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 [math]\displaystyle{ \mu_{\tilde{C}} : \tilde{PF} \times U \to \tilde{T} }[/math] that indicates the degree to which an individual [math]\displaystyle{ i\in U }[/math] is a member of the fuzzy class [math]\displaystyle{ \tilde{C} }[/math], given its fuzzy classification predicate [math]\displaystyle{ \tilde{\Pi}_{\tilde{C}} \in \tilde{PF} }[/math]. Here, [math]\displaystyle{ \tilde{T} }[/math] is the set of fuzzy truth values, i.e., the unit interval [math]\displaystyle{ [0,1] }[/math]. The fuzzy classification predicate [math]\displaystyle{ \tilde{\Pi} _{\tilde{C}}(i) }[/math] corresponds to the fuzzy restriction "[math]\displaystyle{ i }[/math] is a member of [math]\displaystyle{ \tilde{C} }[/math]".

Classification

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.
• A propositional function is "an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition".

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.

μ:PF × 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.

μC(i):=τ(Π(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.

See also

• Fuzzy logic