The calculus of relations is a more modern subject than the
calculus of classes. Although a few
hints for it are to be found in De Morgan^{[20]}, the subject was first developed
by C. S. Peirce^{[21]}. A careful analysis of mathematical reasoning shows (as we
shall find in the course of the present work) that types of relations are the
true subject-matter discussed, however a bad phraseology might disguise this
fact; hence the logic of relations has a more immediate bearing on mathematics
than that of classes or propositions, and any theoretically correct and adequate
expression of mathematical truths is only possible by its means. Peirce and
Schröder have realized the great importance of the subject, but unfortunately
their methods, being based, not on Peano, but on the older Symbolic Logic derved
(with modifications) from Boole, are so cumbrous and difficult that most of the
applications which ought to be made are practically not feasible. In addition to
the defects of the old Symbolic Logic, their method suffers technically (whether
philosophically or not I do not at present discuss) from the fact that they
regard a relation essentially as a class of couples, thus requiring elaborate
formulae of summation for dealing with single relations. This view is derived, I
think, probably unconsciously, from a philosophical error: it has always been
customary to suppose relational propositions less ultimate than
class-propositions (or subject-predicate propositions, with which
class-propositions are habitually confounded), and this has led to a desire to
treat relations as a kind of classes. However this may be, it was certainly from
the opposite philosophical belief, which I derived from my friend Mr G. E.
Moore^{[22]}, that I was led to a different formal treatment of relations. This treatment, whether more
philosophically correct or not, is certainly far more convenient and far more
powerful as an engine of discovery in actual mathematics^{[23]}.(§ 27 ¶ 1)

§ 27 n. 1. Camb. Phil. Trans. Vol. X, On the Syllogism, No. IV, and on the Logic of Relations. Cf. ib. Vol. IX, p. 104; also his Formal Logic (London, 1847), p. 50. ↩

§ 27 n. 2. See especially his articles on the Algebra of Logic, American Journal of Mathematics, Vols. III and VII. The subject is treated at length by C. S. Peirce’s methods in Schröder, op. cit., Vol. III. ↩

§ 27 n. 3. See his article On the Nature of Judgment, Mind, N. S. No. 30. ↩

§ 27 n. 4. See my articles in R. d. M. Vol. VII, No. 2 and subsequent numbers. ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.