Identity

Two notions of identity have to be distinguished: qualitative identity and numerical identity. Qualitative identity can be defined in a relatively simple way: two objects are qualitatively identical if and only if they have all properties in common. Formally, we can write this as follows: x is qualitatively identical with y iff for all properties P, x has P if and only if y has P. It remains to be decided what is the range of the property variable P, but let's put this aside for the moment. However, the notion of numerical identity presents a challenge. It is not trivial to come up with a direct definition of this concept. One suggestion may be to characterize numerical identity as the relation that holds between each object and itself, and only itself. But this definition only appears to be correct. In fact it is not, since the condition that every object is identical only to itself is in fact empty, as it can be presented equivalently as follows: if y is distinct from x, then x is not identical with y. But this last expression is tautologous, hence does not restrict the notion of identity in any way. Consequently, we are left with the characteristic that identity holds between every object and itself, and this is definitely too broad.


Another possible definitional characteristic of numerical identity is similarly bound to fail. Suppose that we stipulate that for x to be numerically identical with y there has to be exactly one object that is both x and y. But the expression "exactly one object" contains hidden reference to the notion of identity. The standard way of interpreting the expression "There is exactly one x such that Px" is "There is an x such that Px and for all y, if Py, then y is identical to x". The proposed definition of identity turns out to be circular.


It is commonly accepted that the relation of numerical identity should satisfy the following conditions:


1. Reflexivity. For all x, x is identical to x
2. Symmetricity. For all x, y, if x is identical to y, y is identical to x
3. Transitivity. For all x, y, z, if x is identical to y, and y is identical to z, then x is identical to z


But obviously not all relations satisfying 1-3 are identities. The conditions 1-3 define what is called equivalence relations. However, equivalence relations are closely connected with identity. It turns out that each equivalence relation can be turned into numerical identity if we change appropriately the domain of the relation. To give an example, let us consider the relation of being parallel defined on the set of all straight lines on a plane. This relation is clearly not identity, but if we introduce the abstract notion of a direction (meant either as the property common to all mutually parallel lines, or as their set), then it can be claimed that this relation is reducible to identity when defined on the set of all directions (two "parallel" directions are actually one and the same direction).


In order to obtain the relation of identity, we have to add to 1-3 the following condition:


4. Identity is the smallest relation satisfying conditions 1-3


where the condition 4 is understood in such a way that the relation of identity must be included in any relation satisfying 1-3.