384 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



which similarly involves the privative symbols I - x l9 1 - x t9 

 . . 1 - x n . But in the records of assemblies, it is not presumed 

 to declare which set of opinions is right or wrong. Hence the 

 functions X i9 X^, . . X m must be solely of the kind above de- 

 scribed. 



7. Now in proceeding, according to the general method, to 

 determine the value of Prob. x l9 we should first equate the func- 

 tions Xi 9 . . X m to a new set of symbols t l9 . . t m . From the 

 equations 



Xi = ti 9 X Z = 2 , X m = t m9 



thus formed, we should eliminate the symbols # 2 , x 39 . . x n9 and 

 then determine x t as a developed logical function of the symbols 

 ti, t 29 . . t m , expressive of events whose probabilities are given. 

 Let the result of the above elimination be 



Xi) = Q', (1) 



E and E' being function of t lt t 29 . . t m . Then 



'' 



Now the functions X 19 X Z9 . . X m are symmetrical with re- 

 ference to the symbols Xi 9 . . x n and 1 - x l9 . . 1 -x n . It is evi- 

 dent, therefore, that in the equation E' must be identical with E. 



Hence (2) gives 



E 



and it is evident, that the only coefficients which can appear in the 

 development of the second member of the above equation are 



- and -. The former will present itself whenever the values 



assigned to ti , . . t m in determining the coefficient of a constituent, 

 are such as to make E = 0, the latter, or an equivalent result, in 

 every other case. Hence we may represent the development 

 under the form 



*'1 C + 1 D - (3) 



C and D being constituents, or aggregates of constituents, of the 

 symbols t l9 t 99 . .t m . 



