Fuzzy quantification in fuzzy description logics

This chapter introduces reasoning procedures for {\qfdl}, a fuzzy description logic with extended qualified quantification \cite (SanchezTettamanzi2004}. The language allows for the definition of fuzzy quantifiers of the absolute and relative kind by means of piecewise linear functions on $\mathbb{N}$ and $\mathbb{Q}\cap[0,1]$ respectively. In order to reason about instances, the semantics of quantified expressions is defined by using method $GD$ \cite{DSV2000:ijar}, which is based on recently developed measures of the cardinality of fuzzy set. The main contribution of this chapter is a procedure to calculate the fuzzy satisfiability of a fuzzy concept, which is a very important reasoning task. The procedure considers several different cases and provides direct solutions for the most frequent types of fuzzy concepts. In order to distinguish between these cases, a novel idea of concept independence is also introduced.