A property is a way something could be. For example, if there is a brown dog named Fido, then it can be said that Fido has the property being brown and lacks the property being blue. This “having” relation between a thing and a property of the thing is sometimes called “exemplification”. Thus Fido exemplifies brownness.

Properties are a kind of universal. A universal is something that could be simultaneously multiply located, exemplified, or instantiated. For example, Donald Trump and I now both have the property being human. One and the same property–humanity–is simultaneously exemplified by billions of people.

Properties are abstract objects. Something is abstract if it is essentially causally impotent. This should be an obvious point. Properties can’t cause things. While it may be that a thing’s having a certain property could cause something, the property itself could not. What effect could, say, being blue have upon anything? All abstract objects are immaterial because any material thing could have impinged on another material thing.

There are two ways of thinking about properties. The first sees them as things that really exist, just as robustly as people and electrons. The second sees property talk as no more than a convenient way of speaking. For example, to say that Fido has the property being brown is just another way of saying Fido is brown. There is no property “brownness” that exists “out there” and stands in some mysterious exemplification relation with Fido.

I will now argue for the view that properties do not exist. If properties exist, then there is the property non-selfexemplification. This is the property something has when it does not exemplify itself. For example, the property being blue is not blue; thus it does not exemplify itself and has the property of non-selfexemplification. On the other hand, the property abstract is itself abstract and so lacks the property of non-selfexemplification. If such a property exists, then everything either has it or lacks it. What, then, about the property itself? Does non-selfexemplification exemplify itself or not? If it does, it doesn’t; and if it doesn’t, it does! This is very unpleasant, to say the least, and it seems to follow from the assumption that properties exist.

If we say instead that there really is no such thing as non-selfexemplification, then its associated problems vanish with it. If you have any doubts about this solution, consider the more down-to-earth example of a barber who shaves all and only those who don’t shave themselves. Does the barber shave himself? If he does, he doesn’t; and if he doesn’t, he does. Is this a problem? Not if we conclude no such barber exists.

Do we lose anything by getting rid of properties? I don’t think so. As abstract objects, they have no effect upon anything. Properties also don’t explain anything. Saying Fido exemplifies the property of brownness does nothing to explain why Fido is brown or why Fido is similar to other brown objects. A scientific explanation of Fido’s brown fur seems sufficient.