Identity of indiscernibles

Both identity and indiscernibility are expressed by the word "same".[2][3] Identity is about numerical sameness, it is expressed by the equality sign ("="). It is the relation each object bears only to itself.[4] Indiscernibility, on the other hand, concerns qualitative sameness: two objects are indiscernible if they have all their properties in common.[1] Formally, this can be expressed as "${\displaystyle \forall F(Fx\leftrightarrow Fy)}$". 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.[3] 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.[3] This entails that "no two distinct things exactly resemble each other".[1] Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic (are nonfirstorderizable). Taken together, they are sometimes referred to as Leibniz's law. Formally, the two principles can be expressed in the following way:

1. The indiscernibility of identicals: ${\displaystyle \forall x\,\forall y\,[x=y\rightarrow \forall F(Fx\leftrightarrow Fy)]}$

   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.

2. The identity of indiscernibles: ${\displaystyle \forall x\,\forall y\,[\forall F(Fx\leftrightarrow Fy)\rightarrow x=y]}$

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

Principle 1 is generally regarded as an a priori logical truth.[1] Principle 2, on the other hand, is controversial; Max Black famously argued against it.[5]

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 (${\displaystyle \rightarrow }$).[6]

Indiscernibility is usually defined in terms of shared properties: two objects are indiscernible if they have all their properties in common.[7] The plausibility and strength of the principle of identity of indiscernibles depend on the conception of properties used to define indiscernibility.[7][8]

One important distinction in this regard is between pure and impure properties. Impure properties are properties that, unlike pure properties, involve reference to a particular substance in their definition.[7] So, for example, being a wife is a pure property while being the wife of Socrates is an impure property due to the reference to the particular "Socrates".[9] Sometimes, the terms qualitative and non-qualitative are used instead of pure and impure.[10] Discernibility is usually defined in terms of pure properties only. The reason for this is that taking impure properties into consideration would result in the principle being trivially true since any entity has the impure property of being identical to itself, which it does not share with any other entity.[7][8]

Another important distinction concerns the difference between intrinsic and extrinsic properties.[8] A property is extrinsic to an object if having this property depends on other objects (with or without reference to particular objects), otherwise it is intrinsic. For example, the property of being an aunt is extrinsic while the property of having a mass of 60 kg is intrinsic.[11][12] If the identity of indiscernibles is defined only in terms of intrinsic pure properties, one cannot regard two books lying on a table as distinct when they are intrinsically identical. But if extrinsic and impure properties are also taken into consideration, the same books become distinct so long as they are discernible through the latter properties.[7][8]

• 1st axiom of a metric
• Disquotational principle – Philosophical assertion about rational thought
• Duck test – Classification based on observable evidence
• Indistinguishable particles – Concept in quantum mechanics of perfectly substitutable particlesPages displaying short descriptions of redirect targets, a similar idea in quantum mechanics
• Identity (philosophy) – Relation each thing bears to itself alone
• Masked-man fallacy – Formal fallacy when one makes an illicit use of Leibniz's law in an argument, a fallacious use of this principle
• Ship of Theseus – Thought experiment about identity over time
• Structural type system – Class of type systems, a similar idea in computer science
• First-order logic § Equality and its axioms

• Leibniz's Law
• Stanford Encyclopedia of Philosophy entry