386 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



Under these circumstances it may easily be shown, that the 

 developed logical value of x l will be 



Xl = o ^ ' ' 7w "*" tl ** ' ' Tm ' ' 4 tmJl ' ' tm ~^ 



+ constitutents whose coefficients are - . 



In the above equation ^ stands for 1 - 15 &c. 



These investigations are equally applicable to the case in 

 which the probabilities of the verdicts of a jury, so far as agree- 

 ment and disagreement of opinion are concerned, form the data 

 of a problem. Let the logical symbol w denote that event or 

 state of things which consists in the guilt of the accused person. 

 Then the functions X 19 X 2 . . X m of the present problem are 

 such, that no change would therein ensue from simultaneously 

 converting w, x l9 x z . . x n into 1>, aii, x 2 , . . x n respectively. 

 Hence the final logical value of w 9 as well as those of x l9 rc 2 ? %n 

 will be exhibited under the same form (3), and a like general 

 conclusion thence deduced. 



It is therefore established, that from mere statistical docu- 

 ments nothing can be inferred respecting either the individual 

 correctness of opinion of a judge or counsellor, the guilt of an 

 individual, or the merits of a disputed question. If the deter- 

 mination of such elements as the above can be reduced within 

 the province of science at all, it must be by virtue either of 

 some assumed criterion of truth furnishing us with new data, or 

 of some hypothesis relative to the connexion or the independence 

 of individual judgments, which may warrant a new form of the 

 investigation. In the examination of the results of different 

 hypotheses, the following general Proposition will be of im- 

 portance. 



PROPOSITION II. 



8. Given the probabilities of the n simple events # 15 .r 2 , . . x n9 

 viz. : 



Prob. Xi = c l9 Prob. x z = c 2 , . . Prob. x n = c n ; (1) 



also the probabilities of the m-l compound events Xi, A" 2 , . . X m . i , 

 viz. : 



Prob. Xi = a t , Prob. X 2 = 2 , . . Prob. X m _, = ,,,.i ; (2) 



