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. 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. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's law. It is considered to be one of his great metaphysical principles, the other being the principle of noncontradiction and the principle of sufficient reason (famously been used in his disputes with Newton and Clarke in the Leibniz–Clarke correspondence). 
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.
Willard Van Orman Quine thought that the failure of substitution 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. 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.
The identity of indiscernibles has been used to motivate notions of noncontextuality within quantum mechanics.
Associated with this principle is also the question as to whether it is a logical principle, or merely an empirical principle.
Identity and indiscernibility.
Both identity and indiscernibility are expressed by the word "same". "Identity" is about "numerical sameness", it is expressed by the equality sign ("="). It is the relation each object bears only to itself. "Indiscernibility", on the other hand, concerns "qualitative sameness": two objects are indiscernible if they have all their properties in common. Formally, this can be expressed as "formula_1". The two senses of "sameness" are linked by two principles: the principle of "indiscernibility of identicals" and the principle of "identity of indiscernibles". The principle of "indiscernibility of identicals" is uncontroversial and states that if two entities are identical with each other then they have the same properties. The principle of "identity of indiscernibles", on the other hand, is more controversial in making the converse claim that if two entities have the same properties then they must be identical. This entails that "no two distinct things exactly resemble each other". Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic (are nonfirstorderizable). Taken together, they are sometimes referred to as "Leibniz's law". Formally, the two principles can be expressed in the following way:
Principle 1 is taken to be a logical truth and (for the most part) uncontroversial. Principle 2, on the other hand, is controversial; Max Black famously argued against it.
If all such predicates ∀F are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology: if "x" is non-identical to "y", then there will always be a putative "property F" 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 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.
The equality relation expressed by the sign "=" is an equivalence relation in being reflexive (everything is equal to itself), symmetric (if "x" is equal to "y" then "y" is equal to "x") and transitive (if "x" is equal to "y" and "y" is equal to "z" then "x" is equal to "z"). The principle of "indiscernibility of identicals" together with the "axiom of reflexivity" can be used to define the equality relation. Other properties of the equality relation, like "symmetry" and "transitivity", follow from these two axioms and therefore no additional axioms are needed to account for them.
Indiscernibility and conceptions of properties.
Indiscernibility is usually defined in terms of shared properties: two objects are indiscernible if they have all their properties in common. The plausibility and strength of the principle of identity of indiscernibles depend on the conception of properties used to define indiscernibility.
One important distinction in this regard is between "pure" and "impure" properties. "Impure properties" are properties that, unlike "pure properties", involve reference to a particular substance in their definition. So, for example, "being a wife" is a pure property while "being the wife of Socrates" is an impure property due to the reference to the particular "Socrates". Sometimes, the terms "qualitative" and "non-qualitative" are used instead of "pure" and "impure". Discernibility is usually defined in terms of pure properties only. The reason for this is that taking impure properties into consideration would result in the principle being trivially true since any entity has the impure property of being identical to itself, which it does not share with any other entity.
Another important distinction concerns the difference between intrinsic and extrinsic properties. A property is "extrinsic" to an object if having this property depends on other objects (with or without reference to particular objects), otherwise it is "intrinsic". For example, the property of "being an aunt" is extrinsic while the property of "having a mass of 60 kg" is intrinsic. Defining indiscernibility in terms of only "intrinsic pure" properties results in the strongest version of the "identity of indiscernibles". According to this version, it would be impossible for two distinct but "intrinsically identical" books to lie next to each other on a table. On a weaker version, taking also "extrinsic pure" properties into consideration, this case is possible since the two books are "discernible" through their "extrinsic properties".
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 provide 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.
Black argues that even relational properties (properties specifying distances between objects in space-time) fail to distinguish two identical objects in a symmetrical universe. Per his argument, two objects are, and will remain, equidistant from the universe's plane 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. A response may be that the argument in the "Meditations on First Philosophy" is that the inability of Descartes to doubt the existence of his mind is part of his mind's essence. One may then argue that identical things should have identical essences.
Numerous counterexamples are given to debunk Descartes' reasoning via "reductio ad absurdum", such as the following argument based on a secret identity: