394 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



algebraic function Xi in the system (4) (5) as expressing the 

 probability of the event denoted by the logical function X it on 

 the supposition that the logical symbols x l9 # 2 , . . x n denote in- 

 dependent events whose common probability is x. On the same 

 supposition (xXi) would denote the probability of the concur- 

 rence of any particular event of the series aj 1? x 29 . - x n with Xi. 

 The forms of Xt 9 (xXi), &c. being determined, the equation (4) 

 gives the value of x, and this, substituted in (5), determines the 

 value of the element c required. Of the two values which its so- 

 lution will offer, one being greater, and the other less, than ^, the 

 greater one must be chosen, whensoever, upon general conside- 

 rations, it is thought more probable that a member of the assembly 

 will judge correctly, than that he will judge incorrectly. 



Here then, upon the assumed principle that the largest of 

 the values a m _i shall be reserved for final comparison in the 

 equation (2), we possess a definite solution of the problem pro- 

 posed. And the same form of solution remains applicable should 

 any other equation of the system, upon any other ground, as that 

 of superior accuracy, be similarly reserved in the place of (2). 



1 1 . Let us examine to what extent the above reservation has 

 influenced the final solution. It is evident that the equation (5) 

 is quite independent of the choice in question. So is likewise 

 the second member of (4). Had we reserved the function X 19 

 instead of X m -i 9 the equation for the determination of x would 

 have been 



*Xi X m 



= ; (o) 



fli a m 



but the value of x thence determined would still have to be sub- 

 stituted in the same final equation (5). We know that were 

 the events Xi 9 #2, . . x n really independent, the equations (4), 

 (6), and all others of which they are types, would prove equi- 

 valent, and that the value of x furnished by any one of them 

 would be the true value of c. This affords a means of verifying 

 (5). For if that equation be correct, it ought, under the above 

 circumstances, to be satisfied by the assumption c = x. In other 

 words, the equation 



| a m (xX m ) . 



" ~ 1 



