Abstract
In this short paper, a critical analysis of the Neutrosophic, Pythagorean and some other novel fuzzy sets theories foundations is provided, taking into account that they actively used for the solution of the decision-making problems. The shortcomings of these theories are exposed. It is stated that the independence hypothesis, which is a cornerstone of the Neutrosophic sets theory, is not in line with common sense and therefore leads to the paradoxical results in the asymptotic limits of this theory. It is shown that the Pythagorean sets theory possesses questionable foundations, the sense of which cannot be explained reasonably. Moreover, this theory does not completely solve the declared problem. Similarly, important methodological problems of other analyzed theories are revealed. To solve the interior problems of the Atanassov’s intuitionistic fuzzy sets and to improve upon them, this being the reason most of the criticized novel sets theories were developed, an alternative approach based on extension of the intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory is proposed. No propositions concerned with the improvement of the Cubic sets theory and Single-Valued Neutrosophic Offset theory were made, as their applicability was shown to be very dubious. In order to stimulate discussion, many statements are deliberately formulated in a hardline form.
Highlights
We present here some basic definitions of the Dempster–Shafer theory of evidence (DST) [18,19] in the extent needed for our analysis
It is known that the probability and possibility theories are in some sense the asymptotic cases of the DST; in [15,22], we have shown that the DST is the useful generalization of the A − IFS, as well as of its intervalvalued extension; in [23], we have proved that the same is true in relation to the hesitant and interval-valued hesitant fuzzy sets; in [24], we have introduced the DST extension of the rule-base evidential reasoning in the intuitionistic fuzzy setting, which was used in [25]
It is shown that the cornerstone hypothesis of the Neutrosophic sets theory, i.e., the assumption of independent components does not conform to common sense and leads to the senseless paradox results in the asymptotic cases of this theory
Summary
During the last 25 years, a number of novel fuzzy sets theories such as Neutrosophic [1,2,3], Pythagorean [4,5], Spherical [6], Picture [7], Cubic [8], etc., with their numerous combinations and modifications were proposed and used mainly in the multiple criteria decision making and the group multiple criteria making but often without previous critical analysis. Articles on these topics were published in journals with relatively low citation indexes, e.g., such as Neutrosophic Sets and Systems or Journal of New Theory, but in recent years, they have increasingly appeared ( in very small numbers) in the reputed scientific journals of the first league such as Applied Soft Computing, KnowledgeBased Systems, Information Sciences, Expert systems with applications, Fuzzy Sets and System, Engineering Applications of Artificial Intelligence, Artificial Intelligence Review, etc In our opinion, such a caution of editors and reviewers of solid old journals is not caused by their conservatism at all, but by the desire to see, in addition to formal definitions of these theories and numerous theorems, the solution of real methodological and practical problems.
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