Universals are things that have a single name but many instances. There are many instances in universals in real life. Humanity, planets, stars, atoms and rocks are all universals. So is unity, redness and straightness. But while the existence of such universals is demonstrable, there is good reason to believe that they do not exist. A universal is contradictory and cannot exist.
The existence of universals is demonstrable. We know that certain things share features. This truth is self-evident. For example, a red box and a red painting both share the same color. This shared feature is a universal. In fact, everything that can be named is a universal in this sense. All concepts can apply, at least potentially, to more than one thing. But anything that can do this is a universal. Therefore, universals are everywhere.
The problem with universals is that they are both one and many. When we say that a box is red and a painting is red, we do not mean that the box has some redness and the painting has the rest. Such a statement would be absurd. Both the box and the painting are wholly red. But the redness of the painting is not the same as the redness of the box. Both the box and the painting are one because they are both red and in the way that they are red there is no distinction between them. Therefore, they are one. But the box and the painting are different things. So each has a redness that is different from the other. If I painted the box blue, then it would not change the color of the painting. But if changing one thing means that the other is unchanged, then they are different things. Therefore, each redness is a different instance of redness.
But what is true for one universal – redness – is true for all universals. This problem can be generalized to apply to every single universal. But since nothing can be both one and many in the same sense, no universal can exist. But since it is demonstrable that universals do exist, then it is true that universals do exist. Therefore, universals both exist and do not exist.