Mathematics Behind Fuzzy Logic

The range of techniques denoted by the term “fuzzy” began with the deceptively simple‐sounding idea of fuzzy set membership advanced by Zadeh as long ago as 1965. It obviously found a niche and has been elaborated into fuzzy logic and fuzzy control as well as fuzzy versions of various statistical and decision‐making procedures, and means of representing uncertainty in expert systems. It is widely agreed that uncertainty can be usefully represented and treated in ways other than by classical probability theory, and the term “fuzzy” has become something of a blanket term to cover the alternatives, of which those that stem directly from Zadeh’s initiative appear to lead the field.

The mathematical treatments associated with fuzzmess have proliferated at an alarming rate, with several journals dedicated to the topic, and with, for example, the relevant symposium within the European Meetings on Cybernetics and Systems Research in Vienna amounting effectively to a conference within a conference and producing papers that are certainly among the most mathematically daunting in the proceedings.

A text book giving access to all this has clearly been needed, and the present text appears to fill the requirement. Fuzzy theory is related to the multi‐valued logic developed by Lukasiewicz, which later gave rise to MV‐algebras (in which “MV” still denotes multi‐valued), and then to a system of fuzzy logic due to the Czech worker Pavelka. The development appears to have been rounded off nicely by the author of the present book, who says in his preface:

The contribution of the present author was to show in 1994 the algebraic connections of Pavelka’s logic, e.g. that in the real unit interval Pavelka style fuzzy logic is axiomatizable if, and only if the set of truth values forms a complete MV‐algebra of [0, 1].

The book is written in an admirably clear style and is claimed to be self‐contained, so that no previous knowledge of algebra or logic is required. It has numerous exercises, with complete answers, and is suitable as a text for undergraduate or postgraduate courses. Of course, as the reference to postgraduate use indicates, there is a lot of material here, and although the style is clear, complete perusal would be a substantial undertaking. Some idea of the coverage can be given by quoting from the notes on the back cover, in which the letters “BL” in BL‐algebra stand for “basic (fuzzy) logic”:

Chapter 1 starts from such basic concepts as order, lattice, equivalence and residuated lattice. It contains a full section on BL‐algebras. Chapter 2 concerns MV‐algebra and its basic properties. Chapter 3 applies these mathematical results on Lukaziewicz‐Pavelka style fuzzy logic, which is studied in detail; besides semantics, syntax and completeness of this logic, a lot of examples are given. Chapter 4 shows the connection between fuzzy relations, approximate reasoning and fuzzy IF‐THEN rules to residuated lattices.

A number of practical applications are discussed. These are fairly simple and it is not clear that so much mathematical sophistication was needed to obtain solutions. On the other hand, simple examples are appropriate in a teaching text and the fuzzy approach has been found by many people to be robust and intuitive, at least to suitably conditioned intuitions. It is rather likely that the inherent robustness of the approach would readily allow scaling‐up of these simple illustrative examples.

I have to admit to having reservations about the often‐assumed exact correspondence of fuzzy techniques to the means by which people operate under uncertainty. There seems to be an unwarranted assumption that human performance is determined linguistically, whereas it may be more appropriate to regard the linguistic expression as merely the “tip of the iceberg” of the thought process. On the other hand, fuzzy techniques have achieved a great deal and are definitely here to stay, and I am very glad to have this little book on my shelves to help me try to make sense of their more esoteric ramifications.