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. 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. 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 two great metaphysical principles, the other being the principle of sufficient reason (both having famously been used in his disputes with Newton and Clarke in the LeibnizClarke 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 strengthand 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 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.^{[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]}
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.
Leibniz's Law can be expressed symbolically as (x)(y) [x=y > (F)(Fx Fy)], which may be read as "for every x and for every y, if x is identical to y, then every property F that is possessed by x is also possessed by y, and every property F that is possessed by y is also possessed by x" (this is the indiscernibility of identicals), and conversely as (x)(y) [(F )(Fx Fy) > x=y], which may be read as "for every x and for every y, if every property F that is possessed by x is also possessed by y, and every property F that is possessed by y is also possessed by x, then x is identical to y" (this is the identity of indiscernibles).
"=" in Leibniz's Law means "quantitative sameness", not merely qualitative sameness. "Identical" is not merely of equal value, or equivalent, or isomorphic, but rather is "x is the same object as y".
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 (are nonfirstorderizable).
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.
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:^{[]}
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 nonidentical 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 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 identicalwhich is a contradiction.
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.^{[4]}
Black's argument appears 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. 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.
As stated above, the principle of indiscernibility of identicalsthat if two objects are in fact one and the same, they have all the same propertiesis 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.^{[5]}
Numerous counterexamples are given to debunk Descartes' reasoning via reductio ad absurdum, such as the following argument based on a secret identity: