Theory of Relatwe Position. 443 
From these it is an easy matter to deduce that 
Po (2122/2) Owe 
while on the hypothesis that PCJ, the converse deduction can 
readily be made. Therefore, we may define simultaneity as that 
relation which holds between w and y when both either follow or 
precede something and neither precedes the other. The second 
property of relations of complete sequence may, then, be inter- 
preted to state that if R is such a relation, then if xRy, y-is- 
simultaneous-with-respect-to-R to z*, and zRw, then «Rw. 
We shall find that most of the properties of relations of the 
sort of complete temporal succession between events follow from 
the two conditions which we have mentioned above—indeed, many 
of the most important ones follow from the second alone—so that 
we shall define a relation of complete succession as one which 
satisfies those two conditions: in other words, we shall make the 
following definition : 
#0014. cs=RISn RIR|(+R=R)|RER} De 
Moreover, as we shall have frequent cause to refer to the relation 
(+P=P)t C*P, and as this expression is rather unwieldy, we shall 
abbreviate it as follows: 
02, P.=(+P+P)[CP Df. 
Now the question arises, how are the members of ces related to 
series? How, for example, is the relation between an event and 
another that completely succeeds it related to the relation between 
an instant and another that follows it? Two methods of procedure 
are open to us; we may define an event as a class of instants, and 
derive succession between events from that between instants, or we 
may define an instant as the class of all the events that occur at it. 
Both these methods seem to have certain inherent disadvantages : 
if we choose the first method, then we cannot consider the possi- 
bility of several events occurring with the same times of beginning 
and ending, whereas if we choose the second alternative, we cannot 
consider the possibility of all the events of one moment happening 
also at another and vice versa. However, we shall choose the 
latter method of procedure, since cs is a more general notion than 
ser. This can be proved as follows: 
+. R|(+R+R)|R=R|[(+R+R)ECR]|R 
=h|@7(a~Ry.y—Ra.a,yeCR)R (1) 
* In this paper, ‘a-is-simultaneous-with-respect-to-R to y’ will be interpreted 
as meaning ¢[C(+R+R) > C‘R] y. 
+ I follow the method of the Principia Mathematica of Russell and Whitehead. 
