The Inconceivability of the Non-Identity of Indiscernibles

The Identity of Indiscernibles may be expressed as (x)(y)(X)((Xx « Xy) ® (x = y)) — for all x and all y, if x has all of the same properties as y then x is identical to y. It is controversial as a logical law as many maintain that there could exist qualitatively indistinct things that are not identical. It is not to be confused with the Indiscernibility of Identicals, which is less controversial. The intuition against the Identity of Indiscernibles was most notably elaborated by Black¹ with his thought experiment of a universe in which the only things that exist are two molecule-for-molecule duplicate iron spheres at some distance from each other. The conceivability of this scenario, taken as its possibility, demonstrates that there could be two things that are qualitatively identical, even as to the relations between them, and yet are numerically distinct.  Black believes that unless there is some contradiction inherent to his account of the spheres, the Identity of Indiscernibles is shown to be false. In the face of this counter-example, many proponents of the Identity of Indiscernibles retreated from a staunch claim of necessary truth of the principle to the status of a contingent truth: a truth local to this possible world, the actual world. Yet, recent interpretations of the peculiarities of quantum mechanics have apparently provided grounds to refute even a contingent version of the principle, along similar lines to Black’s original counterexample. 

Yet, an approach to the problem for proponents of the Identity of Indiscernibles may be adopted other than questioning the possibility of Black’s universe. Arguably, Black’s thought experiment merely shows that bilocation is possible. We could simply stop with the argument that there is no reason why the same entity could not undergo bilocation – there is no significant difference between there being the same entity at different places and there being the same entity at different times. After all, there seems to be nothing contentious about an entity enduring without qualitative change save for Cambridge relations. However, the clarified intuition to be rebutted, that bilocation is miraculous or absurd, remains strong. Some philosophers also respond to the suggestion of bilocation with, “Doesn’t that make individuals into universals?” This probably depends on how you want to define universals. Yet, another way to look at it is to just say, “Yeah, sure.” That worked for Quine², and he was a proponent of the Identity of Indiscernibles. This still seems too strange for some. 

Here, then, begins a positive argument for bilocation. Kant³ most notably argued that existence is not a real predicate and it adds nothing to our concept of a thing. This was in response to Descartes’ version of the ontological argument for the existence of God. Part of Kant’s argument in support of this stance could be paraphrased as follows. In different circumstances a dog might exist or the same dog might not exist. If we were to allow that existence is a property of the dog, so that to exist is to possess the property of existence or instantiate that property, we are still yet to explain how such a property differentiates the two cases. The non-existent dog will be the non-existence of a dog with all of the same properties as the existing dog, including the property of existence. If the non-existent dog was not the non-existence of a dog with the property of existence then the existing thing referred to initially and the thing said not to exist are different, and we are not talking about the non-existence of the same dog. We annihilate the wrong dog. Kant is assuming the less controversial Indiscernibility of Identicals, here. The underlying intuition is that all things that exist contingently are identical to their non-existence at some other time and/or at some other possible world.

Let us return, then, to Black’s counter-example to the Identity of Indiscernibles and add a temporal twist. We begin by considering what would happen to the imagined possible world if all spherical things ceased to exist. We would be left with empty space. We then imagine what would happen to the same possible world if all iron spheres with the same properties as the imagined two ceased to exist. Again, we would be left with an empty space. The question is where to go from here if we want to imagine one sphere ceasing to exist but not the other. There is no description that will only pick out one of the spheres, and there is also no way to fix a reference to one but not the other. As the persuasiveness of the spheres example was based on its conceivability, our inability to conceive of the non-existence of only one of the spheres allows us to conclude that such is impossible. As each sphere has the same non-existence, each sphere is identical, based on the argument that a thing is identical to its non-existence at other times and worlds, and assuming the transitivity of identity. 

It might be objected that, if this is the case then we should not be able to conceive of one of the spheres ceasing to exist while the other persists. Yet, we can certainly conceive of a universe of two spheres becoming a universe of one sphere by imagining the non-existence of the sum of the two spheres. In this way we can say that the sphere has not ceased to exist – which would be a logical contradiction while the same sphere also persisted – but rather the sum has ceased to exist. The non-existence of a sum of two spheres does not entail the non-existence of either one of the constituent spheres in particular, but one must go. Yet, there is then no fact of the matter as to which of the spheres has ceased to exist and which sphere has persisted. To avoid the contradiction of each sphere both existing and not existing after one exists, it must be conceded that the two spheres were, in fact, the same sphere bilocated. Therefore, the Identity of Indiscernibles is vindicated. 

References 

1.  Black, M., ‘The Identity of Indiscernibles’ (1952) 61/242 Mind pp. 153-64 

2.   Quine, W. V. O., ‘Word and Object’ (1960) MIT Press: New York; London pp. 176-9 

3.   Kant, I., ‘Critique of Pure Reason’ trans. Kemp Smith, N. (1961) Macmillan & Co: London; New York A598-602, B626-630