Universals

1) Particulars can be defined as something that exist in space and time (has extension), whereas universal do not exsts in space and time; in the phrase “this apple is red” “this apple” is a particular because it exist in space and time, whereas “red” is a universal because the redness doesn’t exist in space and time; this approach, nonetheless, overmphasizes the importance of space and time; space and time are in some way physical object and not logical objects..
2) Another definition is that particulars are multiple and that universal are unique. Nonetheless, each particular is actually unique: two red apples will always differ for some quality different from “redness”. A way to define multiplicity is that two objects are “copies” if they are identical or more similar than a third object; this means that in fact we should investigate the meaning of “resemblance”; resemblance is a universal, but it should be given to it a status particular among the other universals.
3) Another definition is that of Hegel: the concrete is the unity of all determinations; in terms of particulars and universals, a given particular is the intersection (set) of all the universals related to that given particular. This is in some way a kind of realism.
4) If we join the definition 4 with the nominalist position: universals are set of individuals, we have a quite simple possible solution to the “problem of the universals”; universals are set of particulars, and particulars are sets of universals; the definition is not circular, because the two sets are not the same.
5) This solution is equivalent; or at least can be better understood, if we postulate that there are entities, that we will nam “atoms”, that are neither particulars nor universals. Suppose for instance that we have the atoms A B C D E; suppose the quality α (universal) is the set AB, and that the quality β is the set CDE; there are four possible particulars: a particular a with the qualities αβ, a particular b with only the quality α a particular c with only the quality β and a particular d with no quality. Thus the particular a is composed of the atoms ABCDE, the particular b is composed of the atoms AB, and so on. The construction of the reciprocal sets of particulars and universals in this case is somewhat undeterminate, but can be more rigorous if the process is closed (if the operation of set construction represent what mathematicians call a group).
6) Atoms need not to have qualities (qualities are sets of atoms), they only need to be distinct. Of course, distinct means “with no resemblance”; and therefore the problem of unviersal is fundamentally the problem of resemblance.
7) Anyway, it is actually not necessary to postulate atoms – although I suspect that something like that exists “beyond reality” - they are only an euristic tool for understanding. Universals and particulars are reciprocal sets, and it doesn’t matter of what matter they are made, only the process necessary to construct them.
8) A mathematical method exists – called calibration – that, starting from individuals (particulars), defines classes (universals) and than redefines the individuals – iteratively. I am not able to explain it rigorously, but intuitively it is based on the fact that when we pass from particulars to set of particulars (universals) and then back we give a refined definition of particulars and then we go from these particulars to universals and so on, we must, at each step, change our definitions (the boundaries of our sets) a little. Step after step, this changement can decrease, increase, or remain the same; the only acceptable solution is the first, where the set progressively tend toward a stable solution.
9) In fact, the stable solutions can be many or even infinite, but this is not, in my opinion, a problem, but an enrichment. We can read the myth of Edip at different levels, and levels aer possible different solution of the iterative process of constructing reciprocal sets.
10) Incidentally, this definition of universals and particulars, means that “this apple si red” is true when “this apple” and “redness” belong to reciprocal sets of particulars and universal. This can depend on the context a centaur doesn’t exist in the physical world, but exists in the world of literature – different contexts are different solution in the iterative process of construction of reciprocal sets of universals and particulars.
11) An appealing alternative solution to the problem of unviersals is represented by the theory of tropes. Tropes are unique in quality but multiple in number; only the concepts of multiplicity and quality are necessary (although of course thus the problem of the definition of quality and numebr arises, a very hard but very interesting problem) and particulars are considered a collection of tropes. Coming back to the atoms of point 5), the theory of tropes can be stated as folllowing: AB and CDE are two different tropes, with quality α and β respectively, and particulars are intersections of tropes.
12) tropes shift a logical problem to an ontological problema, which appeals me.
13) I am much indebted for this discussion to the Internet Encyclopaedia of Phylosophy (http://www.iep.utm.edu/universa/ ), whose plain language allows access to difficult phylosophical problem even to non specialists.