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

*162. THE SUM OF THE RELATIONS OF A FIELD.

Summary of *162.
The form of summation defined in *160 cannot be extended beyond
a finite number of summands, since it involves explicit mention of all the
summands. In the present number, we shall be concerned with a form of
summation which is not subject to this restriction. It will be observed that,
since relational summation is not permutative, we cannot define the sum
of a class of relations, for this would not determine the order in which the
summation is to be effected. Our relations must be given as the field
of some relation which orders them; thus the sum appears not as the sum
of a class, but as the sum of a relation, namely of a relation whose field is the
relations to be summed. In the case of two relations Q and R, the sum of
Q 4 R, as defined in the present number, will be equal to Qt-R; similarly for
three, the sum of Q $ R w Q Q S w R 4 S will be equal to Q -R -S, and so on
for any finite number of summands.
As explained in the introduction to this Section, if P is a relation between
relations, we put
2'P = 9'C'P w F;P Df.
It is convenient to suppose that P is serial, and that every member of C'P is
also serial. Then 2'P holds between x and y if either (1) there is a series,
in the field of P, in which x precedes y, or (2) x belongs to a series which is
earlier, in the P-series, than the series to which y belongs. The following
are the chief propositions of this number:
*162'22-23. F. C''P = s'C"C'P = C''P = F"C'P - F2'P
*162-26. I. 2'(P c Q) = -'P w 'Q
*162-3. -. 1'(Q, R) = Q41R
*162-31.. -. 'Q 'R = '(Qt-R)
*162-34. F. 2';P = ',E'P [Associative Law. Cf. *42-1]
*162'35. F: C'f'Q C I'R. D. I'Rt;Q = R;''Q
This is the analogue of *40-38. (Cf. note to *162'35, below.)
*162-4. F. Z'A = A
*162-42. h: [! Z'P..! s'C'P. -. a! C'P- tA