Fuzzy classification

Fuzzy classification is the process of grouping elements into a fuzzy set whose membership function is defined by the truth value of a fuzzy propositional function.

A fuzzy class ~C = { i | ~Π(i) } is defined as a fuzzy set ~C of individuals i satisfying a fuzzy classification predicate ~Π which is a fuzzy propositional function. The domain of the fuzzy class operator ~{ .| .} is the set of variables V and the set of fuzzy propositional functions ~PF, and the range is the fuzzy powerset (the set of fuzzy subsets) of this universe, ~P(U):

~{ .| .}∶V × ~PF ⟶ ~P(U)

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 in the sense of.

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 that indicates whether an individual is a member of a class, given its fuzzy classification predicate ~Π.

μ∶~PF × U ⟶ ~T

Here, ~T is the set of fuzzy truth values (the interval between zero and one). The fuzzy classification predicate ~Π corresponds to a fuzzy restriction "i is R"  of U, where R is a fuzzy set defined by a truth function. The degree of membership of an individual i in the fuzzy class ~C is defined by the truth value of the corresponding fuzzy predicate.

μ~C(i):= τ(~Π(i))

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.