Multiple Valued Logic - Neutrosophic Logic
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To add at dialectic ===Neutrosophy=== [[Florentin Smarandache]] extended the dialectic to [[neutrosophy]] which, roughly speaking, is an exchange of ''propositions'' , ''counter-propositions'' , and (in addition to dialectic) ''neutralities'' (i.e. un-biased propositions, that support neither nor ), resulting in a synthesis of these three assertations (, , and ) or in a qualitative transformation of the dialogue. It is necessary to include these ''neutralities'' because they influence the dialogue. Neutrosophy is a base for [[neutrosophic logic]], [[neutrosophic set]], and [[neutrosophic probability]]. [[Neutrosophic logic]] introduces into fuzzy logic, besides truth and falsehood values, the indeterminacy value which stands for hidden variables; all three components, truth, falsehood, and indeterminacy are standard or non-standard subsets in the non-standard unit interval ]-0, 1+[. In neutrosophic logic a proposition can be, say, between 20-30% true, between 40-50% false, and between 15-25% indeterminant; because the sources that provide information about this proposition may be contradictory or incomplete (they don't know much about the proposition), the sum of superior limits of the components could be greater than 100% (overlapping), as in [[paraconsistent logic]], or less than 100%, as in [[intuitionistic logic]]. One can apply these systems of logic in other areas of math such as [[fuzzy set]] or [[neutrosophic set]] theories. '''Fuzzy sets''' are an extension of the classical [[set theory]]. A fuzzy set is characterized by a ''membership-degree function'', which maps the members of the universe into the real continuous interval [0;1]. The value 0 means that the member is not included in the given set, 1 describes a fully included member (this behaviour corresponds to the classical sets). The values between 0 and 1 characterize fuzzy members. For the universe ''X'' and given the membership-degree function ''f'' : ''X''→[0;1], the fuzzy set ''A'' is defined as ''A'' = {(''x'', ''f''(''x'')) | x ∈ ''X''}. A generalization of the fuzy set is the [[neutrosophic set]] applied in information fusion. Besides the membership-degree and non-membership-degree used in fuzzy set theory, one also considers the indeterminacy-degree. A neutrosophic set is characterized by a ''membership-degree, indeterminacy-degree, and non-membership-degree function'' and it is an extension of the [[fuzzy set]]. The indeterminacy-degree, a new component introduced here besides the two ones of the fuzzy set, leaves room for unknown parameters that may occur in the calculation of the membership-degree or non-membership-degree.
Using non-standard analysis, as in [[neutrosophic logic]], these three components, ''membership-degree'', ''non-membership-degree'', and ''indeterminacy-degree'' are standard or non-standard subsets in the non-standard unit interval ]-0, 1+[. For example, an element x from the universe belongs to a neutrosophic set in the following way: its memebership-degree is, say, between 40-50%, its non-membership-degree is between 3642%, and its indeterminacy-degree between 2-7%. And, similarly, because the sources that provide information about this proposition may be contradictory or incomplete (they don't know much about the proposition), the sum of superior limits of the components could be greater than 100% (overlapping), as in [[paraconsistent set]], or less than 100%, as in [[intuitionistic set]].
Using non-standard analysis, as in [[neutrosophic logic]], these three components, ''membership-degree'', ''non-membership-degree'', and ''indeterminacy-degree'' are standard or non-standard subsets in the non-standard unit interval ]-0, 1+[. For example, an element x from the universe belongs to a neutrosophic set in the following way: its memebership-degree is, say, between 40-50%, its non-membership-degree is between 3642%, and its indeterminacy-degree between 2-7%. And, similarly, because the sources that provide information about this proposition may be contradictory or incomplete (they don't know much about the proposition), the sum of superior limits of the components could be greater than 100% (overlapping), as in [[paraconsistent set]], or less than 100%, as in [[intuitionistic set]].
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