Back when I taught Philosophy A-level, I noticed that students were usually able to explain Plato's Theory of Forms, or at the very least, to tell a story about how all concrete, individual horses all somehow participate in or reflect a perfect, unchangeable 'Form of the Horse'. However, when asked why Plato, or anybody else, would believe such 'forms' to exist, answers proved rare.
Unfortunately, A-level students aren't alone in their confusion. Philosophers tend to agree that the motivation for believing in 'forms' -- which they now call 'universals' -- comes from a perceived need for a solution to a so-called 'Problem of Universals'. However, on top of disagreeing about whether universals exist, they also disagree on the exact nature of the 'Problem' they are supposed to solve.
I have found at least two candidates for being the 'real' Problem of Universals, which I will call the logical and explanatory problem respectively.
The logical problem of universals goes something like this: consider two Lions, Liam and Lionel. Liam is a lion. Lionel is a lion. But Liam is not Lionel. How is this possible? How can Liam and Lionel both 'be' the same thing if they are distinct from each other? After all, we know that if x = y and y = z, then necessarily, x = z (the transitivity of identity). The natural answer is that, when we say that Lionel is a lion, and that Liam is a lion, we mean that despite being distinct, they have something in common. But what is this 'something' which they share? Clearly, it can't be the specific, concrete traits and body-parts which each lion has. Liam doesn't 'share' his mane with Lionel, Liam has his mane, and Lionel has his, they don't take turns to wear the same mane. And so it is for all their other traits and body-parts. Plato and his supporters conclude that the 'something' they share is their exemplification of the universal 'lion'.
Thus understood, the problem of universals is a logical problem, because universals are brought in to dispel an apparent logical contradiction. Universals answer the question of how it is (logically) possible for two distinct things to 'be' the same thing. In this version, the problem is sometimes referred to as the 'One Over Many' problem. Whatever Plato himself thought, many since him have espoused realism (here, the view that universals exist) on the basis that it is necessary to solve the logical problem -- see Bertrand Russell's classic The World of Universals for an example of this.
On the other side of the fence, proponents of nominalism (the view that there are no universals) also often agree that the problem is a logical one (e.g. Rodriguez-Pereyra 2000), though they disagree that other-worldly, abstract universals are needed to solve it. In its simplest form, the nominalist response says that what Liam and Lionel share is a resemblance: they are said to both be lions because they resemble each other. But that clearly isn't enough. Liam the lion also resembles Terry the tiger, but obviously they can't both be said to be lions. Nominalists have put forward many ways to improve the simple response, but the strongest of these, in my opinion, is the one that appeals to tropes. A trope is sometimes referred to as a 'particularized property'. Think of two red balls: the two balls may both be red, but the redness of the first ball is distinct to the redness of the second ball. If I hide the second ball under the table, you can see the redness of the first ball, but not that of the second ball. Thus, we say that each ball has a 'red-trope'. Such tropes are readily accessible and part and parcel of the world around us, not shut away in their own realm. Importantly, they do not share the same red-trope. Rather, they each have their own, though the two red-tropes may resemble each other exactly. This is the trope nominalist solution to the logical problem of universals: it is logically possible for two distinct things to 'be' the same thing if the have tropes that resemble each other. Liam and Lionel are both lions because Liam has a lion-trope and Lionel has a lion-trope, and both lion-tropes resemble each other.
Not everyone is happy with the idea of a trope, some even calling it incoherent (Levinson 2006). But it seems to me that their existence is perfectly plain -- right now as I write, I see the blueness of the sky outside, the redness of the table cloth, the 'rectangleness' of my notebook, and so on. In any case, I think the trope nominalist successfully solves the logical problem of universals, in showing that it is logically possible for two distinct things to 'be' the same thing, without recourse to universals.
As I have said, however, not everyone agrees that the problem of universals is a logical problem. Paul Gould, a proponent of realism, argues that the question that realists and nominalists must answer is the question of " what a metaphysical explanation of qualitative and resemblance facts would be" (2012). The world we live in is one in which different things come under the same names or categories. The explanatory problem of universals is the problem of the "explanation" of this interesting feature of reality. Of course, some philosophers will say that this is just due to a contingent feature of language, i.e. its tendency to regroup different things under the same concepts (this view is held by conceptualists and predicate nominalists). But many other philosophers would insist that our conceptual groupings reflect real categories and resemblance relations. Indeed, both realists and trope nominalists would say that distinct things fall under the same concepts because they really resemble one another in certain ways. Realists argue that this real resemblance between different things is due to them all exemplifying the same universal. Trope nominalists, on their side, contend that it is due to individual things having tropes that themselves resemble one another (e.g. the redness of ball 1 resembles the redness of ball 2).
So, do both realism and trope nominalism solve the explanatory problem of universals? Not, says Gould, to the same degree of satisfaction. To see why, notice that the trope nominalist account explains the resemblance of distinct things by pointing to the resemblance of their respective tropes. But in that case, it cannot really explain why the world contains resemblance at all. Such an explanation would be viciously circular, as it would involve explaining resemblance by appealing to more resemblance. The realist goes further in answering the explanatory problem: the world contains real resemblance relations, because groups of things are linked to the same universals by the exemplification relation. Therefore, while trope nominalism must ultimately regard resemblance as an unexplained brute fact -- because the resemblances between tropes are themselves unexplained -- realism does answer the question of 'why is there resemblance at all?', since the exemplification of universals by concrete things is not itself a resemblance, making the explanation non-circular.
Gould concludes that "realism regarding universals is explanatorily superior to its nominalistic competitors". Of course, the debate between realists and trope nominalists cannot end here, as realism has problems of its own. The most damning of these, I think, have to do with the apparent impossibility of universals disconnected from our world somehow causing the world to exhibit resemblance, and of being known to exist by us (I hope to address this is another entry). But what's interesting for our purposes is that our understanding of the problem of universals, i.e. whether we regard it as a logical or an explanatory problem, bears on the question of whether we ought to think that Plato may have been right after all.
Levinson, J. (2006). Why There Are No Tropes. Philosophy, 81(04), p.563.
Gould, P. (2012). The Problem of Universals, Realism, and God. Metaphysica, 13(2), pp.183-194.
Rodriguez-Pereyra, G. (2000). What is the problem of universals?. Mind, 109(434), pp.255-273.