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# Identity

In philosophy, it is important to distinguish between two senses of identity, qualitative identity and numerical identity.

Arbitrary objects a and b can be said to be qualitatively identical if a and b are duplicates, that is, if a and b are exactly similar in all respects, that is, if a and b have all qualitative properties in common. Examples of this might be two wine glasses made in the same wine glass factory on the same production line (at least, for a relaxed standard of exact similarity), or a carbon atom in one's left hand and a carbon atom in one's right shoulder (perhaps true even for the most strict standard of exact similarity).

a and b can be said to be numerically identical if a and b are one and the same thing, that is, if a just is b, that is, if there is only one thing variously called 'a' and 'b'. For example, Clark Kent is numerically identical with Superman in the sense that there is only one person (who happens to wear different clothes at different times).

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## Logic

In logic, the identity relation is normally, (by definition), the transitive, symmetric, and reflexive relation that holds only between a thing and itself. That is, identity is the two-place predicate, "=", such that for all x, y, "x = y" is true iff x is y.

More usefully, it can be expressed formally in second-order logic or in set theory: For all objects x, y, if for all properties F, Fx iff Fy, then x = y.

It is an axiom of most normal modal logics that for all x, if x = x then necessarily x = x.

(These definitions are of course inapplicable in some area of quantified logic, such as fuzzy logic and fuzzy set theory , and with respect to vague objects .)

## Metaphysics

Metaphysicians, and sometimes philosophers of language and mind, ask other questions:

• What does it mean for an object to be the same as itself?
• If x and y are identical (are the same thing), must they always be identical? Are they necessarily identical?
• What does it mean for an object to be the same, if it changes over time? (Is applet the same as applet+1?)
• If an object's parts are entirely replaced over time, as in the Ship of Theseus example, in what way is it the same?

A traditional view is that of Gottfried Leibniz, who held that x is the same as y if and only if every predicate true of x is true of y as well.

Leibniz's ideas have taken root in the philosophy of mathematics, where they have influenced the development of the predicate calculus as Leibniz's law. Mathematicians sometimes distinguish identity from equality. More mundanely, an identity in mathematics may be an equation that holds true for all values of a variable.

More recent metaphysicians have discussed trans-world identity -- the notion that there can be the same object in different possible worlds.

## Computer programming

In object-oriented programming, object identity is a mechanism for distinguishing different objects from each other. This is based on the philosophical concept of identity, but applied to data structures.

In programs, one frequently may have several variables (or pointers) which refer to the same underlying data structure. An identity predicate allows one to ask whether two variables refer to the same thing. In many languages, identity can be determined more efficiently than equality since the former involves simply a pointer comparison while the latter must traverse data structures.

## Digital identity

The identity of a person, group, thing, process or any other entity is often expressed through a digital identifier within a certain context. For example, a Social Security Number is a digital identity that identifies a person within the context of the US social security administration and the internal revenue service. A library-assigned accession number is a digital identity for a particular book within the context of the library that owns the book.

Contrast with an analog identity, such as a fingerprint.

Recently, various efforts have started to define universal digital identities, i.e. digital identities whose context is global. URLs are an example for universal digital identity for web pages.

A digital identity is often used jointly with one or more credentials that make (credible) assertions about an entity and a digital identity claimed by the entity. For example, a credit card number is a digital identity which is typically used in connection with a credit card slip and the actual credit card (both are credentials) in order to prevent or reduce financial fraud.

## Cultural identity

Cultural identity is the (feeling of) identity of a group or culture, or of an individual as far as she/he is influenced by her/his belonging to a group or culture. Common habits, characteristics, ideas may be clear markers of a shared cultural identity, but essentially it is determined by difference: we feel we belong to a group, and a group defines itself as a group, by noticing and highlighting differences with other groups and cultures.