Toll Sets and Toll Logic

The introduction of the notion of fuzzy set by Zadeh (1965) was mainly motivated by the problem of modelling linguistically expressible categories with unsharp boundaries. Then starting with the usual convention for set characteristic functions: zero (0) for non-membership, one (1) for membership, it leads to take (0,1) as a possible set of intermediary degrees of membership. Very soon after the introduction of fuzzy sets, Goguen (1967) proposed to generalize the idea of a fuzzy set by replacing the real interval [0,1] by some abstract set L equipped with some mathematical structure. Until now, in spite of the theoretical interest of L-fuzzy sets, only [0,l]-valued fuzzy sets are considered in practice.