Identity of indiscernibles

Leibniz's Law can be expressed symbolically as ${\displaystyle \forall x\,\forall y\,[x=y\rightarrow \forall F\,(Fx\leftrightarrow Fy)]}$, which may be read as "for every ${\displaystyle x}$ and for every ${\displaystyle y}$, if ${\displaystyle x}$ is identical to ${\displaystyle y}$, then every property ${\displaystyle F}$ that is possessed by ${\displaystyle x}$ is also possessed by ${\displaystyle y}$, and every property ${\displaystyle F}$ that is possessed by ${\displaystyle y}$ is also possessed by ${\displaystyle x}$" (this is the indiscernibility of identicals), and conversely as ${\displaystyle \forall x\,\forall y\,[\forall F(Fx\leftrightarrow Fy)\rightarrow x=y]}$, which may be read as "for every ${\displaystyle x}$ and for every ${\displaystyle y}$, if every property ${\displaystyle F}$ that is possessed by ${\displaystyle x}$ is also possessed by ${\displaystyle y}$, and every property ${\displaystyle F}$ that is possessed by ${\displaystyle y}$ is also possessed by ${\displaystyle x}$, then ${\displaystyle x}$ is identical to ${\displaystyle y}$" (this is the identity of indiscernibles).

"${\displaystyle =}$" in Leibniz's Law means "quantitative sameness", not merely qualitative sameness. "Identical" is not merely of equal value, or equivalent, or isomorphic, but rather is "${\displaystyle x}$ is the same object as ${\displaystyle y}$".

There are two principles here that must be distinguished (equivalent versions of each are given in the language of the predicate calculus).[1] Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic (are nonfirstorderizable).

1. The indiscernibility of identicals
   • 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.

     ${\displaystyle \forall x\,\forall y\,[x=y\rightarrow \forall F(Fx\leftrightarrow Fy)]}$

2. The identity of indiscernibles
   • For any ${\displaystyle x}$ and ${\displaystyle y}$, if ${\displaystyle x}$ and ${\displaystyle y}$ have all the same properties, then ${\displaystyle x}$ is identical to ${\displaystyle y}$.

     ${\displaystyle \forall x\,\forall y\,[\forall F(Fx\leftrightarrow Fy)\rightarrow x=y]}$

Principle 1 doesn't entail reflexivity of = (or any other relation R substituted for it), but both properties together entail symmetry and transitivity (see proof box). Therefore, Principle 1 and reflexivity is sometimes used as a (second-order) axiomatization for the equality relation.

Principle 1 is taken to be a logical truth and (for the most part) uncontroversial.[1] Principle 2, on the other hand, is controversial; Max Black famously argued against it.[4]

The above formulations are not satisfactory, however: the second principle should be read as having an implicit side-condition excluding any predicates that are equivalent (in some sense) to any of the following:

1. "is identical to x"
2. "is identical to y"
3. "is not identical to x"
4. "is not identical to y"

If all such predicates ∀F are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology: if x is non-identical to y, then there will always be a putative "property F" that distinguishes them, namely "being identical to x".

On the other hand, it is incorrect to exclude all predicates that are materially equivalent (i.e., contingently equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two non-identical objects, because all distinguishing predicates are materially equivalent to at least one of the four given above (in fact, they are each materially equivalent to two of them), the two non-identical objects are identical—which is a contradiction.

• Disquotational principle – Philosophical assertion about rational thought
• Indistinguishable particles, a similar idea in quantum mechanics
• Intensional fallacy
• Identity (philosophy) – Relation each thing bears to itself alone
• Masked-man fallacy, a fallacious use of this principle
• Ship of Theseus – Philosophical question
• Structural type system, a similar idea in computer science
• First-order logic § Equality and its axioms

• Leibniz's Law
• Stanford Encyclopedia of Philosophy entry