Fuzzy classification is the process of grouping elements into a fuzzy set whose membership function is defined by the truth value of a fuzzypropositional function. A fuzzy class ~C = 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 of this universe, ~P: ~∶V × ~PF ⟶ ~P 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 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:= τ
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 = 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 that is, the set of possible subsets: ∶V×PF⟶P Here is an explanation of the logical elements that constitute this definition:
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:=τ In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.
