Identity of indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every 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 snowflakes) 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.
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 Newton and 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 of the principle, the indiscernibility of identicals (described below), which is logically distinct and not to be confused with the identity of indiscernibles.)
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 that 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.^{[1]}
Willard Van Orman Quine thought that the failure of 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.^{[2]} 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.^{[3]}
Associated with this principle is also the question as to whether it is a logical principle, or merely an empirical principle.
Identity and indiscernibility
There are two principles here that must be distinguished (equivalent versions of each are given in the language of the predicate calculus).^{[1]} Note that these are all secondorder expressions. Neither of these principles can be expressed in firstorder logic.
 The indiscernibility of identicals
 For any x and y, if x is identical to y, then x and y have all the same properties.
 For any x and y, if x is identical to y, then x and y have all the same properties.
 The identity of indiscernibles
 For any x and y, if x and y have all the same properties, then x is identical to y.
 For any x and y, if x and y have all the same properties, then x is identical to y.
Proof box  

 
 
 

Principle 1 doesn't entail reflexivity of = (or any other relation R substituted for it), but both properties together entail symmetry and transitivity (see proof box). Therefore, Principle 1 and reflexivity is sometimes used as a (secondorder) axiomatization for the equality relation.
Principle 1 is taken to be a logical truth and (for the most part) uncontroversial.^{[1]} Principle 2, on the other hand, is controversial; Max Black famously argued against it. (see Critique, below).
The above formulations are not satisfactory, however: the second principle should be read as having an implicit sidecondition excluding any predicates that are equivalent (in some sense) to any of the following:
 "is identical to x"
 "is identical to y"
 "is not identical to x"
 "is not identical to y"
If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology: if x is nonidentical to y, then there will always be a putative "property" that distinguishes them, namely "being identical to x".
On the other hand, it is incorrect to exclude all predicates that are materially equivalent (i.e., contingently equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two nonidentical 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 nonidentical objects are identical—which is a contradiction.
Critique
Symmetric universe
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 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.^{[4]}
Black's argument is significant because it shows that even relational properties (properties specifying distances between objects in spacetime) 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.
Indiscernibility of identicals
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 argument), but that he could doubt the existence of his body.
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:
 Entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa.
 Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact.
 Lois Lane thinks that Clark Kent cannot fly.
 Lois Lane thinks that Superman can fly.
 Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
 Therefore, Superman is not identical to Clark Kent.^{[5]}
 Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong. Either:
 Leibniz's law is wrong; or
 A person's knowledge about x is not a predicate of x; or
 The application of Leibniz's law is erroneous; the law is only applicable in cases of monadic, not polyadic, properties; or
 What people think about are not the actual objects themselves; or
 A person is capable of holding conflicting beliefs.
 Any of which will undermine Descartes' argument.^{[3]}
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 (e.g., 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.^{[6]}
See also
 Disquotational principle
 Identical particles, a similar idea in quantum mechanics
 Intensional fallacy
 Indiscernibles
 Identity (philosophy)
 Ship of Theseus
Notes
 1 2 3 Forrest, Peter (Fall 2008). "The Identity of Indiscernibles". In Edward N. Zalta. The Stanford Encyclopedia of Philosophy. Retrieved 20120412.
 ↑ Quine, W. V. O. "Notes on Existence and Necessity." The Journal of Philosophy, Vol. 40, No. 5 (March 4, 1943), pp. 113–127
 1 2 Kripke, Saul. "A Puzzle about Belief". First appeared in, Meaning and Use. ed., A. Margalit. Dordrecht: D. Reidel, 1979. pp. 239–283
 ↑ Metaphysics: An Anthology. eds. J. Kim and E. Sosa, Blackwell Publishing, 1999
 ↑ Pitt, David (October 2001), "Alter Egos and Their Names" (PDF), The Journal of Philosophy, 98 (10): 531–552, 550, doi:10.2307/3649468
 ↑ Carriero, John Peter (2008). Between Two Worlds: A Reading of Descartes's Meditations. Princeton University Press.
References
 Lecture notes of Kevin Falvey / UCSB