On the notion of concept I

It is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory. Recent work on compositionality and concept determination showed that the quantitative solution initially offered by classical fuzzy logic also led to important drawbacks. Several qualitative approaches were thereafter tempted, that aimed at modeling membership through ordinal scales or lattice fuzzy sets. Most of the solutions obtained by these theoretical constructions however are of difficult use in categorization theory. We propose a simple qualitative model in which membership relative to a given concept f is represented by a function that takes its value in a finite abstract set Af equipped with a total order. This function is recursively built through a stratification of the set of concepts at hand based on a notion of complexity. Similarly, the typicality associated with a concept f will be described using an ordering that takes into account the characteristic features of f. Once the basic notions of membership and typicality are set, the study of compound concepts is possible and leads to interesting results. In particular, we investigate the internal structure of concepts, and obtain the characterization of all smooth subconcepts of a given concept.