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 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. 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, and 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 " ∀ F ( F x ↔ F y ) {\displaystyle \forall F(Fx\leftrightarrow Fy)} ". 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". On this principle, for any two different items there must be some discriminable or distinguishable difference, at least in principle, between the two items because were there no difference, they would not be two but rather one. Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic (are nonfirstorderizable). Formally, the two principles can be expressed in the following way:
The indiscernibility of identicals: ∀ x ∀ y [ x = y → ∀ F ( F x ↔ F y ) ] {\displaystyle \forall x\,\forall y\,[x=y\rightarrow \forall F(Fx\leftrightarrow Fy)]}
For any x {\displaystyle x} and y {\displaystyle y} , if x {\displaystyle x} is identical to y {\displaystyle y} , then x {\displaystyle x} and y {\displaystyle y} have all the same properties. The identity of indiscernibles: ∀ x ∀ y [ ∀ F ( F x ↔ F y ) → x = y ] {\displaystyle \forall x\,\forall y\,[\forall F(Fx\leftrightarrow Fy)\rightarrow x=y]}
For any x {\displaystyle x} and y {\displaystyle y} , if x {\displaystyle x} and y {\displaystyle y} have all the same properties, then x {\displaystyle x} is identical to y {\displaystyle y} . The indiscernibility of identicals is usually taken to be uncontroversially true, whereas the identity of indiscernibles is more controversial, having been famously disputed by Max Black. The conjunction of these two principles is sometimes called "Leibniz's Law", although this name has sometimes been used for either of the two other principles, or for other principles. It may be stated as a biconditional:
Biconditional "Leibniz's Law": ∀ x ∀ y [ x = y ↔ ∀ F ( F x ↔ F y ) ] {\displaystyle \forall x\,\forall y\,[x=y\leftrightarrow \forall F(Fx\leftrightarrow Fy)]}
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