# Identity of indiscernibles

(Redirected from Leibniz's law)

The identity of indiscernibles, also known as Leibniz's Law, is an ontological principle first forumlated by German philosopher Göttfried Wilhelm Leibniz. The principle states that if there is no way of telling two entities apart then they are one and the same. That is, two entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa. So "if it looks like a duck, walks like a duck, and quacks like a duck, then it is a duck".

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## Symbolic expression

In predicate calculus, the identity of indiscernibles may be written as

(x)(y)(P)(x=y ↔ (Px ↔ Py))

## Controversial applications

One famous application of the identity of indiscernibles was by René Descartes in his Meditations on First Philosophy. Descartes concluded that he could not doubt the existence of himself (the famous cogito ergo sum argument), but that he could doubt the existence of his body. From this he inferred that the person Descartes must not be identical to his body, since one possessed a characteristic that the other did not: namely, it could be known to exist.

This argument is normally rejected by modern philosophers on the grounds that it derives a conclusion about what is true from a premise about what people know. What people know or believe about an entity, they argue, is not really a characteristic of that entity. Numerous counterexamples are given to debunk Descartes' reasoning via reductio ad absurdum, such as the following:

1. Bill, an elementary-school student who has just learned division, knows the quotient of [itex]49 \over 7[itex].
2. Bill has not learned about exponents, so he cannot know what [itex]\sqrt 49[itex] equals.
3. Therefore, [itex]49 \over 7[itex] has a property that [itex]\sqrt 49[itex] does not: its quotient is known to Bill.
4. Therefore, [itex]49 \over 7[itex] does not equal [itex]\sqrt 49[itex].

## Critique

Max Black has arqued against the identity of indiscernibles by counterexample. He claimed that in the symmetric universe where only two symmetrical spheres exist, the two spheres are two distinct objects, even though they have all the properties in common.1

## Notes and references

• [1] Metaphysics an Anthology. eds. J. Kim and E. Sosa, Blacwell Publishing, 1999
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