I originally wrote this for an ontology reading group in 2016. I thought it worth posting here, warts and all:
What is the Problem of Universals?
If I go outside, there are trees with same shaped leaves. If I go into my kitchen, there are bags of same coloured apples or potatoes. It's a Moorean fact—a pre-analytic datum—that many distinct things have identical properties.
The problem of universals asks how to account for this datum.
What is the difference between a realist and a nominalist?
Recall the categories tree from Lowe's A Survey of Metaphysics:
Given two identically red apples, the believer in universals will say that the red in each of the apples is the exact same entity wholly present in both apples; the nominalist will deny that there are any such entities. The believer in universals says there are either universals, or universals and particulars in reality; the nominalist says there are only particulars.
Can nominalists still believe in properties?
Yes. Although, given our two apples, they would say that apple1 and apple2 each have their own red property, red1 and red2 respectively. They would, in other words, make the properties into particulars.
Wait, so I can believe in properties even if I'm not a “realist”?
Yes. There is a distinction between realism about universals (universals realism) and realism about properties (property realism). In work on the problem of universals, "realism" is usually used to mean realism about universals, but one shouldn't let this convention mislead them.
What about nominalists that don't believe in properties?
Nominalists that don't believe in properties try to give an analysis of our pre-analytic datum some other way (e.g. they say the two red objects just fall under the same words, or the same mental concepts).
What is the difference between transcendent and immanent realism?
There is an important distinction between instantiated and uninstantiated properties. The property white is instantiated when there is at least one white object in the universe. It's uninstantiated when there are no white objects in the universe.
Transcendent realists say that, whether or not there are instantiated properties, there are uninstantiated properties in the world; immanent realists say there are only instantiated properties.
(Transcendent and immanent realists are sometimes also called Platonic and Aristotelian realists respectively.)