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.

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[edit] Symbolic expression

In the language of the predicate calculus, the indiscernibility of identicals may be written as:

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

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

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

[edit] Identity and indiscernibility

There are two principles here that must be distinguished (two equivalent versions of each are given).[1]

  1. The indiscernibility of identicals
    • 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)]
    • 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
    • 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]
    • 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, below).

[edit] 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. Lois Lane thinks that Clark Kent cannot fly.
  3. Lois Lane thinks that Superman can fly.
  4. Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
  5. Therefore, Superman is not identical to Clark Kent.
  6. Since in proposition (5) we came to an absurd result (we know that Clark Kent is only Superman's alter ego), we conclude that proposition (1) is wrong i.e. Leibniz's law is wrong or that person's knowledge about X is not a predicate of X.

[edit] Critique

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles 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. [2]

[edit] Notes and references

  1. ^ ???
  2. ^ ???
  • Metaphysics an Anthology. eds. J. Kim and E. Sosa, Blackwell Publishing, 1999
  • Lecture notes of Kevin Falvey / UCSB

[edit] See also

[edit] External links

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