One part of Leibniz’s law, named by Willard Van Orman Quine (1908-2000), the American mathematical logician.
It says that if what appear to be two or more objects are in fact identical, there can be no property held by one and not by the others.
This must be distinguished from the sub-stitutivity of identicals, which says that two names or true descriptions for the same object can always be intersubstituted; this is false for if Cicero and Tully name the same orator we cannot infer from ‘Smith believes that Cicero was an orator’ to ‘Smith believes that Tully was an orator’.
If we make this distinction (which Leibniz didn’t) then the Cicero/Tully example will not be an objection to Leibniz’s law properly stated.
R Cartwright, ‘Identity and Substitutivity’ in M K Munitz, ed., Identity and Individuation (1971); reprinted in R Cartwright, Philosophical Essays (1987)
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.