Fuzzy classification

Fuzzy classification is the process of grouping elements into fuzzy sets whose membership functions are defined by the truth value of a fuzzy propositional function. A fuzzy propositional function is analogous to an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition. Accordingly, fuzzy classification is the process of grouping individuals having the same characteristics into a fuzzy set. A fuzzy classification corresponds to a membership function that indicates the degree to which an individual is a member of the fuzzy class , given its fuzzy classification predicate . Here, is the set of fuzzy truth values, i.e., the unit interval . The fuzzy classification predicate corresponds to the fuzzy restriction " is a member of ". Intuitively, a class is a set that is defined by a certain property, and all objects having that property are elements of that class. The process of classification evaluates for a given set of objects whether they fulfill the classification property, and consequentially are a member of the corresponding class. However, this intuitive concept has some logical subtleties that need clarification. A class logic is a logical system which supports set construction using logical predicates with the class operator { .| .}. A class C = { i | Π(i) } is defined as a set C of individuals i satisfying a classification predicate Π which is a propositional function. The domain of the class operator { .| .} is the set of variables V and the set of propositional functions PF, and the range is the powerset of this universe P(U) that is, the set of possible subsets: { .| .} :V×PF⟶P(U) Here is an explanation of the logical elements that constitute this definition: An individual is a real object of reference. A universe of discourse is the set of all possible individuals considered. A variable V:⟶R is a function which maps into a predefined range R without any given function arguments: a zero-place function.