14 Mr Wiener, Studies in Synthetic Logie. | : 
Studies in Synthetic Logic. By Norpert WIENER, Ph.D.) 
(Communicated by Mr G. H. Hardy.) | 
[Received 13 July 1914] 
§ 1. In a recent article of mine in the Proceedings of the 
Cambridge Philosophical Society*, 1 showed how we can regard the) 
series of the instants of time as a construction from the non-serial 
relation of complete temporal succession between events in time,’ 
and how only a few simple presuppositions concerning the formal 
character of this relation of complete temporal succession sufficed | 
to establish the seriality of the relation of succession between in-| 
stants ; and, in a foot-note, I showed further how, without making | 
any assumptions concerning the formal properties of a given} 
relation, P, we can construct another relation from P in a perfectly 
determinate manner, so that this latter relation will always be a/ 
series. 
In this article, I wish to extend this method of series-con- 
struction in two different directions. I first mean to bring | 
definitions of order through triadic and tetradic relations under a | 
single very general heading, and to show that Frege’s theory of | ! 
hereditary relations and the theory of series-synthesis developed | 
in my former article can be generalized so as to apply to these. 
Then I shall give an alternative method of constructing series from | 
non-serial relations which bears much the same relation to the 
various series of sensation-intensities that the method of my 
previous article bears to the series of instants that constitutes one | 
sort of extension, time. 
In general, our symbolism will be that of the Principia Mathe- - 
matica of Whitehead and Russell, and we shall take the theorems | 
established in that book for granted. But as we shall have much to 
do with polyadic relations, and as the parts of the Principra which — 
will treat of general polyadic relations are not yet in print, it will 
be necessary for us to develop a symbolism of our own here, 
Such properties of polyadic relations as have precise analogues in — 
the theory of classes we shall take for granted. Moreover, as we — 
shall want to speak of properties of relations among any number - 
of terms, and as in Mr Russell’s system}, relations among m terms - 
belong to different types than relations among n terms, if m=, - 
so that no propositional functions whose arguments range over 
— 
Se 
* «©A Contribution to the Theory of Relative Position,” vol. xvm, Part 9, 
pp. 4419. 
+ See, however, my article, ‘‘A Simplification in the Logic of Relations,” 
Proc. Camb. Phil. Soc., vol. xvir, Part 5, pp. 387—90. The method of this article 
can be extended to n- adic relations i in general. 
