CHAP. XXI.] PROBABILITY OF JUDGMENTS. 383 



Let us suppose the case of a deliberative assembly consisting 

 of n members, no hypothesis whatever being made respecting 

 the dependence or independence of their judgments. Let the 

 logical symbols x i9 x 99 . . x n be employed according to the fol- 

 lowing definition, viz. : Let the generic symbol # denote that 

 event which consists in the uttering of a correct opinion by the 

 i th member, Ai of the court. We shall consider the values of 

 Prob. #1, Prob. x 29 . . Prob. #, as the qucesita of a problem, the 

 expression of whose possible data we must in the next place 

 investigate. 



Now those data are the probabilities of events capable of 

 being expressed by definite logical functions of the symbols Xi , 

 x 2 , . . x n . Let X^ , X 2 , . . X m represent the functions hi question, 

 and let the actual system of data be 



Prob. X l = a l9 Prob. X z = 2 Prob. X m = a m . 



Then from the very nature of the case it may be shown that 

 X 19 X Z9 . . X m , are functions which remain unchanged if 

 a?!, # 2 , x n are therein changed into 1 - x l , 1 - # 2 1 - # 

 respectively. Thus, if it were recorded that in a certain pro- 

 portion of instances the votes given were unanimous, the event 

 whose probability, supposing the instances sufficiently numerous, 

 is thence determined, is expressed by the logical function 



X l X Z . , X n + (1 - X,) (I - # 2 ) . . (1 - Xn), 



a function which satisfies the above condition. Again, let it be 

 recorded, that in a certain proportion of instances, the vote of an 

 individual, suppose Ai 9 differs from that of all the other mem- 

 bers of the court. The event, whose probability is thus given, 

 will be expressed by the function 



X,(l - X 2 ) , . (1 - X n ) + (1 - Xi) Xi, . . Xnl 



also satisfying the above conditions. Thus, as agreement in 

 opinion may be an agreement in either truth or error ; and as, 

 when opinions are divided, either party may be right or wrong ; 

 it is manifest that the expression of any particular state, whether 

 of agreement or difference of sentiment in the assembly, will 

 depend upon a logical function of the symbols x l9 x Z9 . . x n9 



