Definition by Abstraction

a method of describing (isolating, “abstracting”) nonsensory (”abstract”) properties of objects by specifying a certain equivalence relation (identity) in the subject domain. Such a relation, which possesses the properties of reflexivity, symmetry, and transitivity, divides the subject domain into mutually exclusive classes (classes of abstraction or equivalence classes), in which the elements belonging to the same class are indistinguishable with respect to the property so defined. For example, in political economy, “value” is defined by abstraction (by the relation of exchangeability of goods), and in set theory, the “capacity” of a set is defined by this method (by the relation of set-theoretic equivalence).

Definition by abstraction always (although usually implicitly) relies on the principle of abstraction, which states that for every property there is a corresponding class (set) of objects possessing that property. In practice, this principle is very convenient, natural, and productive. However, postulation of the principle of abstraction as a universal methodological law leads to difficulties, which manifest themselves primarily in the form of paradoxes in logic and in set theory.