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| The '''identity of indiscernibles''' is an [[ontology|ontological]] principle which states that there cannot be separate [[object (philosophy)|object]]s or [[entity|entities]] that have all their [[property (philosophy)|properties]] in common. That is, entities ''x'' and ''y'' are identical if every [[predicate (logic)|predicate]] possessed by ''x'' is also possessed by ''y'' and vice versa; to suppose two things indiscernible is to suppose the same thing under two names. It states that no two distinct things (such as [[snowflake]]s) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below.
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| A form of the principle is attributed to the German philosopher [[Gottfried Wilhelm Leibniz]]. It is one of his two great metaphysical principles, the other being the [[principle of sufficient reason]]. Both are famously used in his arguments with [[Isaac Newton|Newton]] and [[Samuel Clarke|Clarke]] in the [[Leibniz–Clarke correspondence]]. Because of its association with Leibniz, the principle is sometimes known as '''Leibniz's law'''. (However, the term "Leibniz's Law" is also commonly used for the [[Converse (logic)|converse]] of the principle, the indiscernibility of identicals (described below), which is logically distinct and not to be confused with the identity of indiscernibles.)
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| Some philosophers have decided, however, that it is important to exclude certain predicates (or purported predicates) from the principle in order to avoid either triviality or contradiction. An example (detailed below) is the predicate which denotes whether an object is equal to ''x'' (often considered a valid predicate). As a consequence, there are a few different versions of the principle in the philosophical literature, of varying logical strength—and some of them are termed "the strong principle" or "the weak principle" by particular authors, in order to distinguish between them.<ref name=Forrest2008>{{cite encyclopedia | last=Forrest | first=Peter | title=The Identity of Indiscernibles | encyclopedia=The Stanford Encyclopedia of Philosophy | date=Fall 2008 |editor=Edward N. Zalta | url=http://plato.stanford.edu/archives/fall2008/entries/identity-indiscernible/ |accessdate=2012-04-12}}</ref>
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| [[Willard Van Orman Quine]] thought that the failure of [[substitution (logic)|substitutivity]] in intensional contexts (e.g., "Sally believes that ''p''" or "It is necessarily the case that ''q''") shows that [[modal logic]] is an impossible project.<ref>Quine, W. V. O. "Notes on Existence and Necessity." ''The Journal of Philosophy'', Vol. 40, No. 5 (March 4, 1943), pp. 113–127</ref> [[Saul Kripke]] holds that this failure may be the result of the use of the [[disquotational principle]] implicit in these proofs, and not a failure of substitutivity as such.<ref name="belief">Kripke, Saul. "A Puzzle about Belief". First appeared in, ''Meaning and Use''. ed., A. Margalit. Dordrecht: D. Reidel, 1979. pp. 239–283</ref>
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| Associated with this principle is also the question as to whether it is a [[logical]] principle, or merely an [[empirical]] principle.
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| ==Identity and indiscernibility==
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| There are two principles here that must be distinguished (equivalent versions of each are given in the language of the predicate calculus).<ref name=Forrest2008 /> Note that these are all [[Second-order logic|second-order]] expressions. Neither of these principles can be expressed in [[first-order logic]].
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| #The indiscernibility of identicals
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| #*For any ''x'' and ''y'', if ''x'' is identical to ''y'', then ''x'' and ''y'' have all the same properties.
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| #*:<math>\forall x \forall y[x=y \rightarrow \forall P(Px \leftrightarrow Py)]</math>
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| #The identity of indiscernibles
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| #*For any ''x'' and ''y'', if ''x'' and ''y'' have all the same properties, then ''x'' is identical to ''y''.
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| #*:<math>\forall x \forall y[\forall P(Px \leftrightarrow Py) \rightarrow x=y]</math>
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| {| align="right" class="wikitable collapsible collapsed"
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| ! colspan=3 | Proof box
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| {|
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| | colspan=3 | Abbreviations:
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| | '''Indsc''': || align=right | ∀''x'',''y''. || (''xRy'' → ∀''P'' (''Px''→''Py''))
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| |-
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| | '''Refl''': || align=right | ∀''x''. || ''xRx''
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| |-
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| | '''Symm''': || align=right | ∀''x'',''y''. || (''xRy'' → ''yRx'')
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| |-
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| | '''Trans''': || align=right | ∀''x'',''y'',''z''. || (''xRy'' ∧ ''yRz'' → ''xRz'')
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| |}
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| |-
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| | colspan=2 |
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| {| class="collapsible collapsed"
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| ! colspan=3 | '''Indsc''' ∧ '''Refl''' → '''Symm'''
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| | colspan=3 | '''Proof:'''
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| | colspan=3 | Given ''x'', ''y'' such that ''xRy'',
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| | colspan=3 | define ''P''(''v'') = ''vRx'', then
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| | || ''xRx'' || by '''Refl'''
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| | → || ''Px'' || by Def. of ''P''
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| |-
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| | → || ''Py'' || by '''Indsc''', since ''xRy''
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| |-
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| | → || ''yRx'' || by Def. of ''P''
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| |}
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| |-
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| | colspan=2 |
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| {| class="collapsible collapsed"
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| ! colspan=3 | '''Indsc''' ∧ '''Refl''' → '''Trans'''
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| | colspan=3 | '''Proof:'''
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| | colspan=3 | Given ''x'', ''y'', ''z'' such that ''xRy'' and ''yRz'',
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| | colspan=3 | define ''P''(''v'') = ''vRz'', then
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| | || ''yRz'' || assumed
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| | → || ''Py'' || by Def. of ''P''
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| | → || ''Px'' || by '''Indsc''', since ''yRx'' (from ''xRy'' and '''Symm''', see above)
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| |-
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| | → || ''xRz'' || by Def. of ''P''
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| |}
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| | colspan=2 |
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| {| class="collapsible collapsed"
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| ! colspan=3 | '''Indsc''' ⇸ '''Refl'''
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| | colspan=3 | '''Proof:'''
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| | colspan=3 | The empty relation satisfies '''Indsc''',
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| | colspan=3 | but not '''Refl'''.
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| |}
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| |}
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| Principle 1 doesn't entail [[reflexive relation|reflexivity]] of = (or any other relation ''R'' substituted for it), but both properties together entail [[Symmetric relation|symmetry]] and [[Transitive relation|transitivity]] (see proof box). Therefore, Principle 1 and reflexivity is sometimes used as a (second-order) [[Axiomatic system|axiomatization]] for the equality relation.
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| Principle 1 is taken to be a [[logical truth]] and (for the most part) uncontroversial.<ref name=Forrest2008 /> Principle 2, on the other hand, is controversial; [[Max Black]] famously argued against it. (see [[#Critique|Critique]], below).
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| The above formulations are not satisfactory, however: the second principle should be read as having an implicit side-condition excluding any predicates which are equivalent (in some sense) to any of the following:
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| #"is identical to ''x''"
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| #"is identical to ''y''"
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| #"is not identical to ''x''"
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| #"is not identical to ''y''"
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| If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a [[tautology (logic)|logical tautology]]: if ''x'' is non-identical to ''y'', then there will always be a putative "property" which distinguishes them, namely "being identical to ''x''".
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| On the other hand, it is incorrect to exclude all predicates which are [[materially equivalent]] (i.e., [[Contingency (philosophy)|contingent]]ly equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two non-identical objects, because all distinguishing predicates are materially equivalent to at least one of the four given above (in fact, they are each materially equivalent to two of them), the two non-identical objects are identical—which is a contradiction.
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| ==Critique==
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| ===Symmetric universe===
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| [[Max Black]] has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provides a [[Model theory|model]] in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common.<ref>''Metaphysics: An Anthology''. eds. J. Kim and E. Sosa, Blackwell Publishing, 1999</ref>
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| Black's argument is significant because it shows that even relational properties (properties specifying distances between objects in space-time) fail to distinguish two identical objects in a symmetrical universe. The two objects are, and will remain, equidistant from the universe's line of symmetry and each other. Even bringing in an external observer to label the two spheres distinctly does not solve the problem, because it violates the symmetry of the universe.
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| ==Indiscernibility of identicals==
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| As stated above, the principle of indiscernibility of identicals—that if two objects are in fact one and the same, they have all the same properties—is mostly uncontroversial. However, one famous application of the indiscernibility of identicals was by [[René Descartes]] in his ''[[Meditations on First Philosophy]]''. Descartes concluded that he could not doubt the existence of himself (the famous ''[[Cogito ergo sum|cogito]]'' argument), but that he ''could'' doubt the existence of his body.
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| This argument is criticized by some modern philosophers on the grounds that it allegedly derives a conclusion about what is true from a premise about what people know. What people know or believe about an entity, they argue, is not really a characteristic of that entity. Numerous counterexamples are given to debunk Descartes' reasoning via ''[[reductio ad absurdum]]'', such as the following argument based on a [[secret identity]]:
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| #Entities ''x'' and ''y'' are identical if and only if any predicate possessed by ''x'' is also possessed by ''y'' and vice versa.
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| #Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact.
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| #[[Lois Lane]] thinks that [[Clark Kent]] cannot fly.
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| #Lois Lane thinks that [[Superman]] can fly.
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| #Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
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| #Therefore, Superman is not identical to Clark Kent.<ref>{{citation | url=http://www.calstatela.edu/faculty/dpitt/Egos.pdf | title=Alter Egos and Their Names | first=David | last= Pitt | journal=The Journal of Philosophy | volume=98 |issue=10 | date=October 2001 | pages=531–552, 550}}</ref>
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| #Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong. Either:
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| #* Leibniz's law is wrong; or
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| #* A person's knowledge about ''x'' is not a predicate of ''x''; or
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| #* The application of Leibniz's law is erroneous; the law is only applicable in cases of monadic, not polyadic, properties; or
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| #* What people think about are not the actual objects themselves; or
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| #* A person is capable of holding conflicting beliefs.
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| :::Any of which will undermine Descartes' argument.<ref name="belief" />
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| A response may be that the argument in the ''[[Meditations on First Philosophy]]'' isn't that Descartes cannot doubt the existence of his mind, but rather that it is beyond doubt, such that no being with understanding could doubt it. This much stronger claim doesn't resort to relational properties, but rather presents monadic properties, as the foundation for the use of Leibniz's law. One could expound an infinite list of relational properties that may appear to undermine Leibniz's law (i.e., Lois Lane loves Clark Kent, but not Superman. etc.) but nonetheless any approach focused on monadic properties will always produce accurate results in support of Descartes' claim.<ref>{{cite book |url=http://books.google.com/books/p/princeton?id=8sN_sdfBpGIC&printsec=frontcover&cd=1&source=gbs_ViewAPI&hl=en#v=onepage&q&f=false | title=Between Two Worlds: A Reading of Descartes's Meditations | first=John Peter | last=Carriero | publisher=Princeton University Press | year=2008}}</ref>
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| ==See also==
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| *[[Disquotational principle]]
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| *[[Indistinguishable particles|Identical particles]], a similar idea in [[quantum mechanics]]
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| *[[Intensional fallacy]]
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| *[[Indiscernibles]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Lecture notes of Kevin Falvey [http://www.philosophy.ucsb.edu/fac_profiles/falvey.html / UCSB]
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| ==External links==
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| * http://www.consciousentities.com/tactics.htm
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| * [http://plato.stanford.edu/entries/identity-indiscernible/ Stanford Encyclopedia of Philosophy entry]
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| {{Gottfried Wilhelm Leibniz}}
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| {{DEFAULTSORT:Identity Of Indiscernibles}}
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| [[Category:Identity|indiscernibles]]
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| [[Category:Ontology]]
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| [[Category:Gottfried Leibniz]]
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| [[Category:Principles]]
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| [[Category:Concepts in logic]]
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| [[Category:Concepts in metaphysics]]
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| [[pl:Identyczność przedmiotów nierozróżnialnych]]
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| [[fi:Huomaamattomien identiteetti]]
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| [[sv:Leibniz lag]]
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