24 Mr Wiener, Studies in Synthetic Logic. ' 
a eg 
y, and P| P,.| PGP. inst‘P is therefore here also a series, and 
nothing would seem more natural than for us to call it the series 
of sensation intensities. i 
But there are serious objections against this method of pro- 
cedure, and here a genuine logical problem arises. For, although 
it is natural to regard a sensation-intensity as a class of sense- 
objects—the class of sensations ‘of a certain intensity ’—we ' 
naturally consider the intensity of a given sensation as uniquely | 
determined, and the relations between two sensations, # and ¥, 
when « is of the same intensity as y, as a transitive, symmetrical, || 
reflexive relation. Now, in general, rp is not a class of mutually ; 
exclusive classes, and the relation between two terms which belong ! 
to the same member of 7p is not transitive. The fact that a certain | 
river was flowing during the Siege of Troy, and is flowing while 
I am writing this article, does not mean that I was writing this - 
article during the Siege of Troy, yet if we take P as the relation 
between one event and another which completely follows it, my 
writing this article and the flowing of the river will both belong 
to some member of tp; the Siege of Troy and the flowing of the | 
river will both belong to some other member of tp. So we have | 
the definite mathematical problem before us: given a relation, P, | 
fulfilling certain conditions, not sufficient to make it a series, we wish | 
to construct from it a serial relation in such a manner that the 
terms of this series shall form a class of mutually exclusive classes. 
I shall first give the method by which this series may be~ 
derived from the relation between x and y when z is of noticeably 
greater intensity than y; then I shall state a set of conditions — 
sufficient to secure the serial character of the derived relation, and | 
finally I shall interpret conditions and results. Perhaps the best 
method logically would be first to formulate all the conditions to » 
which the original relation must be subject, and then to treat the — 
problem as a purely formal one, but the logical gain would hardly 
zompensate us for the loss in clarity. So I first make the follow- — 
ing definitions : 
, 
Vv 
> — 
(15) P,=(Pee| Pre) [ C&P Df. 
(16) Kee ee, DE 
(17) int —OP(Q—|e5(P.| Py wae 
If P is the relation between x and y when z is, say, noticeably 
brighter than y, then P,, is the relation between two things which — 
are not distinguishable as concerns their brightness, and P, is 
the relation between two things possessing brightness when 
each of the things which is indistinguishable from the one in _ 
brightness is also indistinguishable from the other, and wice versa. 
