The Fuzzy Nature Of Concepts: Stochastic Objects
Megaklis Th. Sotiropoulos, Theodore E. Simos, George Psihoyios, Ch. Tsitouras · AIP conference proceedings · 2010
Concept is every assignment of a prototype to an icon, whatever may be the prototype or the icon. We call the prototype “object” and the icon “attributes”. Concepts are couples of sets O and A, that is assignments, of the object O (a set of none, or one or more elements ‐there is no real difference), to the set A of (their common) attributes. The objects change according to the sequence of attributes. So, only couples of objects and attributes, enriched with the proper operations, are adequate for our Knowledge Space. Concepts are proved to have the structure (order) of a Boolean Algebra(Lattice), which is more complex than linear or hierarchical ones. The lattice is created by two algebraic operations (“intersection of concepts” as the multiplication and “symmetric‐difference(!) of concepts” as the addition (!)). There are two other operations (the “union of two concepts” and the “complement of a concept”). Intersection and union(which cannot play the role of multiplication) express similarities, while the other two operations express dissimilarities. Union, intersection and the complement are used for the definition of the symmetric‐difference. The complement cannot be expressed by the two predefined operations union and intersection, which have the meaning of “common”. Besides, it is not a deterministic function: the complement Oc may have attributes inside the complement Ac of attributes (and if yes, we do not know, from the beginning, which of them). Now, the situation becomes stochastic: if Ω′ is the set of n attributes we are interested in (e.g., in m sequential experiments), suppose that, in each experiment, some attributes appear(are detected) and the rest from Ω′ do not appear. Everyone from the m experiments consists of n Bernulli trials (one for every attribute) and, consequently, we get m stochastically changing objects (concepts) and the (complete, if we are very lucky) lattice of the m concepts. Everyone from the n attributes corresponds to a B(m,pii = 1,2,…,n. E.g., diverse appearances of one germ(like influenza…). But, knowledge is comparison, distance, classification. In other M experiments, we get another lattice: what do we decide? Then, we use the fuzzy logical operators “at most” (or, simplified, maximum) as the union and “at least”(or, simplified, minimum) as the intersection, by which we get a “final”, “clever” lattice.