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

SECTION E]

INDUCTIVE RELATIONS

573

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Thus minp'('T, where T is any power of P (including IrC'P) is the
generation of P corresponding to T; thus the whole class of generations
is minp"(C"Potid'P. Hence we put
gen'P = minp""'Potid'P Df,
where "gen" stands for "generation."
The notation "min " will not be much used until we come to series, but
then it will be constantly used. At present, we shall only give such properties
of mine as are necessary for our immediate purposes, but in Part V (on series)
we shall devote a number (*205) to its properties.
~In this number we also introduce the notation "xBP" for "x e D'P - (P."
" xBP" may be read " x begins P." If there is a single beginning of P, this
-4
is B'P; otherwise the class of beginnings is B'P, which = D'P-(I'P.
Thus if P is the relation of father and son, B'P = Adam; if P is the relation
-> v
of parent and child, B'P=Adam and Eve. B'P will be the end of P, if
-> v
there is one; generally, B'P will be the class of ends, i.e. ('P - D'P. The
first generation of P is B'P. If P e 1 -> Cls, any generation of P is T"B'P,
where T is the corresponding power of P.
The field of a relation consists, in general, not only of the generations of
P, but also of another part, the part in which, however far we go backwards,
we never reach a beginning. This part is p'(I"Pot'P. The two parts
s'gen'P and p'(I"Pot'P are mutually exclusive, and together exhaust C'P.
The two next numbers, *94 and *95, are hardly ever relevant in subsequent
propositions, and may therefore be omitted by any reader who is not interested
in their subject-matter. *94 deals with powers of relative products. It is
only used in the following number (*95), on "equi-factor relations." The
matter to be dealt with in this number may be explained as follows. In
dealing with correlations and similar topics, we often wish to consider the
series of relations
R, P R Q, P2 R Q2, P3 R Q3, etc.
Now we have not yet at our command a definition of Pv, where v is any finite
number; thus we cannot define a general term of this series as Pv R Qv.
We need therefore a different method of definition. We have
P R Q= (P 1 Q)'R,
p2 RIQ =(PIIQ)'R,
and so on. Thus if T is any power of (P 1) (I Q), a general term of our series
is T'R. For convenience of notation, we put
P*Q = sg'(P Q)* Df.