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In philosophy, identity is the quality of being "the same as". It is of particular interest to logicians and metaphysicians.

1 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 .)

2 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 apple_t the same as apple_t+1?)
• If an object's parts are entirely replaced over time, as in the Ship of TheseusThe Ship of Theseus is a replacement paradox also known as Theseus's paradox . According to Greek legend as reported by Plutarch, :"The ship wherein Theseus and the youth of Athens returned [from Crete had thirty oars, and was preserved by the Athenians d example, in what way is it the same?

A traditional view is that of Gottfried LeibnizGottfried Wilhelm von Leibniz ( July 1, 1646 in Leipzig November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. Leibniz is credited with the term " function" ( 1694), which he use, 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 mathematicsPhilosophy of mathematics is that branch of philosophy which attempts to answer questions such as: "why is mathematics useful in describing nature?", "in which sense, if any, do mathematical entities such as numbers exist?" and "why and how are mathematic, where they have influenced the development of the predicate calculus as Leibniz's law. Mathematicians sometimes distinguish identity from equalitySee also the disambiguation page title equality. In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. This defines a binary predicate, equality denoted " "; x y iff x and y are equal. Equivalence in th. More mundanely, an identity in mathematicsMathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of "figures and numbers". In the formalist view, it is the investigation of axiomatically defined abstract structures 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.