On a New View of a Fuzzy Set
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Enric Trillas
1
,
Adolfo De Soto
2
1 - Department of Mechanical, Informatics and Aeroespacial Engineering, Visiting Professor, University of Le´on, Le´on, Spain.
2 - Department of Mechanical, Informatics and Aeroespacial Engineering, University of Le´on, Le´on, Spain.
کلید واژه: Fuzzy Set Theory, Fuzzy Logic, Meaning,
چکیده مقاله :
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.
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.
[1] A. R. de Soto, E. Trillas and S. Cubillo, Classes of fuzzy sets with the same material conditional, International Journal of Approximate Reasoning, 16(3-4) (1997), 225-233.
[2] P. R. Halmos, Naive set theory, Dover Books, (1960).
[3] E. Trillas, On the logos: A nave view on ordinary reasoning and fuzzy logic, Springer, (2017).
[4] E. Trillas, On the use of words and fuzzy sets, Information Sciences, 176(11) (2003), 1463-1487.
[5] E. Trillas and L. Eciolaza, Fuzzy logic. An introductory course for engineering students, Springer, (2015).
[6] E. Trillas, S. Termini and M. E. Tabacchi, Reasoning and language at work. A critical essay, Springer, (2022).
[7] L. Wittgenstein, Philosophical investigations, Basil Blackwell, (1953).
[8] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
