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

$C=\left\{i|\mathrm{\Pi <!--\; \Pi \; -->}\right(i\left)\right\}$

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:

$\{\cdot <!--\; \cdot \; -->|\cdot <!--\; \cdot \; -->\}:V\times <!--\; \times \; -->PF\to <!--\; \to \; -->P\left(U\right)$

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 $V:\to <!--\; \to \; -->R$ 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.

$\mathrm{\mu <!--\; \mu \; -->}:PF\times <!--\; \times \; -->U\to <!--\; \to \; -->T$

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.

$\mathrm{\mu <!--\; \mu \; -->}C\left(i\right):=\mathrm{\tau <!--\; \tau \; -->}\left(\mathrm{\Pi <!--\; \Pi \; -->}\right(i\left)\right)$

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