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
is the set of fuzzy truth values, i.e., the unit interval
[6] 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
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 { .| .}
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.
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.
In classical logic the truth values are certain.
Therefore a classification is crisp, since the truth values are either exactly true or exactly false.