In philosophy, identity, from Latin: identitas (“sameness”), is the relation each thing bears just to itself.[1][2] The notion of identity gives rise to many philosophical problems, including the identity of indiscernibles (if x and y share all their properties, are they one and the same thing?), and questions about change and personal identity over time (what has to be the case for a person x at one time and a person y at a later time to be one and the same person?).
It is important to distinguish the philosophical concept of identity from the more well-known notion of identity in use in psychology and the social sciences. The philosophical concept concerns a relation, specifically, a relation that x and y stand in just in case they are one and the same thing, or identical to each other (i.e. just in case x = y). The sociological notion of identity, by contrast, has to do with a person's self-conception, social presentation, and more generally, the aspects of a person that make them unique, or qualitatively different from others (e.g. cultural identity, gender identity, national Identity, online identity and processes of identity formation.)
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In logic, the identity relation (also called "equality") is normally defined as the binary relation that holds only between a thing and itself. That is, identity is the two-place predicate, "=", such that for all x and y, "x = y" is true if x is the same thing as y. Identity is transitive, symmetric, and reflexive. It is an axiom of most normal modal logics that for all x and y, if x = y then necessarily y = x. That is, identity does not hold contingently, but of necessity.
Put in these terms (above), it can seem rather trivial, but the point, of course, is that the same number, object, or person, can have different "guises", and it is far from trivial if two or more guises refer to the same same number, object, or person. For example, 1+1=2 because '1+1' and '2' are different "guises" for the same number. Similarly, for personal identity over time, "me today" and "me yesterday" are different guises for the same person.
Metaphysicians, and sometimes philosophers of language and mind, ask other questions:
The Law of identity originates from classical antiquity. The modern formulation of identity 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. Hegel argued that things are inherently self-contradictory and that the notion of something being self-identical only made sense if it were not also not-identical or different from itself and did not also imply the latter. In Hegel's words, "Identity is the identity of identity and non-identity." More recent metaphysicians have discussed trans-world identity -- the notion that there can be the same object in different possible worlds. An alternative to trans-world identity is the counterpart relation in Counterpart theory. It is a similarity relation that rejects trans-world individuals and instead defends an objects counterpart - the most similar object.
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 hand.
Alternatively, a and b can be said to be numerically identical if a and b are one and the same thing, that is, if a 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). This relationship is expressed in mathematics with the "=" symbol, e.g., a = b, or Clark Kent = Superman.
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