CHARLES GRADY MORGAN AND FRANCIS JEFFRY PELLETIER SOME NOTES CONCERNING FUZZY LOGICS* ABSTRA~. Fuzzy logics are systems of logic with infinitely many truth values. Such logics have been claimed to have an extremely wide range of applications in linguistics, computer technology, psychology, etc. In this note, we canvass the known results concerning infinitely many valued log&; make some suggestions for alterations of the known systems in order to accommodate what modern devotees of fuzzy logic claim to desire; and we prove some theorems to the effect that there can be no fuzzy logic which will do what its advocates want. Finally, we suggest ways to accommodate these desires in finitely many valued log&. I. HISTORICAL NOTES The concept of a fuzzy set and the related systems of fuzzy set theory and fuzzy Zogic have been claimed to have an extremely wide range of applica- tions: reasoning involving inexact concepts (Goguen, 1969; Zadeh, 1972), psychological classification (Heider, 1971; Dreyfus et al., 1975; Kochen, 1975; Kochen and Badre, 1975), threshold phenomena (Nakamura, 1962a,b; 1963a,b,c; 1964,1965), pattern recognition (Bellman etal., 1966), computer learning (Samuel, 1967), and the provision of an adequate description of certain linguistic phenomena (Lakoff, 1971, 1972). The purpose of this note is to point out certain shortcomings inherent in fuzzy logics which demonstrate their unsuitability for these purposes. Presentations of fuzzy logics have thus far been semantic in nature and the syntax has been ignored.’ It seems to be assumed by most of these writers that an adequate syntax, one in which exactly the (semantically) valid arguments are provable, can easily be constructed and is ‘merely a task for logicians’. (For a dissenting view, see Goguen (1969; p. 326) who says that it is unlikely that we should find a purely syntactic logic for inexact concepts. We shall show that Goguen’s intuition here is correct.)2 The basic semantic intuition behind a fuzzy propositional logic is that propositions cannot only take the two values 0 (false) and 1 (true), but any of * We would like to express our thanks to Stan Peters for helpful conversations. The method in the Appendix is due to David Lewis. ’ That is to say, the usual presentations give an account-informal and intuitive in Lakoff, more formal in Goguen (1969) and Zadeh (1965, 1971) - of the truth of certain kinds of sentences and validity of certain arguments. The syntax of fuzzy logic - a specification of the axioms and rules of inference - has been left unspecified. (Compare the discussion in Goguen, Sect. I.) * Also Lakoff p. 216 seems to feel that no axiomatic treatment will be forthcoming. For some reason, however, he seems to think this is good and that it shows the superiority of fuzzy logics.