Identity of indiscernibles

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The identity of indiscernibles is an ontological principle that states that if there is no way of telling two entities apart then they are one and the same entity. That is, entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa.

The principle is also known as Leibniz's law since a form of it is attributed to the German philosopher Gottfried Wilhelm Leibniz.

1 Symbolic expression
2 Controversial applications
3 Critique
4 See also
5 External links
6 Notes and references

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Symbolic expression

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

<math>\forall x \forall y(x=y \leftrightarrow \forall P(Px \leftrightarrow Py))</math>

Note that this is a second-order expression. The principle cannot be expressed in first-order calculi.

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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. Entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa

2. Bill, an elementary-school student who has just learned division, knows the quotient of <math>49 \over 7</math>.

3. Bill has not learned about exponents, so he cannot know what <math>\sqrt{49}</math> equals.

4. Therefore, <math>49 \over 7</math> has a property that <math>\sqrt{49}</math> does not: its quotient is known to Bill.

5. Therefore, <math>49 \over 7</math> does not equal <math>\sqrt{49}</math>.

6. Since in proposition (5) we came to an absurd result, we conclude that proposition (1) is wrong i.e. Leibniz's law is wrong.

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Critique

Max Black has argued 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

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