On a New View of a Fuzzy Set

With this paper the authors try to newly reflect on Zadeh's concept of a fuzzy set‎. ‎The departing point is the fact that not only fuzzy sets originate in Language‎, ‎but that they are just 'linguistic entities' genetically different from the concept of 'crisp sets' whose origin is either in a physical collection of objects‎, ‎or in a list of them‎.Thus‎, ‎a new definition of a fuzzy set is presented by means of two magnitudes‎: ‎A qualitative one‎, ‎a graph‎, ‎the basic magnitude‎, ‎and a quantitative one‎, ‎a scalar magnitude‎. ‎If the first reflects the language's relational ground of the fuzzy set‎, ‎the second‎ - ‎and thanks to 'measuring the meaning of words'‎ -, ‎reflects the (numerical) extensional state in which it currently appears‎.‎Since the second‎, ‎the scalar magnitude‎, ‎is essential for the applications‎, ‎it is also introduced the concept of a 'working fuzzy set' by taking into account the numerical function‎, ‎the meaning's measure or the membership function‎. ‎The working fuzzy set‎, ‎that enlarges the corresponding fuzzy set‎, ‎allows us to see the 'same fuzzy set' with different membership functions‎, ‎that is‎, ‎the same graph appearing in different extensional states‎. ‎Notice that a 'working crisp set' is but the same crisp set‎.