In this paper we introduce a new method of generating fuzzy implications via known fuzzy implications. We focus on the case of generating fuzzy implications via a fuzzy connective and at least one known fuzzy implication. We present some basic desirable properties of fuzzy implications that are invariant via this method. Furthermore, we suggest some ways of preservation or violation of these properties, based in this method. We show how we can generate not greater or not weaker fuzzy implications with specific properties. Finally, two subclasses of any fuzzy implication arise, the so called T and S subclasses.

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.

In this paper, we introduce and study the GQL′-operations. We prove that this class is a hyper class of the known class of QL′-operations. Similar to QL′-operations, GQL′-operations are not always fuzzy implications. On the other hand, we present and prove a necessary but not sufficient condition that leads to the generation of a GQL′-implication. Our study is completed by studying the satisfaction or the violation of some basic properties of fuzzy implications, such as the left neutrality property, the exchange principle, the identity principle and the left ordering property. Our study also completes the study of the aforementioned basic properties for QL′-implications and leads to a new connection between QL-operations and D′-operations.

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In this paper, generalized R ′ -implications are introduced and studied. Although they are not always fuzzy implications, the sufficient and necessary condition such that a generalized R ′ -operation is a fuzzy implication is discovered and proved. The satisfaction or violation of some basic properties of fuzzy implications, such as the left neutrality property, the identity principle, and the left ordering property, is also investigated and presented. The intersections among the classes of generalized R ′ -implications (respectively, operations), R-implications, and R ′ -implications (respectively, operations) are also studied. It is demonstrated that the generalized R ′ -implications class is a hyper class of the known R- and R ′ -implications classes. This research also completes the investigation into R ′ -operations and improves the outcomes of the intersection of R- and R ′ -implications.