By S. L. Parsonson
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The definition of fuzzy connectives and the study of their properties. 1 Introduction Forty years have gone by since Zadeh’s pioneering paper introducing fuzzy sets and fuzzy logic [94]. During this period, numerous papers have been published on fuzzy topics, the field has experienced an enormous growth, and many of Zadeh’s seminal concepts have naturally evolved in different directions. In particular a variety of set theories have been defined such as: L-fuzzy sets [46], flou sets [45], type-2 fuzzy sets [61,95], interval-valued fuzzy sets [75], intuitionistic fuzzy sets [12], twofold fuzzy sets [34], fuzzy rough sets [35], vague sets [44] or loose sets [81].
This paper aims to review the main issues related to the construction of fuzzy set theories from the Zadeh’s initial conceptions to some newer proposals. It is an extended and revised version of [72], and is organized as follows. Section 2 presents a general discussion of the construction of fuzzy set theories. Section 3 proposes a general definition for the concept of fuzzy set theory, and Sect. 4 recalls the definition and the main properties of one of the most popular and important theories, so-called standard fuzzy set theories.
Gq : L × L → L and N1 , . . , Nr : L → L such that for any A, B in FL (X) and for any x ∈ X: (A ∩1 B)(x) = F1 (A(x), B(x)), . . , (A ∩p B)(x) = Fp (A(x), B(x)) (A ∪1 B)(x) = G1 (A(x), B(x)), . . , (A ∪q B)(x) = Gq (A(x), B(x)) Ac1 (x) = N1 (A(x)), . . , Acr (x) = Nr (A(x)) Functionally expressible theories are very useful for practical purposes for two main reasons. First, it is clearly much easier to build functionally expressible than non-functionally expressible operations. Second, the study of which properties are fulfilled by the operations ∩, ∪ and c is enormously simplified in the first case, since it relies on the properties of the underlying functions F , G and N , which are much easier to determine.