Fuzzy classification is the process of grouping elements into fuzzy sets[1] whose membership functions are defined by the truth value of a fuzzy propositional function.[2] [3] [4] A fuzzy propositional function is analogous to[5] an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition.[6]
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[7] is a logical system which supports set construction using logical predicates with the class operator . A class
C=\{i|\Pi(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 x PF → P(U)
Here is an explanation of the logical elements that constitute this definition:
In contrast, classification is the process of grouping individuals having the same characteristics into a set. A classification corresponds to a membership function μ that indicates whether an individual is a member of a class, given its classification predicate Π.
\mu:PF x U → T
The membership function maps from the set of propositional functions PF and the universe of discourse U into the set of truth values T. The membership μ of individual i in Class C is defined by the truth value τ of the classification predicate Π.
\muC(i):=\tau(\Pi(i))
In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.