CHAP. XXI.] PROBABILITY OF JUDGMENTS. 387 



the latter events X^ . . X m .\ being distinct and mutually exclusive ; 

 required the probability of any other compound event X. 



In this proposition it is supposed, that X 19 X 2 , . . X m . l9 as 

 well as X 9 are functions of the symbols a?i, # 2 , . . x n alone. 

 Moreover, the events X 19 X 29 . . X m . l9 being mutually exclusive, 

 we have 



X l X 2 = 0, . . Xi Xm., = 0, X 2 X 3 = 0, &c. ; (3) 



the product of any two members of the system vanishing. Now 

 assume 



Xity, X.-,-^-,, X = t. (4) 



Then t must be determined as a logical function of x l , . . x n , 

 *u ' t m 



Now by (3), 



all binary products of t l9 . . t m . lt vanishing. The developed ex- 

 pression for t can, therefore, only involve in the list of constitu- 

 ents which have 1, 0, or - for their coefficients, such as contain 

 some one of the following factors, viz. : 



TI standing for 1 - ^ , &c. It remains to assign that portion of 

 each constituent which involves the symbols #!..#; together 

 with the corresponding coefficients. 



Since Xi = ti (i being any integer between 1 and m- \ inclu- 

 sive), it is evident that 



from the very constitution of the functions. Any constituent 

 included in the first member of the above equation would, there- 

 fore, have - for its coefficient. 

 Now let 



-\r i -y -y /*7\ 



and it is evident that such constituents as involve TI . ~t m . l9 as 

 a factor, and yet have coefficients of the form 1,0, or -,must be 



2 c 2 



