Fuzzy Implications (Studies in Fuzziness and Soft Computing)

Nothing new had been done in Logic since Aristotle! -KurtGodel ] (1906-1978) Fuzzyimplicationsareoneof themain operationsinfuzzy logic.Theygeneralize the classical implication, which takes values in {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. In classical logic the implication canbede?nedindi?erentways.Threeofthesehavecometo assumegreatert- oreticalimportance, viz. the usual materialimplication from the Kleene algebra, the implication obtained as the residuum of the conjunction in Heyting algebra (also called pseudo-Boolean algebra) in the intuitionistic logic framework and the implication (also called as 'Sasaki arrow') in the setting of quantum logic. Interestingly, despite their di?ering de?nitions, their truth tables are identical in classical case. However, the natural generalizations of the above de?nitions in the fuzzy logic framework are not identical. This diversity is more a boon than a bane and has led to some intensive research on fuzzy implications for close to three decades. It will be our endeavor to cover the various works churned out in this period to su?cient depth and allowable breadth in this treatise. In the forewordto Klir andYuan's book[147], ProfessorLot?A. Zadehstates the following: "The problem is that the term 'fuzzy logic' has two di?erent meanings.

Fuzzy Implications (FIs) generalize the classical implication and play a similar important role in Fuzzy Logic (FL), both in FL_n and FL_w in the sense of Zadeh. Their importance in applications of FL, viz., Approximate Reasoning (AR), Decision Support Systems, Fuzzy Control (FC), etc., is hard to exaggerate. This treatise is perhaps the first attempt at dealing exclusively with this class of operations.

The book begins with an introduction to FIs along with their desirable properties and investigates their inter-relationships. The reminder of the book is divided into 3 parts. Part I is an analytical study of FIs, detailing the different ways of obtaining FIs, generally called as families of FIs, the various properties they satisfy and the overlaps that exist among them. Part II is an algebraic exploration into the structures that exist on the set of all FIs, while the final part highlights the effecting role FIs with their myriad properties play in the inference schemes in AR.

This book is intended for any researcher in FL operations and it can serve as an auxiliary textbook for different courses on FL or FC. Parts of this book may also be of interest to practitioners in applications of FL, especially in AR or FC, where FIs play a central role.