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. It is one of his two great metaphysical principles, the other being the Principle of sufficient reason. Both are famously used in his arguments with Newton and Clarke in The Leibniz-Clarke Correspondence.

Contents

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

Symbolic expression

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

[\forall x \forall y[forall P(Px leftrightarrow Py) rightarrow x=y]]

(For any x and y, if x and y have all the same properties, then x is identical to y.)

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

Identity and Indiscernibility

There are two principles here that must be distinguished (two equivalent versions of each are given).²

1. The Indiscernibility of Identicals

(a) For any x and y, if x is identical to y, then x and y have all the same properties.

[\forall x \forall y[x=y rightarrow forall P(Px leftrightarrow Py)]]

(b) For any x and y, if x and y differ with respect to some property, then x is non-identical to y.

[\forall x \forall y[neg forall P(Px leftrightarrow Py) rightarrow x neq y]]

2. The Identity of Indiscernibles

(a) For any x and y, if x and y have all the same properties, then x is identical to y.

[\forall x \forall y[forall P(Px leftrightarrow Py) rightarrow x=y]]

(b) For any x and y, if x is non-identical to y, then x and y differ with respect to some property.

[\forall x \forall y [x neq y rightarrow neg forall P(Px leftrightarrow Py)]]

Principle 1. is taken to be a logical truth and (for the most part) uncontroversial. Principle 2. is controversial. Max Black famously argued against 2. (see Critique section).

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 [49 \over 7].

3. Bill has not learned about exponents, so he cannot know what [\sqrt] equals.

4. Therefore, [49 \over 7] has a property that [\sqrt] does not: its quotient is known to Bill.

5. Therefore, [49 \over 7] does not equal [\sqrt].

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.

Critique

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that 2. is false, it is sufficient that one provide a model in which there are two distinct (non-identical) things that have all the same properties. 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

External links

http://www.consciousentities.com/tactics.htm
[Stanford Encyclopedia of Philosophy entry]

Notes and references

Metaphysics an Anthology. eds. J. Kim and E. Sosa, Blackwell Publishing, 1999
Lecture notes of Kevin Falvey [/ UCSB]

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