On a natural foundation of fuzzy sets

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  • ファジィ集合の自然な基礎づけについて
  • ファジィ シュウゴウ ノ シゼン ナ キソズケ ニ ツイテ

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Abstract

In these 40 years there has been much development on fuzzy set theory especially in the application area, but on basic theory there has not been a uniform and consistent theory which cover most part of fuzzy set theory. It is partly because that fuzzy sets were not defined exlicitly in the pioneering paper of Zadeh. But the primary intention of Zadeh is considered to distinguish the notion of fuzzy set from its membership function and to develop a theory of fuzzy sets as an extention of usual set theory. Now our natural interpretation is to give a satisfactory explanation by giving a model of fuzzy set theory on which ordinary set theory can be naturally extended. Here we present the basic part of our natural interpretation of fuzzy sets. We interpret fuzzy sets in a cumulative Heyting valued model for intuitionistic set thoery. With the interpretation various notions and properties in fuzzy set theory can be acquired consistently. As far as fuzzy set theory is considered as an extension of usual set theory, this interpretation seems to be most natural.

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