Editing Identity (philosophy) (section)

==Metaphysics of identity==<!-- This section is linked from [[Don Quixote]] -->
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Metaphysicians and 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<SUB>''t''</SUB> the same as apple<SUB>''t''+1</SUB>?)
* If an object's parts are entirely replaced over time, as in the [[Ship of Theseus]] example, in what way is it the same?
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 (logic)|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 [[Identity of indiscernibles|Leibniz's law]]. Mathematicians sometimes distinguish identity from [[equality (mathematics)|equality]]. More mundanely, an ''identity'' in [[mathematics]] may be an ''[[equation]]'' that holds true for all values of a [[Variable (mathematics)|variable]]. [[Hegel]] argued that things are inherently self-contradictory<ref>{{Cite journal|last=Siemens|first=Reynold L.|date=1988|title=Hegel and the Law of Identity|url=https://www.jstor.org/stable/20128696|journal=The Review of Metaphysics|volume=42|issue=1|pages=103–127|jstor=20128696|issn=0034-6632}}</ref><ref>{{Cite journal|last=Bole|first=Thomas J.|date=1987|title=Contradiction in Hegel's "Science of Logic"|url=https://www.jstor.org/stable/20128487|journal=The Review of Metaphysics|volume=40|issue=3|pages=515–534|jstor=20128487|issn=0034-6632}}</ref> 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 object's counterpart{{mdash}}the most similar object.

Some philosophers have denied that there is such a relation as identity. Thus [[Wittgenstein|Ludwig Wittgenstein]] writes (''[[Tractatus Logico-Philosophicus|Tractatus]]'' 5.5301): "That identity is not a relation between objects is obvious." At 5.5303 he elaborates: "Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing." [[Bertrand Russell]] had earlier voiced a worry that seems to be motivating Wittgenstein's point (''[[The Principles of Mathematics]]'' §64): "[I]dentity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?" Even before Russell, [[Frege|Gottlob Frege]], at the beginning of "[[On Sense and Reference]]," expressed a worry with regard to identity as a relation: "Equality gives rise to challenging questions which are not altogether easy to answer. Is it a relation?" More recently, [[C. J. F. Williams]]<ref>C.J.F. Williams, ''What is identity?'', Oxford University Press 1989. {{page missing|date=September 2022}}</ref> has suggested that identity should be viewed as a second-order relation, rather than a relation between objects, and [[Kai Wehmeier]]<ref>Kai F. Wehmeier, "How to live without identity—and why," ''Australasian Journal of Philosophy'' 90:4, 2012, pp. 761–777.</ref> has argued that appealing to a binary relation that every object bears to itself, and to no others, is both logically unnecessary and metaphysically suspect.