Ordinal fuzzy sets

Fuzzy set theory has been used as a framework for interpreting imprecise linguistic expressions. In general, a linguistic term is described by the compatibility ordering induced in some universe of discourse (UoD). A membership function in fuzzy set theory serves to reflect this ordering by assignment of values in [0, 1] for objects in UoD. When we compute the meaning of a linguistic expression such as "young and tall" using fuzzy membership functions, two implicit assumptions are made. First, we assume the membership values have quantitative meaning so that they can be quantitatively manipulated, for example, by adding or subtracting (the extensive scale assumption). Second, we assume that the scales of the membership values used in describing the different linguistic terms are comparable and the same (the common scale assumption). In many cases, these assumptions cannot be justified. Some proposals have been made to address the first issue by using ordinal scale in defining fuzzy membership functions. However, the second issue has not been properly investigated. In this paper, we propose a framework that does not depend on both of these assumptions. Such framework will facilitate our understanding and investigation of qualitative reasoning without the extensive scale and common scale assumptions.