The identity of indiscernibles
Leibniz's law · indiscernibility of identicals · principle of identity of indiscernibles
Leibniz's principle that no two distinct things can share every property — if they are indiscernible, they are one. Its converse, that identical things share all properties, is uncontroversial. The principle itself is not.
In practice
Two records with identical values in every field are the same record or the data model is missing a field. Which of those it is has to be decided, and a surrogate key decides it by fiat.
The common mistake
Running the two directions together. That identicals are indiscernible is trivially true. That indiscernibles are identical is a substantial metaphysical claim with serious counterexamples, and only the second is disputed.
Two directions, usually conflated, with very different standing.
The indiscernibility of identicals. If A and B are the same thing, they have all the same properties. This is uncontroversial to the point of being definitional — there is only one thing there, so there is only one set of properties.
The identity of indiscernibles. If A and B have all the same properties, they are the same thing. This is Leibniz's principle, and it is a substantial claim about what the world can contain.
The second says there are no perfect duplicates anywhere. Not that we could not tell them apart — that there cannot be two.
Why Leibniz believed it
It follows from his wider system rather than standing alone, which is worth knowing before dismissing it.
Every substance, on his account, is a monad that reflects the entire universe from its own point of view. Two monads reflecting the universe from the same point of view would be the same monad; from different points of view, they differ in their perceptions, and that is a difference in properties.
It also follows from the principle of sufficient reason, which is Leibniz's deepest commitment. If two things were exactly alike, there could be no reason for God to have placed this one here and that one there rather than the reverse. A universe containing indiscernibles would contain a fact with no explanation, and his metaphysics does not allow one.
The counterexample
Max Black, in 1952: imagine a universe containing nothing but two perfect iron spheres, alike in every quality, a mile apart.
The intuition that there are two of them is strong, and nothing distinguishes them. Position does not help — in a universe containing only the spheres, each is truthfully described as "a mile from a sphere exactly like me", and that description is shared.
The available replies show what the principle costs.
Deny the possibility. Such a universe cannot exist. This preserves the principle and treats a case we can apparently describe coherently as incoherent, which requires an argument the defender has to supply.
Count relations as properties. If being a mile from sphere A is a property, the spheres differ. But naming them presupposes they are two, so the argument is circular unless the property can be specified without the names.
Accept haecceities. Each sphere has a primitive thisness, non-qualitative and unshareable — Scotus's answer, which abandons the principle in its interesting form, since thisness is not a property that could be shared and the claim becomes trivial.
The version that survives
Most philosophers now accept the principle restricted to purely qualitative properties and reject it in general, which amounts to saying that particulars are not exhausted by their qualities.
That has consequences well beyond the example. If two things can differ with no qualitative difference, then what makes a thing the one it is cannot be entirely a matter of what it is like, which is the principle of individuation and pulls toward taking particulars as basic rather than constructing them from properties.
It also settles a question about bundle theory. If a thing simply is its properties bundled, then two things with the same properties are the same bundle and Black's case is impossible. So the bundle theorist is committed to Leibniz's principle whether they want to be or not, and Black's spheres are aimed precisely there.
Physics has been enlisted on both sides. Two electrons in the same quantum state are genuinely indistinguishable, in a stronger sense than any classical case — whether that refutes the principle or shows that electrons are not individuals in the relevant sense is exactly the disagreement, and it has not been resolved by knowing more physics.
Concept web
Open the full webQuestions
What is the identity of indiscernibles?
Leibniz's principle that no two distinct things can share all their properties. Its converse, that a thing shares all properties with itself, is trivially true, and only the principle in this direction is disputed.
What is Max Black's objection?
A universe containing only two exactly similar iron spheres a mile apart. They appear to be two things with no differing property. Position does not distinguish them, since each is truthfully described as a mile from a sphere just like itself.
Does quantum physics refute the identity of indiscernibles?
It is contested. Two electrons in the same state are indistinguishable in a stronger sense than any classical case. Whether that refutes the principle or shows electrons are not individuals in the relevant sense is the disagreement itself.