The fact that many concept categories are graded and therefore not amenable to treatment by classical logic has led to the introduction of fuzzy set theory and to later developments of fuzzy logic and a great volume of fuzzy system theory with important practical applications. In the Introduction to the present book it is observed that the value of fuzzy set theory and fuzzy logic was recognized originally by psychologists as a means of representing and dealing with imprecise concepts but the approach was largely abandoned in the early 1980s. The abandonment is attributed by the editors to erroneous conclusions in the paper of Osherson and Smith (1981), and the present work is meant to set the record straight and to reexamine the usefulness of fuzzy logic in psychological studies.
After a short introductory chapter by the editors there are tutorial chapters, one on concepts from a psychologist's point of view and one on fuzzy logic by the editors. Both chapters contain acknowledgements that the coverage has to be incomplete in the available space. The treatment of concepts as such is under three main headings, namely exemplar theory, prototype theory and what are termed “theory theories”. The first two are compatible with a fuzzy‐set approach and differ only in how the primary representation of a concept is formed but the third includes reference to theoretical treatments having a quite different character and viewing concepts as components in a larger process underlying intelligent behaviour.
I have to admit to having been dismissive of over‐reliance on concepts as a basis for intelligence, and hence of fuzzy approaches, in Andrew (2009). I have argued that the part played by continuous processing in intelligent systems tends to be overlooked and that the origins of intelligence must be pre‐linguistic. This of course is not to deny that linguistic and concept‐based communication and processing play a large part in human mentation and are a legitimate and valuable field of study but it does suggest that treatment in their terms must inevitably be incomplete. It can be considered that there has been evolution of the concept of a concept (Andrew, 1981).
The tutorial chapter on fuzzy logic is admirably clear and as comprehensive as its length allows. It is emphasized that it nevertheless covers only a small part of the totality of theory that has been developed. In the next‐but‐one chapter the editors take Osherson and Smith to task and claim to show that their conclusions are erroneous. It is asserted that one of the mistakes of O. and S. is in regarding fuzzy set theory as a theory of concepts rather than as purely mathematical with no fixed interpretation. Belohlavek and Klir favour this latter view, reminiscent of the formalist view of mathematics associated with Hilbert, though rather at odds with the customary way of introducing fuzzy theory, with reference to some imprecise concept such as tallness.
A main argument put forward by O. and S. was based on experiments in which it was found that subjects could recognize a picture of a “striped apple” more quickly than they recognised either an apple or a striped object. This does not agree with the expectation from applying the usual method of fuzzy combination to the findings but it is shown in the chapter that use of a different combination formula removes the discrepancy. Other work is quoted in support of the contention that the combination method used by O. and S. is not applicable when the combination is of an adjective and a noun rather than of two nouns. Other arguments of O. and S. are refuted similarly. It is difficult, however, to avoid feeling that O. and S. have at least shown that it is only with difficulty that fuzzy theory can be reconciled with all of the experimental data.
The identification and delineation of difficulties can be either a source of discouragement or a spur to further effort and clearly the editors hope for the latter. Even so it is surprising, in view of the avowed aim of the book's compilation, that almost all of the remaining chapters give some support to this weakened version of the contentions of O. and S. Exceptions are a brief Epilogue by the editors, and a chapter by Belohlavek in which he reports on his work on formal concept analysis, both classical and fuzzy. He describes methods that have been developed for the processing of tables of objects and their associated attributes so as to derive super‐concepts and to perform factor analysis by methods that are claimed to be more transparent than those currently employed by statisticians under the heading. The methods have been implemented in efficient computer programs.
The central theme of the book, namely the place of fuzzy logic in psychology, is treated very fully and thoughtfully in a chapter by Eleanor Rosch of the University of California. She points out that: “People have different modes of using their concepts, and they are quite flexible in switching among them” and insists that: “Models of the conjunction of fuzzy concepts need more than degree of membership to produce their results”.
An observation that is rather curiously not made explicitly in the book, but is consistent with the first of these quotes from Rosch, is that the ready acceptance of the arguments of Osherson and Smith by experimental psychologists may be partly attributable to subconscious “wishful thinking” and the cast‐iron excuse offered for ignoring the formidable body of mathematics associated with fuzzy approaches. Acceptance of this argument is a double‐edged sword as far as the overall theme of the book is concerned, since in one way it supports the view that O. and S. have exerted undue influence, but at the same time it illustrates a fundamental difficulty of the fuzzy approach, namely the fact that subjective assignment of truth values (in this case associated with validity of the O. and S. viewpoint) can be affected by considerations well removed from those that come immediately to mind as relevant.
The book has three chapters not so far mentioned, one of them by a group with affiliations in Amsterdam and New York, and two by James Hampton of the City University, London. All of these discuss problems that arise in associating psychological variables with fuzzy membership functions, arising largely from vagueness about what these functions represent. One of my own reasons for being dismissive of fuzzy methods (apart from the evolutionary consideration already mentioned) is that applications generally require use of a membership function drawn rather arbitrarily by the experimenter. The precise form of the function does not seem to be critical, but since the ultimate justification is that it “works” it has to be seen as numerical fine‐tuning that dilutes the purity of the fuzzy‐logic approach. Hampton raises essentially the same objection on page 250 of the book where he observes that the membership function must itself be vague. It could therefore be represented by fuzzy variables with their membership functions, implying the possibility of an indefinite nesting of fuzziness. Hampton reports psychological experiments designed to study perception of vagueness.
All of the chapters give lists of selected references. These include further contributions by Osherson and Smith (1982, 1997), the latter of which is a reply to a claim by Kamp and Partee (1955) that the difficulty of reconciling the O. and S. results with fuzzy theory can be removed by introducing a principle of “supervaluation”. This principle is discussed by Hampton on page 248 of the new book. Reference is also made to papers by Zadeh (1978, 1982), the latter of which bears directly on the O. and S. contentions, with the conclusion that the difficulties pointed out “by no means disqualify the theory of fuzzy sets as a basis for a theory of prototypes”. Zadeh considers it “unlikely that an adequate theory of prototypes can be constructed without an explicit use of fuzzy sets and related concepts”.
What can be seen as an “adequate” theory has to depend on what is expected, which in turn depends on the extent to which intelligence is judged to be essentially verbal. My own feeling is that Zadeh overestimates this, as does Maturana also in his many references to “languaging”. However, these are deep issues and the new book gives a welcome and valuable survey of relevant viewpoints, lucidly presented and mostly having a solid basis in experimental psychology.
