| = | 
| Mr Wiener, Studies in Synthetic Logic. 20 
independent of the value of a. Since u is just noticeably brighter 
than #, the strength of stimulus produced by w will be 
a(1+c)(1+c)=a(1+2c+4+c’). 
But since w is only subliminally different from y in brightness, 
the strength of the stimulus produced by w is less than 
a(1+2c+ c’). 
Hence, we are landed in the contradiction, 
a(l+2ce+0)<a(1+2c4+). 
ik little reflection will convince the reader that any other way of 
violating the condition, P | P,, € P,.| P, would likewise be incom- 
patible with Weber's law. 
This seems the proper place to call attention to the fact that 
if P be the relation of complete precedence between the events in 
time, P|\P,.€ P..|P is false. For suppose that at this present 
moment two events begin, one of which lasts five minutes and 
‘the other ten. It is clear that neither event can be simultaneous 
with an event which wholly precedes the other: that is, neither 
bears to the other the relation P,.|P. Now suppose that one 
minute after the shorter event is ended, some event begins. This 
bears the relation P,, to the longer event, and the shorter event 
bears to it the relation P. Therefore, the shorter event bears 
to the longer event the relation P|, = 1a P. So we have 
proved nothing in this article which entitles us to say that if 
P is the relation of complete precedence among the instants of 
time, int‘P is a series. And, as a matter of ‘fact, it is not a 
‘series. If, however, we limit the field of P to events, say, 
that last exactly five minutes, then P| P,, © P.. | P, and int“P 
Is a series. 
In case P ig the relation, ‘noticeably brighter than,’ one 
ean readily see that int‘P is not only a series, but the series we 
mean when we speak of the series of brightnesses. For, if 
Weber’s law is true, or even if some quantitatively different law 
of the same general form is true, P; is exactly the relation which 
holds between two things of the same brightness, for #P,y says, 
practically, the limina of distinguishability from # are the limina 
of distinguishability from y, and it can be deduced from this and 
Weber’s law that this is true when and only when w and y 
produce stimuli of the same intensity, and hence it follows further 
from Weber’s law, « and y must be of the same sensation-intensity. 
Xp is therefore the class of all classes containing all the things of 
the same brightness as a given thing, and hence can be fittingly 
called the class of all brightnesses; and what could be more 
“natural than to say that a given brightness is greater than another 
when and only when a thing of the first brightness is brighter 
than a thing of the second ? 
