Abstract
The crucial role that fuzzy implications play in many applicable areas was our motivation to revisit the topic of them. In this paper, we apply classical logic’s laws such as De Morgan’s laws and the classical law of double negation in known formulas of fuzzy implications. These applications lead to new families of fuzzy implications. Although a duality in properties of the preliminary and induced families is expected, we will prove that this does not hold, in general. Moreover, we will prove that it is not ensured that these applications lead us to fuzzy implications, in general, without restrictions. We generate and study three induced families, the so-called D ′ -implications, QL ′ -implications, and R ′ -implications. Each family is the “closest” to its preliminary-“creator” family, and they both are simulating the same (or a similar) way of classical thinking.
Highlights
In classical logic, the implication is uniquely determined, in fuzzy logic, there are several formulas and families of fuzzy implications
There are generation methods of fuzzy implications from known fuzzy implications [4, 13]. We focus on these fuzzy implications, in which the formula contains at least a t-norm or a t-conorm
There are other known families of fuzzy implications, such as (i) Yager’s f-generated and g-generated implications as they are defined by Yager [12], (ii) h-implications as they are defined by Jayaram [9, 25, 26], (iii) h-implications as they are defined by Massanet and Torrens [10], and (iv) Fuzzy implications through fuzzy negations as they are defined by Souliotis and Papadopoulos [11]
Summary
In classical logic, the implication is uniquely determined, in fuzzy logic, there are several formulas and families of fuzzy implications. Ere are fuzzy implications that are constructed by generalizations of classical tautologies, such as (S, N)-implications, QL-implications, D-implications [1, 3,4,5,6], and (T, N)-implications [2, 7, 8]. Ere are fuzzy implications that are constructed by function generators with specific properties [4, 9,10,11,12]. The central idea is to apply De Morgan’s laws and, if necessary, the classical law of double negation in known formulas of fuzzy implications and investigate the results. Any case must be studied individually from the beginning
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