OF TESTIMONIES OR JUDGMENTS, 611 
and expressing the unknown value of Prob.  (#, y,z . . ) by #, we form the logical 
equations :— 
Pil@ye.-)=s by (mye.- )at ke, 
V (@,y,2..)=w 
and hence, determining w as a developed logical function of s,t . . we havea 
result of the form 
0 1 
w=A+0B+5C+ 5D ; , F : (1) 
Here A, B, C, D are logical combinations of the simple events, s, ¢, &c., and the 
connection in which they stand to the event ~ and to each other is the following: 
A expresses those combinations of s, ¢, &c., which are entirely included in »,—z.2., 
which cannot happen without our being permitted to say that happens. B re- 
presents combinations which may happen, but are not included under w; so that 
when they happen, we may say that 7 does not happen. C represents those com- 
binations, the happening of which leaves us in doubt whether w happens or not. 
D, those combinations, the happening of which is wholly interdicted. 
Thus far we have only translated our problem into a language in which its 
data are the probabilities of simple events, viz. :— : 
Prob. s=p Prob. t=q, &e. F : (2) 
The condition, founded.on definition, to which these simple eee are subject is, 
D=0 
or, which amounts to the same thing, 
A+B+C=1 
indicating that the combinations expressed by A, B,and C can alone happen. If 
we represent A+B+C by V, we have 
w=A+e0 sigol Ao eelinles oilt yd) 
with the condition 
V=1 batt. (4) 
Of these equations, the latter expresses the conditions to ae the ey events, 
s, t,&c., are subject; the former expresses w as a logical combination of those 
events. 
We now, in accordance with the hypothesis, ascend to a new scheme of simple 
events, s,' ¢,/ &c., unrestricted by any condition, and possessed of unknown proba- 
bilities, »,’ g’, which are to be so determined that when s,’¢’ . . are subjected to 
the same condition (4) to which s,¢..are subject, they will have the same 
probabilities as s,¢ . . The system of equations to which we are thus led, and 
which contains the implicit solution of the problem, is the following (Laws of 
Thought, cap. xvii., p. 267) :-— 
V; uy: Vv; om A+C 
Dive (Oo yc4 
V, being formed by selecting those terms from V, which contain s as a factor; V 
Sy ee ee 
