The Law of Identity

The classical law of identity is commonly interpreted as ∀x(x=x), meaning that everything is identical to itself (Britannica). I argue that this formula does not adequately express the principle as described by Aristotle, that everything is identical Only to itself, because it presupposes the law of identity as one of the theorems of first order logic (FOL). Since the law of identity is fundamental - it is an intrinsic property of meaning or logical sense - we cannot escape this circularity, but we can express the law more explicitly, as if the symbol of identity (=) did not presuppose it.

Aristotle has famously argued that it is impossible to think of anything if we do not think of something definite (Aristotle 1984, 1006b). “To single X out is to isolate X in experience; to determine or fix upon X in particular by drawing spatio-temporal boundaries and distinguishing it in its environment from other things and unlike kinds...” (Wiggins 2001, 6) If we are able to identify two things (a, b) or two occurrences of a thing, there must be some difference between them (Leibniz’s Law of the Identity of Indiscernibles), otherwise it would make no sense to say that there are two things, or two occurrences of a thing, but one thing.

It is sufficient for two things to be distinguished only by modal, dispositional or counterfactual properties. If, for example, two or more things were at the same time and in every respect identical, except their spatial location, this could only mean identity of a ‘kind’, in which case only the relevant kind would have to be identical ‘only to itself’. Two individuals (a,b) belong to the same kind iff “a has to b the relation of identity as restricted to things that f” (Wiggins 2001, 17). Another way, (the same) f is a property or part of both a and b, but a is not absolutely identical to b. This is affirmed by Aristotle as a case of equivalence of a and b with respect to all properties except absolute identity:

“When A belongs to the whole of B and to C and is affirmed of nothing else, and B also belongs to every C, it is necessary that A and B should be convertible; for since A is said of B and C only, and B is affirmed both of itself and of C, it is clear that B will be said of everything of which A is said, except A itself.” (Aristotle 1984, 68a)

A possible objection to this thesis is that a single particle x could hypothetically occupy two spatial locations simultaneously. I contend that by virtue of the assertion that x is a single particle (or the same particle) we would imply that it is in fact a simple extended object - an object that occupies a range of locations - therefore a singular identity. Another way, we cannot consistently assert existence of a single particle in two locations AND also claim that these are two indiscernible particles; we have to commit to the existence of either One particle, in which case it just occupies two locations, or two particles in different locations. Neither of these cases violates the principe of the identity of indiscernibles.

It follows that for anything to have identity it must have a unique identity (discernibility), otherwise it would be unclear which of the objects that ambiguously share an identity is being referred to. This is not equivalent to the case where multiple objects are unambiguously referred to under some collective identity, which must in turn be unique as a group, a composite or a kind. The condition of discernibility is not explicitly captured by ∀x(x=x). “[Self-identity] is certainly a relation formally or logically speaking, but it also holds trivially, it's trivially true of everything...” (Strawson 2013)

In contrast, G.E. Moore (1901) argued that the common definition of the law of identity is not trivial but, rather, implicitly contradictory:

“When we say, ‘This is identical with itself’, the truth of which we are thinking seems to belong to the class of truths of which the general form is, ‘This is identical with that’, and it seems as if in all such cases ‘this’ and ‘that’ must have some difference from one another, and therefore that, in this case, the thing must be different from itself in order to be identical.”

Moore’s and Strawson’s objections do not arise once the condition of indiscernibility is taken into account, whereby the meta-language form ‘this=that’ (an equivocation between the identity of different linguistic terms and what they are intended to identify) is transformed into a strict object-language form ‘this=this’. In order to accomplish this result it is necessary to declare uniqueness and therefore singularity of the term in question, which is just what Moore (Ibid.) was concerned about: “The Law of Identity asserts of everything that it belongs to a certain class: let us say, the class of subjects... We want to say not only that it is a subject like other things, but which subject it is.”

Consider the following example. There is a reservoir containing 100 tennis balls. A variable (x) ranges over 1000 cases of a single ball being drawn. After the draw is made, the self-identity of the selected ball is noted and the ball is returned to the reservoir. Since the number of choices greatly exceeds the number of physical balls to choose from, we are likely to draw the same ball more than once, but when this occurs the formula ∀x(x=x) does not explicitly relate the present selection to any previous one; it is explicit only insofar the presently selected ball is identical to itself. In order to express the principle of identity explicitly, as defined by Aristotle, we must include a feature which would tell us that no matter how many times we choose the same ball, irrespective of our capacity to empirically determine that a particular ball is selected for the second time, we can be sure without constructing further proofs that there is only one ball in existence which is identical to this one, and it is just this ball.

The formula ∀x(x=x) tells us explicitly that ‘everything is identical to itself’ which obviously implies that ‘nothing is not-identical to itself’, but it is neither explicit nor obvious that ‘nothing is identical to not-itself’: ¬∃x(¬x=x). Uniqueness of identity is only implied by the identity relation (=), but this is also what the formula is meant to express. The fact that there is no scenario in which the sentence ‘there is only one ball in existence that is identical to this one, and it is just this ball’ is false has no bearing on whether the symbolic expression captures that fact, or whether that fact is explicit and obvious enough not to require further proofs.

The law of identity can be expressed more explicitly as ∀x∃!y(y=x), which translates ‘among all things (x) there is only one thing that is identical to any particular thing’, or simply ‘everything is identical Only to itself’. We can prove this sentence by demonstrating that its negation implies contradiction:

1) ¬∀x∃!y(y=x) → ∃x¬∃!y(y=x)...

2) → ∃x¬∃y(y=x) ∨ ∃x∃y∃z(x=y.x=z.¬y=z)

3) ∃x¬∃y(y=x) → ¬∃x(x=x) → ∀x(¬x=x) → ⊥

4) ∃x∃y∃z(x=y.x=z.¬y=z) → ∃x(¬x=x) → ⊥

5) ∴ ¬∀x∃!y(y=x) → ⊥

Since FOL presupposes the law of non-contradiction, which implies that ‘nothing is identical to not-itself’ ¬∃x(¬x=x), then uniqueness of identity is already one of the theorems of FOL, implied by the identity symbol (=). ∀x(x=x) is therefore logically equivalent to ∀x∃!y(y=x) but it expresses uniqueness of identity only implicitly whereas ∀x∃!y(y=x) does so explicitly and as such is a more faithful statement of the law of identity as formulated by Aristotle.

An alternative way of expressing the law of identity: ∀x(X={x})∃!y(y∈X.y=x)

• Aristotle. The Complete Works of Aristotle. Princeton: Princeton University Press, 1984.

• Moore, G. E. Identity. Proceedings of the Aristotelian Society, 1901.

• Strawson, Galen. 'Self-intimation'. Phenomenology and the Cognitive Sciences, 2013.

• Wiggins, David. Sameness and Substance Renewed. Cambridge University Press, 2001.