Identity of indiscernibles

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 "${\displaystyle \mathrm{\forall }F(Fx\leftrightarrow Fy)}$". 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: ${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[x=y\to \mathrm{\forall }F(Fx\leftrightarrow Fy)]}$

  For any ${\displaystyle x}$ and ${\displaystyle y}$, if ${\displaystyle x}$ is identical to ${\displaystyle y}$, then ${\displaystyle x}$ and ${\displaystyle y}$ have all the same properties.

• The identity of indiscernibles: ${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[\mathrm{\forall }F(Fx\leftrightarrow Fy)\to x=y]}$

  For any ${\displaystyle x}$ and ${\displaystyle y}$, if ${\displaystyle x}$ and ${\displaystyle y}$ have all the same properties, then ${\displaystyle x}$ is identical to ${\displaystyle y}$.

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": ${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[x=y\leftrightarrow \mathrm{\forall }F(Fx\leftrightarrow Fy)]}$

  For any ${\displaystyle x}$ and ${\displaystyle y}$, it is true that ${\displaystyle x}$ is identical to ${\displaystyle y}$ if, and only if, ${\displaystyle x}$ and ${\displaystyle y}$ have all the same properties.

Some logicians have regarded this principle as essential to identity and equality: Alfred Tarski listed it among the logical axioms governing the notion of identity, and Rudolf Carnap defined the equals sign for identity (=) in terms of this biconditional.

In a universe of two distinct objects A and B, all predicates F are materially equivalent to one of the following properties:

• IsA, the property that holds of A but not of B;
• IsB, the property that holds of B but not of A;
• IsAorB, the property that holds of both A and B;
• IsNotAorB, the property that holds of neither A nor B.

If ∀F applies to all such predicates, then the second principle as formulated above reduces trivially and uncontroversially to a logical tautology. In that case, the objects are distinguished by IsA, IsB, and all predicates that are materially equivalent to either of these. This argument can combinatorially be extended to universes containing any number of distinct objects.

The equality relation expressed by the sign "=" is an equivalence relation in being reflexive (everything is equal to itself), symmetric (if x is equal to y then y is equal to x) and transitive (if x is equal to y and y is equal to z then x is equal to z). The indiscernibility of identicals and identity of indiscernables can jointly be used to define the equality relation. The symmetry and transitivity of equality follow from the first principle, whereas reflexivity follows from the second. Both principles can be combined into a single axiom by using a biconditional operator (${\displaystyle \leftrightarrow }$) in place of material implication (${\displaystyle \to }$).