The identity of indiscernibles is an ontological principle which 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 name '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 which 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 substitutivity in intensional contexts (e.g., "Sally believes that p," "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.
Associated with this principle is also the question as to whether it is a logical principle, or merely an empirical principle.
Read more about Identity Of Indiscernibles: Identity and Indiscernibility, Indiscernibility of Identicals
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