What exactly is the distinction between a universal and a particular? Universals are often said to be repeatable entities, ones-over-many or ones-in-many. Particulars, then, are unrepeatable entities. Now suppose the following: there are universals; there are particulars; particulars instantiate universals; first-order facts are instantiations of universals by particulars.
One and the same universal, F-ness, is repeated in the following facts: Fa, Fb, Fc. But isn't one and the same particular repeated in Fa, Ga, Ha? If so, particulars are as repeatable as universals, in which case repeatability cannot be the mark of the universal. How can it be that all and only universals are repeatable? I stumbled upon this problem the other day. But Frank Ramsey saw it first. See his "Universals," Mind 34, 1925, 401-17.
Instantiation as holding between particulars and universals is asymmetric: if a instantiates F-ness, then F-ness does not instantiate a. (Instantiation is not in general asymmetric, but nonsymmetric: if one universal instantiates a second, it may or may not be the case that the second instantiates the first.) The asymmetry of first-level instantiation may provide a solution to the Ramsey problem. The asymmetry implies that particulars are non-instantiable: they have properties but cannot themselves be properties. By contrast, universals are properties and have properties.
So we can say the following. The repeatability of a universal is its instantiability while the unrepeatability of a particular is its non-instantiability. So, despite appearances, a is not repeated in Fa, Ga, and Ha. For a is a particular and no particular is instantiable (repeatable).
Solve a problem, create one or more others. I solved the Ramsey problem by invoking the asymmetry of instantiation. But instantiation is a mighty perplexing 'relation' (he said with a nervous glance in the direction of Mr. Bradley). It is dyadic and asymmetric. But it is also external to its terms. If a particular has its properties by instantiating them, then its properties are 'outside' it, external to it. Note first that to say that a is F is not to say that a is identical to F-ness. The 'is' of predication is not the 'is' of identity. (For one thing, identity is symmetric, predication is not.) It would seem to follow that a is wholly distinct from F-ness. But then a is connected to F-ness by an external relation and Bradley's regress is up and running. But let's set aside Bradley's regress and the various responses to it to focus on a different problem.
If a and F-ness are external to each other, then it is difficult to see how a could have any intrinsic (nonrelational) properties. Suppose a is an apple and that the apple is red. Being red is an intrinsic property of the apple; it is not a relational property like being in my hand. But if a is F in virtue of standing in an external instantiation relation to the universal F-ness, then it would seem that F-ness cannot be an intrinsic property of a. So an antinomy rears its ugly head: a is (intrinsically) F and a is not (intrinsically) F.
Call this the Problem of the Intrinsically Unpropertied Particular. If there are particulars and universals and these are mutually irreducible categories of entity, then we have the problem of bringing their members together. Suppose it is contingently true that a is F. We cannot say that a is identical to F-ness, nor, it seems, can we say that a and F-ness are wholly distinct and connected by the asymmetric, external tie of instantiation. Is there a way between the horns of this dilemma?
David Armstrong at the end of his career suggested that instantiation is partial identity. The idea is that a and F-ness overlap, are partially identical. This bring a and F-ness together all right, but it implies that the connection is necessary. But then the contingency of the connection is lost. It also implies that instantiation is symmetrical! But then Ramsey is back in the saddle.