396 Bulletin

on this subject. Therefore, another ongoing project is focused on theoretical development and applications of fuzzy clustering methods, which is sponsored by the National Science Council of Taiwan, Republic of China.

Hsiao-Fan Wang Department of Industrial Engineering National Tsing Hua University Hsinchu, Taiwan, ROC

Book Reviews

Vagueness: An Investigation into Natural Languages and the Sorites Paradox

This book by Linda Claire Burns was published by Kluwer Academic Publishers (1991 ; ISBN 0-7923-1489-1 ).

The Sorites paradox arises when there is a continuous range of cases where a description applies at one end and not at the other but it is not clear exactly where it ceases to apply. Thus, when we throw stones into a pond, we start with a pond which becomes gradually shallower but at some point we wil l agree that there is now no pond but a pile of stones.. It appears as a paradox when it is claimed that the borderline case is simultaneously a heap and not a heap. Similarly, if there are three shades of red A, B and C with A indistinguish- able from B and B from C though A and C are distinguishable, does this mean that B is of two distinct shades at the same time? Obviously, this is an important problem for fuzzy set theorists and for all artificial intelligence workers who wish to capture this aspect of reasoning in computer systems.

Burns' book is a monograph steeped in the conven- tions of analytical and linguistic philosophy. It starts with Frege's characterization of vagueness and surveys the work of several modern philosophers such as Dummett who have discussed the issue and their various solu- tions in terms of tolerance rules for predication, rejection of the induction hypothesis for vague predicates and even the retreat into idealism (where there are no heaps). The survey is both penetrating and comprehen- sive though no pre-analytic contributions are dealt with. This is slightly disappointing since this has been a topic for Western philosophers from Herakleitos to Hegel. This neglect of history shows most clearly in the part of the discussion which suggests that it is necessary to decide cleanly whether the Sorites is an epistemological or semantic (i.e. objective) phenomenon. Some non- analytic thinkers would argue that this distinction cannot be made since knowledge is related dialectically to its content and object. Also the assumption throughout is that negation applies to predicates and that a mathe- maticization is possible. The result is that fuzzy set theory is squarely demolished as an approach to vague- ness and if you agree with the assumptions then you wil l give up fuzzy sets and indeed all multivalent logics forthwith, for the arguments themselves can't be faulted.

The chief reason is that assigning a truth value of 0.5 is equally as problematical as assigning a value of 1 or 0 when it comes to borderline cases. The arguments and assumptions remind me of Haack's criticism of deviant logics. Burns' programme is to rescue the classical laws of logic such as the excluded middle, and she does this using several reductio arguments (which of course depend on this very law). At least three reactions are possible: one can say that fuzzy sets is mere engineer ing-so it doesn't matter; find no reason to believe the classical laws of logic in the first place; or suggest that the way the problem is posed is wrong.

Most of the book discusses other people's solutions. In fact it is slightly annoying that the author makes the reader wait to the end before her solution is revealed. This makes a second reading almost imperative. Let me give the game away by telling you that the solution is found in the context of utterances or the pragmatics of language. What constitutes a heap is determined only in context. This is something of an over-simplification of course and Burns argues the mathematical consistency of her solution over the weaknesses she finds in her predecessors' formulations very well and clearly.

This is an important book that should not be ignored by theorists interested in the foundations of fuzzy sets. It is a serious contribution to our understanding of vague- ness and needs to be answered more thoroughly than is either appropriate or possible in a review. One further point; even though it assumes a good deal of back- ground in modern Philosophy, it is extraordinarily well written.

lan Graham

Fuzzy Set Theory and its Applications

This book by Hans-J~irgen Zimmermann was pub- lished in a second edition by Kluwer Academic Pub- lishers (Dordrecht, 1991; 399 pages).

From the purely mathematic view point, the evolution of fuzzy logic (soft logic) has been very exciting, but complex. Many scientific theories start by borrowing notions from the already developed areas of mathe- matics, but in this case, Professor Lotfi A. Zadeh intro- duced the basic notion of vagueness which has no sharp morphology and which is so common in human thought processes. This notion of vagueness is modelled using the notion of graded membership in fuzzy logic. Since the introduction of the theory, the applications of it can be found in diverse fields such as artificial intelligence, expert systems, medical diagnosis, control systems, operations research, and management science.

This book on "Fuzzy Set Theory and its Application" by Hans Zimmermann is the revised second edition (1991). It introduces the basic notion of fuzzy set theory using simple to complex examples and easy to under- stand mathematical steps, and then leads the readers to