OF LOGICAL CLASS-FEEQLTENCIES, ETC. 
113 
It should be noticed that from x = 0 y = 0 we can in the 2 ^>'esent case, infer 
z > 0"2, i.e., from “ no A’s are B” and “ no A’s are C ” infer “ some (at least half) of 
the B’s are C.” But the reason why the ordinary rule of logic, “ from two negative 
premises no conclusion can he drawn,” comes into play is, however, fairly oijvious. If 
Pi + P. + As < 1 
the plane « disappears behind the origin of co-ordinates, and the rear of the surface 
is bounded solely by the co-ordinate planes. The value of z may then he anywhere 
between zero and or at the point x = 0 y = 0, i.e., there is, a priori, no 
inference. 
I do not propose to enter into the discitssion of numerically indefinite inferences — 
i.e., the “particular atfirmadive ” and “particular negative” conclusions of the 
ordinary syllogistic treatment. I woidd, however, suggest that such numerically 
indefinite inferences may he regarded as mere degradations, owing to the truncation 
of the tetrahedron, of the series of exact or definite inferences possible in the last 
case. 
§ 19. Case (3). 
= 0‘45 Pi~ P-i — 
The equations to the hounding planes are 
.x=0 P=0 z=U 
X = 0'4 y = 0’4 2 0'4 
X -j“ y ~h 2 = 0‘25 (ot) 
X y — z = 0’45 (/3) 
X — y + z 0-4 (y) 
— X y +2=0-4 (8). 
The form of the surface is shown in fim 3. 
o 
Only one edge of the primitive tetrahedron is 
now untruncated, viz., GK or yS. Hence only 
one of the original six series of definite in¬ 
ferences is left, viz.— 
Given. Inferred. 
2 = 0'4 X = y. 
In addition to this infinite series there are, 
however, two special cases of inference corre¬ 
sponding respectively to the points IT and M 
of the figure. 
VOL. CXCVII.-A. o 
