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 toRussel, B. (1919). ''Introduction to Mathematical Philosophy''. London: George Allen & Unwin, Ltd., S. 155 an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition.Zadeh, L. A. (1975). Calculus of fuzzy restrictions. In L. A. Zadeh, K.-S. Fu, K. Tanaka, & M. Shimura (Hrsg.), Fuzzy sets and Their Applications to Cognitive and Decision Processes. New York: Academic Press.

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 $\mu_ : \tilde \times U \to \tilde$ that indicates the degree to which an individual $i\in U$ is a member of the fuzzy class $\tilde$, given its fuzzy classification predicate $\tilde_ \in \tilde$. Here, $\tilde$ is the set of fuzzy truth values, i.e., the unit interval ,1/math>. The fuzzy classification predicate $\tilde _(i)$ corresponds to the fuzzy restriction "$i$ is a member of $\tilde$".

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 logicGlubrecht, J.-M., Oberschelp, A., & Todt, G. (1983). Klassenlogik. Mannheim/Wien/Zürich: Wissenschaftsverlag. is a logical system which supports set construction using logical predicates with the class operator . A ''class''

C =

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

References

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Fuzzy logic