One part of Leibniz’s law, saying that if what appear to be two or more things have all their properties in common they are identical and so only one thing.
In its widest and weakest form, the properties concerned include relational properties such as spatiotemporal ones and self-identity.
A stronger version limits the properties to non-relational properties (that is, qualities), and would therefore imply that there could not be, for example, two exactly similar ball-bearings.
Even the weaker version faces objections if we envisage two ball-bearings alone in an otherwise empty universe, or in corresponding positions in the two halves of a symmetrical universe: what property would one of them have and the other lack? (To try to distinguish them by their relations to each other would presuppose that we could already distinguish them, and the same holds of the halves of the symmetrical universe.)
Also see: principle of sufficient reason
M J Loux, ed., Universals and Particulars (1970)
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 (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