On numerical and linguistic quantification in linguistic approximation

Interpretation of outputs of fuzzy systems often involves the use of linguistic approximation that assigns a linguistic label to a fuzzy set based on the predefined primary terms, linguistic modifiers and linguistic connectives. More formally linguistic approximation can be considered in the terms of retranslation rules that correspond to the translation rules in explicitation in computing with words such as the simple, modifier, composite, quantification and qualification rules. However most existing methods of linguistic approximation use the simple, modifier and composite retranslation rules only. Although these methods can provide a sufficient approximation of simple fuzzy sets the approximation of more complex ones that are typical in many practical applications of fuzzy systems may be less satisfactory. In particular quantification may be desirable in situations where the conclusions interpreted as quantified linguistic propositions can be more informative and natural. Quantification in linguistic approximation can provide an additional information about the scope of the linguistic labels assigned to a given fuzzy set enhancing the interpretability of the conclusions. This paper presents some aspects of linguistic approximation in the context of the retranslation rules and proposes an approach to linguistic approximation with the use of the quantification rules. Two methods of the quantification in linguistic approximation are considered with the use of the numerical and linguistic quantifiers. These quantifiers are based on the concepts of the non-fuzzy and fuzzy cardinalities of fuzzy sets, respectively. A number of examples are provided to illustrate the proposed approach.