Mr Wiener, Studies in Synthetic Logie. 25 
‘It follows at once from the definition of P, that it is transitive, 
symmetrical, and reflexive, whatever P may be, and hence in 
‘this respect it satisfies the requirements we have set up for the 
relation between two members of a sensation-intensity. 
Xp is the class of brightness-intensities, where P 1 ee the relation 
‘noticeably brighter than.’ Since 2X is defined as DP, it follows 
that it must always be a class of mnutoally ue classes ; for 
suppose that two members of Xp, say P. ‘x and Ps y, had the 
term Zz in common. ‘Then we would have zP,a and zPsy. 
From the definition of P, it is symmetrical, so we get wP,z and 
Py, which, on accom) of the transitivity of 12, gives us 
aPyy, and, hence, P. wn us In just the same way, we get 
P. a G PE x, or, finally, P. “oy = Ps x. int‘P is the relation between 
‘wo members of Xp when a member of one is in the relation P,, | P 
with a member of the other. Whatever P is, int‘PGJ. For 
suppose that a (int‘P)a. Then, since a must belong to Ap, every 
term of a stands in the relation P, to every term of a. However, 
from the definition of int‘P, there must be two terms of a, # 
and y, such that #P,,| Py. This may be written as 
GBc@ew. Zea. 
From this and the definition of P,,, we get (q2z)»@P6o212~ Prey, 
> > 
or P,,‘v+ P..“y, which may be written «+P,y. Thus, the 
assumption that ~(int‘P G./) is self-contradictory. 
A condition which will ensure the transitivity of int ‘P is 
P,,|Petrans. For it follows from the definitions of P;, Xp, and 
int that if a(int*PY B, alfe UE | Je P.,| | ez, e522 Belek 
Now, 
fell loca |) bree = a2 (qa, 4 ys oi Poly 0 Pen YP 0 25 
= #2 \(qy) iE Gy — pee -Z€ aR = /Po 
> => 
ee theretore, P..| P|P..| P..|Pse|P is simply P..|P|P..|P. Tf 
P,,|P is transitive, then we find that a(int‘P)?@ implies that 
al{e%(P..|P)} [Ap] 8, which is simply a(int‘P)8. A hypothesis 
which will make P,,|P transitive is P| P,,| PGP. This is the 
same condition which we found to suffice for the transitivity of 
inst‘P. 
When will int‘P be connected? Under what conditions, that 
is, will it be true that 
a, Be Cont*P .a+B.d.9:4 pon es Vi) (int*P) a ? 
