A possibility is a way that the world could be. Some philosophers hold that there cannot be distinct possibilities that are qualitatively identical. "Qualitatively identical" is a technical term. Think of it this way: two things are qualitatively identical if no matter how thoroughly we investigated, and no matter how good the tools we used in such an investigation, we couldn't tell the two objects apart.
I take it as a datum that there are possibilities in which there are qualitative duplicates. For instance, imagine a Max Black world. In the Max Black world there are only two objects, both are perfect spheres made of iron exactly three inches in diameter. No other objects exist in the Max Black world. If the two spheres of iron are just exactly alike, then they are qualitatively identical.
The question that I want to address is whether there are qualitatively identical possibilities. Sure, there can be qualitatively identical iron spheres. But are there different ways the world can be that are qualitatively identical? I think I have an argument that are qualitatively identical but distinct possibilities.
Imagine a Max Black world with two iron spheres. Bow imagine that there is a two-sided red arrow exactly between the balls. The world ends up looking something like this: [ O <--> O ].
Now imagine that the arrow has the following feature. At time T one of the sides of the arrow will randomly turn blue. So, at and after T one of the sides of the arrow is red and the other side of the arrow is blue. Let us agree to call the iron sphere closest to the red arrow the Red Sphere, and the iron sphere closest to the blue arrow the Blue Sphere. So here is one possibility. It could be that the by random the side of the two-sided arrow closest to the Blue Sphere turns blue.
But which side of the arrow turned blue was a random matter. It could just have easily been that the other side of the arrow turned blue. That is, here is another possibility. It could have been that by random the side of the two-sided arrow closest to the Red Sphere turned blue. This is a different possibility. At least it had better be. When considering the world in which the blue side of the arrow is closest to Blue Sphere we need to be able to truly say: it could have been the case, but actually isn't, that the blue side of the arrow was closest to the Red Sphere.
But now notice that we have two possibilities that are qualitatively identical. In both worlds we have two qualitatively identical iron spheres, and a two-sided arrow between them which is half red and half blue. Both worlds look something like this, where 'b' indicates the blue side of the arrow:
[ O <-->b O].
One interesting thing about this argument is that it suggests that there can be non-qualitative differences among possibilities originating from the identities of non-actual things like our non-actual iron sphere.