A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. (I). General argumentation

The generalization of the concept of a classical (or crisp) preference structure to that of a fuzzy preference structure, expressing degrees of strict preference, indifference and incomparability among a set of alternatives, requires the choice of a de Morgan triplet, i.e., of a triangular norm and an involutive negator. The resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences. More specifically, one of the degrees of strict preference, indifference or incomparability should always be unconstrained to the preference modeller. This intuitive requirement is violated when choosing a triangular norm without zero divisors, since in that case fuzzy preference structures reduce to classical preference structures, and hence none of the degrees can be freely assigned. Furthermore, it is shown that the choice of a continuous non-Archimedean triangular norm having zero divisors is not compatible with our basic requirement: the sets of degrees of strict preference, indifference and incomparability in [0, 1[ are always bounded from above by a value strictly smaller than 1. These fundamental results imply that when working with a continuous triangular norm, only Archimedean ones having zero divisors are suitable candidates. These arguments sufficiently support our plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures.