OF TESTIMONIES OR JUDGMENTS. 611 



and expressing the unknown value of Prob. «\> (x,y,z . . ) by u, we form the logical 

 equations : — 



</>! {x,y,z . . ) = * (p 2 (x, y, z . . ) = t, &c. 

 •v}/ {x,y,z . . ) = iv 

 and hence, determining w as a developed logical function of s, t . . we have a 

 result of the form 



w/=A + 0B + [jc + Jd .... (1) 



Here A, B, C, D are logical combinations of the simple events, s, t, &c, and the 

 connection in which they stand to the event w and to each other is the following : 

 A expresses those combinations of s, t, &c, which are entirely included in w, — i.e., 

 which cannot happen without our being permitted to say that w happens. B re- 

 presents combinations which may happen, but are not included under w; so that 

 when they happen, we may say that w 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, &c. .... (2) 



The condition, founded on definition, to which these simple events are subject is, 



D=0 

 or, which amounts to the same thing, 



A+B+C=l 

 indicating that the combinations expressed by A, B, and C can alone happen. If 

 we represent A + B + C by V, we have 



«/ = A + qC (3) 



with the condition 



V=l . . . . (4) 



Of these equations, the latter expresses the conditions to which the simple events, 

 5, 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,' t,' &c, unrestricted by any condition, and possessed of unknown proba- 

 bilities, p,' q, which are to be so determined that when s,' t' . . are subjected to 

 the same condition (4) to which s, t . . are subject, they will have the same 

 probabilities as s, t . . 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, = V, = A ± _C =V .... (5 ) 



p q u y 



Y s being formed by selecting those terms from V, which contain s as a factor ; V 



