache extended neutrosophy to neutrosophic logic (or Smarandache logic),neutrosophic sets, and so forth.In [[bivalent logic]], the truth value of a [[proposition]] is given by either one (true), or zero(false). Neutrosophic logic is a [[multi-valued logic]], in which the truth values are given byan amount of truth, an amount of falsehood, and an amount of indeterminacy. Each of thesevalues is between 0 and 1. In addition, the values may vary over time, space, hidden parameters, etc. Further, these values can be ranges.In the neutrosophic logic every logical variable x is described by an ordered triple x = (T, I,F) where T is the degree of truth, F is the degree of false and I the level of indeterminacy.  (A) To maintain consistency with the classical and fuzzy logics and with probability there isthe special case where T + I + F = 1.(B) But to refer to intuitionistic logic, which means incomplete information on a variable proposition or event one has T + I + F < 1.(C) Analogically referring to Paraconsistent logic, which means contradictory sources of information about a same logical variable, proposition or event one has T + I + F > 1.Thus the advantage of using Neutrosophic logic is that this logic distinguishes in philosophy between relative truth that is a truth in one or a few worlds only noted by 1 and absolutetruth denoted by 1+. Likewise neutrosophic logic distinguishes between relative falsehood,noted by 0 and absolute falsehood noted by -0 in non-standard analysis.For example, a neutrosophic answer to the question Is the pope a Catholic? might be 80-90% true, 36-42% false, and 2-7% indeterminate . Note that these values need not sum to100%. Smarandache claims that it can serve as a generalization of many other logics, suchas: [[fuzzy logic]], [[intuitionistic logic]], [[paraconsistent logic]], [[boolean logic]], etc.In neutrosophic set theory, propositions of the form x is an element of S are answered interms of neutrosophic truth values. Hence, each element has a membership-degree, anindeterminacy-degree, and a non-membership degree. These are claimed to generalise[[paraconsistent set]]s and [[intuitionistic set]]s, amongst others.== Applications ==As examples of application of neutrosophy in information fusion in finance there are some papers by Dr. M. Khoshnevisan, Dr. S. Bhattacharya and Dr. F. Smarandache, where thefuzzy theory doesn't work because fuzzy theory has only two components, truth andfalsehood, while the neutrosophy has three components: truth, falsehood, and indeterminacy(or <A>, <Anti-A>, and <Neut-A>), papers about investments which are: Conservative andsecurity-oriented (risk shy), Chance-oriented and progressive (risk happy), or Growth-oriented and dynamic (risk neutral). See the paper Fuzzy and Neutrosophic Systems andTime Allocation of Money , pp. 5-23, in their book Artificial Intelligence and ResponsiveOptimization at www.gallup.unm.edu/~smarandache/ArtificialIntelligence-book2.pdf.Proponents of neutrosophy claim that in any field where there is indeterminacy, unknown,hidden parameters, imprecision, [[sorites]] paradoxes, high conflict between sources of information, non-exhaustive or non-exclusive elements of the frame of discernment, etc.,then neutrosophy could in theory be applied.More applications of neutrosophics:<br>Fuzzy Cognitive Maps (FCMs) are fuzzy structures that strongly resemble neural networks,and they have powerful and far-reaching consequences as a mathematical tool for modelingcomplex systems. Neutrosophic Cognitive Maps (NCMs) are generalizations of FCMs, and  their unique feature is the ability to handle indeterminacy in relations between two conceptsthereby bringing greater sensitivity into the results. <br>Some of the varied applications of FCMs and NCMs include: modeling of supervisorysystems; design of hybrid models for complex systems; mobile robots and in intimatetechnology such as office plants; analysis of business performance assessment; formalismdebate and legal rules; creating metabolic and regulatory network models; traffic andtransportation problems; medical diagnostics; simulation of strategic planning process inintelligent systems; specific language impairment; web-mining inference application; childlabor problem; industrial relations: between employer and employee, maximizing production and profit; decision support in intelligent intrusion detection system; hyper-knowledge representation in strategy formation; female infanticide; depression in terminallyill patients and finally, in the theory of community mobilization and women empowermentrelative to the AIDS epidemic.See Dr. W. B. Vasantha Kandasamy from Indian Institute of Technology in Madras and Dr.F. Smarandache's book ''Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps'' athttp://www.gallup.unm.edu/~smarandache/NCMs.pdf.The concept of only fuzzy cognitive maps are dealt which mainly deals with the relation /non-relation between two nodes or concepts but it fails to deal the relation between twoconceptual nodes when the relation is an indeterminate one. Neutrosophic logic is the toolknown to us, which deals with the notions of indeterminacy. Suppose in a legal issue the jury or the judge cannot always prove the evidence in a case, in several places we may not be able to derive any conclusions from the existing facts because of which we cannot makea conclusion that no relation exists or otherwise. But existing relation is an indeterminate. Soin the case when the concept of indeterminacy exists the judgment ought to be very carefullyanalyzed be it a civil case or a criminal case. FCMs are deployed only where the existenceor non-existence is dealt with but however in the Neutrosophic Cognitive Maps we will dealwith the notion of indeterminacy of the evidence also. Thus legal side has lot of  Neutrosophic (NCM) applications. Also, NCMs can be used to study factors as varied asstock markets, medical diagnosis, etc.==Some Simple Advantages Of Reasoning In Intuitionistic Neutrosophic Logic==The traditional form of reasoning in logic and automated reasoning is severely limited inthat it cannot be used to represent many circumstances. In this paper, we demonstrate twosimple examples of the superiority of intuitionistic neutrosophic logic in representing thedata of the real world.===The Definition of Intuitionistic Neutrosophic Logic===Intuitionistic neutrosophic logic is an extension of fuzzy logic, where the elements areassigned a four-tuple (t, i, f, u) representation of their truth value. t is the value of truth, i thevalue of indeterminacy, f the value of false and u is the degree to which the circumstancesare unknown. The sum of the four terms is 1.0 and all are greater than or equal to zero,which maintains consistency with the classical and fuzzy logics. The logical connectives of   and (/\), or (\/) and not () can be defined in several ways, but here we will use the definitionsused by Ashbacher to define INL2[1].Definition 1:(t1, i1, f1, u1) = (f1, i1, t1, u1)(t1, i1, f1, u1) /\ (t2, i2, f2, u2) = ( t = min{t1 ,t2 }, i = 1 &#8211; t &#8211; f &#8211; u, f = max{f1 ,f2 }, u = min{ u1 ,u2 } )(t1, i1, f1, u1) \/ (t2, i2, f2, u2) = (t = max{t1 ,t2 }, i = 1 &#8211; t &#8211; f &#8211; u, f = min{f1 ,f2 }, u = min{ u1 ,u2 } )It is easy to verify that the elements of INL2 are closed with respect to these definitions of the basic logical connectives. Furthermore, many of the algebraic properties such as theassociative and commutative laws also hold for these definitions.===An Example of Clauses In Automated Reasoning===In automated reasoning, facts are defined by stating instances of a predicate. For example, inWos[2], the clause:FEMALE(Kim)is used to represent that Kim is a female. A set of clauses is then developed which stores theknowledge of all persons who are female. Clauses such as:MALE(John)are used to represent that John is a male. A query to the database of facts will have a formsimilar to:FEMALE(Kim)?which is asking the question, &#8220;Is Kim female?&#8221; In standard reasoning, theresponse would be a yes or a true if the database of facts contains a clause of the form:FEMALE(Kim)or there is a line of reasoning that leads to the conclusion that Kim is female.In the case where there is no such fact or line of reasoning, the response would be no or false. Therefore, a negative response could be a no that was inferred from the data or a casewhere Kim does not appear in the database of females. The difference between these twoconditions is substantial and the INL2 allows for them to be distinguished. If any form of knowledge can be inferred about the query, the value returned would be computed from thevalues. In the case where there is no information about the clause, the value returned by the
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