Fuzzy classification

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

${\displaystyle C=\{i|\Pi (i)\}}$

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:

${\displaystyle \{\cdot |\cdot \}:V\times PF\rightarrow P(U)}$

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 ${\textstyle V:\rightarrow R}$ 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".^[5]

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.

${\displaystyle \mu :PF\times U\rightarrow T}$

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.

${\displaystyle \mu C(i):=\tau (\Pi (i))}$

In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.

• Fuzzy logic