B. Russell: Relaciones y universales

Abstract

(A) The main conclusion of my argument is that Russell, in his early writings, took relations to be universals, and better still, uninstantiated universals since he thought that to deny this would be to deny that there are relations. And it is well known that this was one of the main thesis within his system as against Leibniz’s and Bradley’s doctrines which both denied the independent being of relations. So Russell’s contention that relations have no instances or in a better terminology, that they are unparticularized universals is much more than a “very curious doctrine” as Weitz has it.
Russell’s argument is put forward in The Principles of Mathematics, sect. 55 and has as a starting assumption that there are relations. But once this is granted, it is relevant to ask for the ontological status of them, i.e., to ask whether they are:

a) Universals
or b) Particulars, in which case no two occurrences of them would be the same.

(B) Russell starts his discussion in sect. 55 with a dilemma that I take as follows:
Either (I) The general concept difference occurs in each one of the propositions of the form “xDy”;
or (II) each case in which the variables “x”, “y” take different values, there will be a different D in propositions of the form “xDy”.
Russell now considers the hypothesis that the D in (II) is different in each one of its occurrences owing to a complexity in the relation itself and so he takes a difference to be compounded of:

i) difference (the general concept)
ii) ”some special quality distinguishing a particular difference from every other particular difference.”

Then he argues to the effect that the specific quality cannot be a quality of the terms of the relation since “If the quality be not a relation, it can have no special connection with the difference of A and B, which it was to render distinguishable from bare difference, and if it fails in this it becomes irrelevant.” (sect. 55)
So Russell now takes the specific quality to be another relation (a specific one) holding between the two terms A and B, over and above difference, and goes on to say, if this is so, that “we shall have to hold that any two terms have two relations, difference and a specific difference, the latter not holding between any other pair of terms.” (Id.) It is this last hypothesis, of there being one specific difference (relation) between every two different terms, that Russell will oppose to the one which holds that the general concept difference occurs instead. Russell’s discussion is divided in three parts as follows:
(1) If the specific differences do differ among themselves, their differences will also differ giving rise, in this way, to an endless process, but, says Russell, this is no objection against the thesis of specific differences since “in the present work, it will be maintained that there are no contradictions peculiar to the notion of infinity, and that an endless process is not to be objected to unless it arises in the analysis of the actual meaning of a proposition. In the present case, the process is one of implications, not one of analysis; it must therefore be regarded as harmless.” (Id.)
(2) Now, if the objection is raised against the thesis that the general concept difference holds between terms A and B by pointing out that the analysis of the proposition “A differs from B” does not preserve the unity of the proposition but leaves, instead, a mere list of terms, the same, says Russell, holds true for the opposing thesis and suggests a tentative answer to this problem: “it remains tenable that, as was suggested to begin with, the true solution lies in regarding every proposition as having a kind of unity which analysis cannot preserve, and which is lost even though it be mentioned by analysis as an element in the proposition.” (Id.)
(3) Russell’s central objection against the thesis of specific differences (and, in general, against the thesis of specific relations) is that, as he puts it, “even if differences did differ, they would still have to have something in common. But the most general way in which two terms can have something in common is by both having a given relation to a given term. Hence if no two pairs of terms can have the same relation, it follows that no two terms can have anything in common, and hence different differences will not be in any definable sense instances of difference.” (Id.)
What Russell’s argument points out, which might be viewed as an application of his Pinciple of Abstraction, is that if each one of the different differences (relations) is unique, there’s not even the possibility of talking about differences (relations); there would be no differences (relations). Hence we should have to accept one of two doctrines: monadism (Leibniz) or monism (Bradley). But Russell has rejected both of them by showing that there are relations and the argument of sect. 55 shows that they must be universals.
C) After the above conclusion we can locate Russell’s argument within the general context of his philosophy in the following way:

(i) As a starting point we would have Russell’s pluralism;
(ii) There are relations among the many different constituents of the world and they have a different kind of being to that of the terms they relate (against Leibniz and Bradley).
(iii) Relations are not specific and unique in each one of their occurrences (i.e., they are unparticularized).

If one denies (iii), Russell’s sect. 55 tends to show (as I have interpreted it) that this would also be a rejection of (ii).
So, by granting (iii) we must conclude

(iv) Relations are universals.

In latter writings, Russell didn’t change his view about identifying relations with universals, although he did change his position in other related points. In this paper I have tried to give a reason why this was so.