Identity and Indiscernibility

What logical relations connect qualitative identity (indiscernibility) and numerical identity? First of all, there is Leibniz's law, or the principle of the indiscernibility of identicals, which states that numerical identity implies indiscernibility. More formally, this principle can be written as follows: For all x, y, if x = y, then for all P, if x has P, then y has P. (If you are worried that this formalization can only ensure that all x's properties will be y's properties, note that you can swap x with y and get the conclusion that all y's properties are x's properties as well). Alternatively, we may put this idea like that: if x has a property that y doesn't have, x must be distinct from y. Leibniz's law seems to be rather uncontroversial; however some possible counterexamples to it are conceivable. Let us consider the property of number 8 that it is necessarily greater than 6 (as a matter of mathematical necessity, 8 has to be greater than 6). But 8 is numerically identical to the number of planets in the solar system (excluding Pluto). But is it really necessary that the number of planets is greater than 6? (We will analyse this problem in more detail later in the course.) This example teaches us that perhaps not every proposition true of a given object picks out its property. The expression 'it is necessary that' is an example of an intensional context, which means that the truth of a statement built with its help depends not only on the reference of the terms included in the statement, but also on their meaning (the meaning of 'number 8' is obviously different from the meaning of 'the number of planets in the solar system').

A more controversial principle connecting numerical identity with qualitative identity is the principle of the identity of indiscernibles (PII for short). It states that all indiscernible objects are identical, or more formally that for all x and y, if x and y have the same properties, x is numerically identical with y. Alternatively, PII can be presented as the principle of the discernibility of distinct, i.e. if x is not numerically identical with y, x and y are discernible (there is at least one property that they don't have in common). It is important to notice that the meaning and status of PII depends on how broad the category of properties which can discern individual objects is. For instance, suppose that we agree that for any object a being identical with a is a property of a. In such a case PII becomes trivially satisfied, for if a and b agree with respect to all their properties, b will have to possess the property of being identical with a which settles the issue of their numerical identity. In order to avoid this trivialisation, we should restrict the range of acceptable properties to so-called qualitative properties (properties that do not involve numerical identity, or any form of 'labelling'). On the other hand, it is acceptable to discern objects with the help of relational properties (for instance, being at a given distance from a third object c).

At first sight it looks like PII should be satisfied at least by all material, spatiotemporal objects, for numerically distinct things are discernible by their different space-time locations. This conclusion can be accepted if we assume that material objects are impenetrable, i.e. that they cannot occupy the same space at the same time. In spite of this argument, it turns out that plausible counterexamples to PII can be presented. One of them is the famous Max Black example involving two spheres. Suppose that the universe consists only of two objects which happen to be two perfect spheres made of pure iron of exactly the same dimensions. In such a case the spheres seem to be indiscernible even by their location, because there is no third object in this entirely symmetrical universe which would stand in different spatiotemporal relations to the spheres. Black's counterexample does not show that PII is actually false (obviously our universe is not even similar to this impoverished world), but at least proves that PII can be only contingently true (that the falsity of PII is conceivable).

But a stronger claim can be made that PII is actually false in our world. According to modern physics, subatomic particles do not possess definite trajectories, thus their discernibility in terms of individual locations is not guaranteed. Moreover, quantum physics introduces the so-called symmetrization postulate which puts severe constraints on possible states of many-particles systems of the same type. The postulate requires that the state of many particles be either a symmetric, or an antisymmetric function* (the first case is when the particles in questions are bosons, e.g. photons; the second when they are fermions). From the postulate it follows that all measurable properties of particles of the same type assume the same values (either deterministic or probabilistic). This fact is interpreted by many as a clear sign that PII is violated in the case of elementary particles of the same type.

*) A symmetric function is a function that gets transformed onto itself when we swap its arguments. An antisymmetric function changes its sign under this operation.

Appendix

Recently an argument in favour of the discernibility of quantum particles has been proposed which involves the notion of weak discernibility. Two objects x and y are said to be weakly discernible if there is an irreflexive relation R (a relation that does not hold between an object and itself) which connects these objects. It can be argued that fermions are connected by an irreflexive relation "having opposite spin directions". But it is doubtful whether weak discernibility presents a case of genuine discernibility. It can be easily verified that the weak discernibility of two objects x and y can only guarantee that they can be discerned in the following way: the relation R connects x with y, but does not connect x with x, hence x discerns relationally x from y. But this is the same sort of 'discernibility' as in the case of the property of being identical with x, and therefore should not be accepted.