identity

The abstract counterpart of unity is the co-called identity (or identity element). Specifically, let the symbol * denote a (binary) operation on some set S. An element e in S is called an identity relative to the operation * if a*e = e*a =a. If * is a noncommutative operation, that is, if there are elements a and b such that a*b ≠ b*a, then we define a right identity to be an element e_r such that a*e_r for all a in S and a left identity to be an element e_r such that e_l*a = a for all a in S. If more than one operation is defined on the set S, then we speak of identities relative to the individual operations (for example, In the case of multiplication we speak of a multiplicative identity and in the case of addition of an additive identity).

a fundamental concept in logic, philosophy, and mathematics; it is used in the language of scientific theories to formulate definitions, laws, and theorems.

In mathematics, an identity is an equation that is satisfied identically—that is, an equation that holds true—for all admissible values of its variables. In logic, identity is a predicate that is represented by the formula x = y (to be read as “x is identical to y, ” or “x is the same as y”); the predicate’s corresponding logical function is true when the variables x and y denote different occurrences of “one and the same” object, and false in the opposite case. In philosophy, or epistemology, identity is a relationship based on ideas or judgments about the meaning of what it is to be “one and the same” object of reality, perception, or thought.

The logical and philosophical aspects of identity are complementary: the former provides a formal model of the concept of identity, while the latter provides grounds for the model’s application. The first aspect includes the concept of “one and the same” object, but the formal model does not depend for its meaning on the content of this concept; the method of identification is ignored, as is the dependence of the results of identification on the conditions or methods of identification and on the abstractions that are explicitly or implicitly assumed in the given case. In the second, or philosophical, aspect, applications of logical models of identity are considered in relation to the methods and tests by which objects are identified; such considerations, in effect, depend on the point of view, conditions, and means of identification.

The distinction between the logical and philosophical aspects of identity goes back to the well-known statement that identity as a concept is not one and the same as a judgment about the identity of objects (see Plato, Soch., vol. 2, Moscow, 1970, p. 36). It is essential, however, to emphasize the independence of and lack of contradiction between these two aspects. The meaning of the concept of identity is settled by its corresponding logical function; the concept is not deduced or extracted from the actual identity of objects, but rather it is an abstraction achieved under “suitable” experiential conditions or, in theory, by means of assumptions, or hypotheses, about actually allowable identifications. At the same time, in the case of a substitution carried out, as in axiom (4) below, in an appropriate domain of abstraction of identification, “within” this domain the actual identity of objects precisely coincides with identity in the logical sense.

Because of the importance of the concept of identity, the need arose for specific theories of identity. The method most commonly used in constructing these theories is the axiomatic method. For example, all or some of the following axioms may be applicable:

(1) x = x

(2) x = y ⊃ y = x

(3) x = y & y = z ⊃ x = z

(4) A(x)⊃(x = y ⊃ A(y))

Here A(x) is an arbitrary predicate containing x free and free for v, while A(x) and A(y) differ only in their occurrences (even if only one) of x and y.

Axiom (1) postulates reflexivity as a property of identity. In traditional logic, reflexivity was considered the only logical law of identity—a law to which axioms (2) and (3) were customarily added as “nonlogical postulates” (in arithmetic, algebra, and geometry). Axiom (1) may be regarded as epistemologically justified, inasmuch as it is sui generis a logical expression of individuation. It is such individuation, in turn, that forms the basis for the experiential “givenness” of objects, or makes it possible for objects to be recognized: in order to speak of an object as of a given, it is necessary to isolate it in some fashion, to distinguish it from other objects, and to maintain this distinction at all times. In this sense identity, based on axiom (1), is the particular relation of “self-identity” in which each object is bound only to itself, and not to any other object.

Axiom (2) postulates symmetry as a property of identity. According to this axiom, the result of any identification is independent of the order in pairs of objects being identified. This axiom is also justified to some extent by experience. For example, when a weight and a piece of goods are placed on weighing scales, their respective position—on the left or on the right—is different for the customer and the salesperson facing each other, but the result—in this case, equal weight—is the same for both.

Axioms (1) and (2) jointly are an abstract expression of the principle of the identity of indiscernibles—a theory in which the notion of “one and the same” object is based on the factual non-observability of differences. Essentially, this theory depends on the criteria of differentiation—the means, or instruments, that distinguish one object from another; it depends, in the final analysis, on the abstraction of indiscernibility. Inasmuch as there is an inherent dependence in practice on the “threshold of differentiation, ” the notion of identity that is satisfied by axioms (1) and (2) is the only natural result that can be obtained in any test.

Axiom (3) postulates the transitiveness of identity. It states that the composition of the identity relation with itself is again the identity relation, and it is the first nontrivial assertion concerning the identity of objects. The transitiveness of identity is either an “idealization of experience” under conditions of “decreasing accuracy” or an abstraction that expands upon experience and “creates” a new meaning of identity, other than indiscernibility. Indiscernibility is guaranteed by identity only in the domain of abstraction of indiscernibility, whereas the latter is not linked to fulfillment of axiom (3). Axioms (1), (2), and (3) jointly are an abstract expression of the theory of identity as an equivalence.

Axiom (4) postulates as a necessary condition for the identity of objects that their properties coincide. From a logical point of view, this axiom is self-evident: all the properties of an object belong to “one and the same” object. This axiom, however, is not a trivial one, inasmuch as the idea of “one and the same” object must inevitably rest on assumptions or abstractions of a particular kind. It is impossible to verify this axiom “in general”—that is, for all conceivable properties; it can only be verified in certain fixed domains of the abstractions of identification or indiscernibility. This is precisely the way in which this axiom is used in practice: objects are compared and identified not with respect to all conceivable properties, but only with respect to the fundamental (primitive) properties of a theory in which one wants to have the notion of “one and the same” object based upon these properties and on axiom (4). In such instances the scheme of axiom (4) is replaced by a finite list of “intensional” axioms of identity of the same form. For example, the Zermelo-Fraenkel axiomatic set theory substitutes the axioms

(4.1) z ∊ x ⊃ (x = y ⊃ z ∊ y)

(4.2) x ∊ z ⊃ (x = y ⊃ y ∊ z)

Assuming that the universe contains only sets, these axioms determine the domain of abstraction in the identification of sets by “the sets that belong to them” and by “the sets to which they belong, ” with the mandatory addition of axioms 1–3, determining identity as equivalence.

Axioms 1–4 belong to the category of the laws of identity. By means of the rules of logic, one can deduce from them many other laws, unknown in premathematical logic. The difference between the logical and epistemological, or philosophical, aspects of identity is not important in the context of general or abstract formulations of the laws of identity. The difference, however, is substantive when these laws are used to describe real states. In defining the concept of “one and the same” object, the axiomatics of identity necessarily influence the formation of the universe “within” the corresponding axiomatic theory.