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

## 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 {⋅|⋅}{\textstyle \{\cdot |\cdot \}}. 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:
$\{\cdot |\cdot \}:V\times PF\rightarrow P(U)$
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 V:→R{\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".
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.
$\mu :PF\times U\rightarrow T$
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.
$\mu C(i):=\tau (\Pi (i))$
In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.