On fuzzy-qualitative descriptions and entropy

• A new general representation of linguistic descriptions by unifying ordinal and fuzzy perspectives.
• The construction of the extended set of fuzzy-qualitative labels over a well-ordered set.
• A theorem characterizing commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy sets.
• The definition of the fuzzy-qualitative descriptions of a set as L-fuzzy sets, and the proof of its lattice structure.
• The entropy of a fuzzy-qualitative description providing a unified framework for the discrete and continuous cases.

This paper models the assessments of a group of experts when evaluating different magnitudes, features or objects by using linguistic descriptions. A new general representation of linguistic descriptions is provided by unifying ordinal and fuzzy perspectives. Fuzzy-qualitative labels are proposed as a generalization of the concept of qualitative labels over a well-ordered set. A lattice structure is established in the set of fuzzy-qualitative labels to enable the introduction of fuzzy-qualitative descriptions as L-fuzzy sets. A theorem is given that characterizes finite fuzzy partitions using fuzzy-qualitative labels, the cores and supports of which are qualitative labels. This theorem leads to a mathematical justification for commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy sets. The information of a fuzzy-qualitative label is defined using a measure of specificity, in order to introduce the entropy of fuzzy-qualitative descriptions.