Graduality, Uncertainty and Typicality in Formal Concept Analysis

There exist several proposals for extending formal concept analysis (FCA) to fuzzy settings. They focus mainly on mathematical aspects and assume generally a residuated algebra in order to maintain the required algebraic properties for the definition of formal concepts. However, less efforts have been devoted for discussing what are the possible reasons for introducing degrees in the relation linking objects and properties (which defines a formal context in the FCA sense), and thus what are the possible meanings of the degrees and how to handle them in agreement with their intended semantics. The paper investigates three different semantics, namely i) the graduality of the link associating properties to objects, pointing out various interpretations of a fuzzy formal context; ii) the uncertainty pervading this link (in case of binary properties) when only imperfect information is available and represented in the framework of possibility theory; and lastly, iii) the typicality of objects and the importance of definitional properties within a class. Remarkably enough, the uncertainty semantics has been hardly considered in the FCA setting, and the third semantics apparently not. Moreover, we provide an algorithm for building the whole fuzzy concept lattice based on Gödel implication for handling gradual properties in a qualitative manner.