' 
442 Mr Norbert Wiener, A Contribution to the 
called relations of complete sequence? and (2) what is the nature 
of the connection between relations of complete sequence and | 
series ? } 
One very general property which belongs to relations of the | 
sort we have called relations of complete sequence is that they | 
never hold between a given term and itself. This property—that | 
of being contained in diversity—they share with series proper, | 
Writing cs for the class of relations of complete sequence, we can | 
represent this fact in the symbolism of the Principia Mathematica | 
of Whitehead and Russell by the formula 
es CRI‘. 
Another property they share with series is that of transitivity. ‘ 
If, for example, the event x wholly precedes the event y, while _ 
the event y wholly precedes the event z, the event « wholly | 
precedes the event z. But they possess another property more | 
powerful logically, which may be called a generalized form of | 
transitivity. If the event x wholly precedes the event y, and 
the event y neither wholly precedes nor wholly follows the 
event z, while the event z wholly precedes the event w, then : 
the event # will wholly precede the event w. All the other | 
relations which we have mentioned as examples of relations of | 
complete precedence will be found to possess the same property, 
| 
which, moreover, will be satisfied by all those relations which we | 
would naturally call relations of complete precedence. We may,. 
then, so define “relations of complete precedence” as to regard 
this as a property common to all such relations. In symbols, we | 
shall then have F 
t csCR{R|(+R+R)|RER. 
The relation (+ R+ R), with its field limited to that of R, is 
what we ordinarily know as simultaneity. In most theories of | 
time and of relations of complete precedence, it has been thought | 
necessary to treat precedence and simultaneity as codrdinate | 
primitive ideas. Nevertheless, those who hold such theories have | 
to assume such propositions as the following, in order to make 
simultaneity and precedence possess the appropriate formal | 
properties* : i 
b.SAP=A, 
b. Su Pw P= fp ors, 
b. SES, 
. OS = OP. 
* In the following list of propositions, § stands for ‘is simultaneous with,’ 
and P for ‘ precedes.’ 
