Il Trimestrale. The Lab's Quarterly, 1, 2008
For example Pedrycz (1993; 68:70) after having presented the problem of individuating which kind of scale (absolut, ratio, ordinal) may be used for estimating membership functions, exposes two general classes of methods used in formalizing membership functions relative to finite discourse universes of significative pratical interestcontext that makes these classes of methods interesting for social sciences as well. The two classes are identified by Pedrycz as an horizontal and a vertical approach. The horizontal approache consists in the collection of informations about the membership degree of determined elements of the universe of discourse in which the fuzzy set has to be defined. By counting the positive (“yes”) answers to questions like “May the x0 element be considered compatible with the concept of the fuzzy set A?” it becomes possible to determine the ratio n(x0) of positive answers versus the total number of answers, and from this ratio the membership degree of the single element (x0) can be derived with the formula
Where μAx0 is the membership degree of the element x0 to the set A. This way the standard deviation of answers can be obtained with the formula
Standard deviation so obtained can be used to establish an acceptation criterion for the membership degree of the x0 element to A fuzzy set, for example accepting A(x0) as membership degree if standard deviation is not superior to a certain λ limit, tipically chosen as small respect to A(x0) values. The vertical approach consists in idntifying an α level of membership, asking to subjects to identify a collection of items that meet the concept represented by A fuzzy set with a membership degree not inferior to α. In this approach, A fuzzy set is built starting from its α-cuts.