Definitions

# Identity of indiscernibles

The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical (are one and the same entity), if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa. A related principle is the indiscernibility of identicals, discussed below.

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.

Philosophers have to come to recognise, however, that it is important to exclude certain predicates - or purported predicates - from this principle. This is necessary to avoid either triviality or contradiction. For example - as detailed below - the predicate which denotes whether an object is equal to x (often considered a valid predicate), and purported predicates denoting what someone believes about an object (which some philosophers have considered to be valid predicates), may both be excluded. As a consequence, there are a few different versions of the principle in the philosophical literature, of varying logical strength - and some of them are termed "the strong principle" or "the weak principle" by particular authors, in order to distinguish between them.

Associated with this principle is also the question as to whether it is a logical principle, or merely an empirical principle.

## Identity and indiscernibility

There are two principles here that must be distinguished (two equivalent versions of each are given in the language of the predicate calculus). Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic.

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.
• :$forall x forall y\left[x=y rightarrow forall P\left(Px leftrightarrow Py\right)\right]$
• For any x and y, if x and y differ with respect to some property, then x is non-identical to y.
• :$forall x forall y\left[neg forall P\left(Px leftrightarrow Py\right) rightarrow x neq y\right]$
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.
• :$forall x forall y\left[forall P\left(Px leftrightarrow Py\right) rightarrow x=y\right]$
• For any x and y, if x is non-identical to y, then x and y differ with respect to some property.
• :$forall x forall y \left[x neq y rightarrow neg forall P\left(Px leftrightarrow Py\right)\right]$

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).

The above formulations are not satisfactory, however: the second principle should be read as having an implicit side-condition excluding any predicates which are equivalent (in some sense) to any of the following:

1. "is identical to x"
2. "is identical to y"
3. "is not identical to x"
4. "is not identical to y"

If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology: if x is non-identical to y, then there will always be a putative "property" which distinguishes them, namely "being identical to x".

On the other hand, it is incorrect to exclude all predicates which are materially equivalent (i.e. contingently equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two non-identical objects, because all distinguishing predicates are materially equivalent to at least one of the four given above (in fact, they are each materially equivalent to two of them), the two non-identical objects are identical - which is a contradiction.

## Critique

### Symmetric universe

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.

## Indiscernibility of identicals

As stated above, the principle of indiscernibility of identicals - that if two objects are in fact one and the same, they have all the same properties - is mostly uncontroversial. However, one famous application of the indiscernibility of identicals 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 argument based on a secret identity:

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. Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact.
3. Lois Lane thinks that Clark Kent cannot fly.
4. Lois Lane thinks that Superman can fly.
5. Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
6. Therefore, Superman is not identical to Clark Kent.
7. Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong. Either:
• Leibniz's law is wrong; or else
• A person's knowledge about x is not a predicate of x, thus undermining Descartes' argument.

## References

• Lecture notes of Kevin Falvey / UCSB