On Quantified Linguistic Approximation

Most fuzzy systems including fuzzy decision support and fuzzy control systems provide out­ puts in the form of fuzzy sets that represent the inferred conclusions. Linguistic interpretation of such outputs 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 generally, linguistic approximation can be formalized in the terms of the re-translation rules that correspond to the translation rules in ex­ plicitation (e.g. simple, modifier, composite, quantification and qualification rules) in com­ puting with words [Zadeh 1996]. However most existing methods of linguistic approximation use the simple, modifier and composite re-translation 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. Therefore the question arises why not use in linguistic ap­ proximation also other re-translation rules corre­ sponding to the translation rules in explicitation to advantage. In particular linguistic quantifica­ tion may be desirable in situations where the conclusions interpreted as quantified linguistic propositions can be more informative and natu­ ral. This paper presents some aspects of linguis­ tic approximation in the context of the re­ translation rules and proposes an approach to linguistic approximation with the use of quantifi­ cation rules, i.e. quantified linguistic approxima­ tion. Two methods of the quantified linguistic approximation are considered with the use of lin­ guistic quantifiers based on the concepts of the non-fuzzy and fuzzy cardinalities of fuzzy sets. A number of examples are provided to illustrate the proposed approach.