In the usual approaches to fuzzy set theory the notions of grade of membership, possibility, and so on are taken as given and axioms are put forward to govern the behaviour of these quantities; by judicious choice of the axioms a mathematically satisfactory theory can be obtained. However, such an approach gives no indication of how one is to decide what particular numerical value to assign to a grade of membership (possibility, etc.) in a given situation, or of how one should use such values in (for instance) decision-making. As a result, the grounds for application of the resulting theory are, to say the least, very insecure. To overcome this deficiency it is necessary to attach an exact empirical meaning to the terms, possibility and grade of membership, as well as to the concepts, proposition and fuzzy set, on which they depend. A proposition is represented by a “testprocedure” which yields one of two possible outcomes, and the meaning of an assignment of a grade of membership or possibility value is defined in terms of bets on the outcome of the test-procedure. The properties of the defined concepts then follow from these definitions and from simple principles of rational betting, and axioms become unnecessary and in fact inadmissible. The same treatment is accorded to the logical connectives, and, or and not, which are defined by relating the meaning of a compound proposition to those of its components. Several approaches are considered. Although all are classically equivalent they differ greatly both in scope and in their mathematical properties. A dialogue definition, studied previously by the author, provides the widest scope. Finally, the possibility that some entirely different solution of the problem might exist is considered. Reasons are given for believing that the possibilities here are rather limited. © 1982, Academic Press Inc. (London) Limited. All rights reserved.
Giles, R. (1982). Semantics for fuzzy reasoning. International Journal of Man-Machine Studies, 17(4), 401–415. https://doi.org/10.1016/S0020-7373(82)80041-2