Conceptual fuzzy sets and their connectives

Abstract

The evaluation of element pertinence to sets and connecting the sets themselves when these are not crisp may cause serious epistemological and practical problems. Fuzzy Set Theory (FST) solved some of these problems, with noticeable success since its inception by Zadeh [40] in the 1960s. But that theory now is meeting with difficulties for penetrating AI or, more generally, Cognitive Science. This is mainly due to the characteristics of the membership (characteristic) function and, by reflection, of the connectives MIN-MAX. According to these operators, set connections are performed by diffusing numerical data, i.e. values of membership degree, rather than qualitative data, such as meaning of the data themselves [42], so that the meaning itself can be taken into consideration. In this paper we present an interpretation of element pertinence to sets which yields a more adequate type of set connectives [3]. The general principles assumed to develop the subject and to show its consistency as an algebraic structure form a theory of conceptual sets (c-sets) [9]. A c-set consists of elements collected as function of a specific concept; to each element there is associated a number — element typicality — which expresses the goodness by which the element itself represents the whole set. The way in which the said typicality is computed permits conceiving how to connect c-sets so that the symbolic aspects of the connected sets are accounted for. This is realized by formally relating the properties characterizing every single element to those of all other elements that form the whole set, rather than the subjectively assessed membership function of the element itself, as performed in FST. We develop c-set theory by formalizing the regularities that can be found in concepts, as described in Cognitive Science literature, beginning with the work of E. Rosch (for example, see [15, 24, 28, 36]). The matter reported herein essentially regards the theoretical basis for the formalization of c-sets; besides it outlines a first approach to codify adequate connectives among the c-sets themselves. In addition to typicality, other important numerical and symbolic c-sets parameters are considered in the paper: variable relevance, c-set specificity, indication clarity, c-set prototype and type of c-set linearity. Among these, relevance quantifies the importance of each variable in describing elements, with respect to a specific c-set; c-set specificity expresses the semantical concordance within the elements of a c-set. The importance of the latter parameter was pointed out by Yager [39] and analysed also by other scholars. All the parameters mentioned enable us to cope with the c-set multiformity, as, for example, the complexity and non-linearity that may be faced in possible applications. By the development we perform a tool for machining knowledge learning and concept computation is achieved. Also, an outlook on a possible abductive c-set logic is introduced.