Just as there is a classical definition of probability, and an axiomatic definition followed thereafter, perhaps fuzziness too should have an axiomatic definition. Please give your comments.
I am not aware of this kind of definition. I think that the reason may be as follows:
Probability is a very old concept. At that time, set theory was not established. So, people at that time defined it as it is in the classical definition. When set theory was discovered, people have to reform the old definitions with this new perspective. Whereas fuzzy set theory is quite a recent idea. So, people directly defined it in terms of a universal set and an existence of a membership function. Similarly, fuzzy number and fuzzy logic are also defined keeping in mind the modern concepts of set theory and function.
Dear Sir,
Yes I will definitely send you comments after a few days.
With Regards,
Rajesh Sanghvi
This is a very good question with more than one answer.
A detailed introduction to fuzzy axioms is given in
Marco Cerami, Fuzzy Description Logics from a Mathematical Fuzzy Logic point of view, Universitat de Barcelona (UB)
Facultat de Matem`atiques
Departament de Probabilitat, L`ogica i Estadstica
Programa de doctorat en L`ogica Pura i Aplicada, Ph.D. thesis, 2012:
http://www.tdx.cat/bitstream/handle/10803/113374/MARCO_CERAMI_THESIS.pdf?sequence=1
See Section 2.2.2, starting on page 34, for the fuzzy axioms.
On RG itself, see
https://www.researchgate.net/profile/Shrikant_Chaudhari/publication/263878818_ON_INTERRELATIONS_BETWEEN_FUZZY_CONGRUENCE_AXIOMS/links/53ff0cb00cf283c3583c5110.pdf?origin=publication_detail
An axiomatization of fuzzy sets is given by Dubois and Prade in
Didier Dubois – Henri Prade, Fuzzy sets and probability:
www.ibrarian.net/.../Fuzzy_sets_and_probability___Misunderstandings_...
Article On interrelations between fuzzy congruence axioms
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