Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*174. THE ASSOCIATIVE LAW OF RELATIONAL MULTIPLICATION.

Summary of *174.
In the present number, we have to prove the associative law for [I and
for Prod, i.e. we have to prove (with a suitable hypothesis)
I'Il;P smor nfl'EP
and Prod'Prod;P smor Prod'Z'P.
The first of these requires P e Rel2 excl and either P C J or
QPQ. Q. C'Q e 0 v 1;
the second requires not only this, but also 2'PeRel excl. When both P
and U'P are relations of mutually exclusive relations, we call P an arithmetical relation, which we denote by "Rel3 arithm." Arithmetical relations
serve exactly analogous purposes to those served by arithmetical classes in
cardinal arithmetic.
The proof of the associative law for H consists in showing that, under
a suitable hypothesis, s D (with its converse domain limited) is a correlator
of II''P and I'FJ;P (*174-221-23). To prove this, we first prove
*174-17. H: P e Rel2 excl. "..D"C'II'1;P = C'H''P
and
*174-19.: P e Rel2excl. D. (s D) C'II;P e- 1- 1
This gives what we may call the cardinal part of the proof, i.e. it shows that
(s D) C'I;P is a ad cardinal correlator of the fields of 1' 'P and HII' P.
We then prove that if M and N belong to the field of IH'I;P, they have the
relation l'LH;P when the relational sums of their domains have the relation
I'r'P. Here, in addition to the hypothesis P e Rel2 excl, we require that if
any relation Q has the relation P to itself, then C'Q is not to have more than
one term. Thus we have
*174-215. F:. P e Rel2 excl: QPQ. DQ. C'Q e 0 v 1::
M ('II;P) N.. M, N e F4'n "'P. (D 'D'M) (f s'Y'P) (h'D'N)
The hypothesis QPQ. DQ. C'Q e0 1 is verified if P G J(*174-216); thus
for most purposes it is more convenient to substitute the simpler hypothesis
P C J for QPQ. Q. C'Q eO v 1. We shall, however, have occasion to use
the hypothesis QPQ. DQ. C'Q e 0 1 in *182'42'43'431, where our P is a
relation whose field consists entirely of relations of the form Q, Q, whose