CHAP. XXI.] PROBABILITY OF JUDGMENTS. 385 



Passing then from Logic to Algebra, we have 



cC 

 Frob. x l = -77 = c, 



o 



the function V of the general Rule (XVII. 17) reducing in the 

 present case to C. The value of Prob. x l is therefore wholly ar- 

 bitrary, if we except the condition that it must not transcend 

 the limits and 1. The individual values of Prob. x Z9 . . Prob.# m , 

 are in like manner arbitrary. It does not hence follow, that 

 these arbitrary values are not connected with each other by ne- 

 cessary conditions dependent upon the data. The investigation 

 of such conditions would, however, properly fall under the me- 

 thods of Chap. xix. 



If, reverting to the final logical equation, we seek the inter- 

 pretation of c, we obtain but a restatement of the original pro- 

 blem. For since C and D together include all possible consti- 

 tuents of t l9 2 . . t m , we have 



C + D = 1 ; 



and since D is affected by the coefficient - , it is evident that on 



substituting therein for t 19 t 2 , . . t m , their expressions in terms of 

 # 15 ff 2J . . # , we should have D = 0. Hence the same substitution 

 would give C = 1 . Now by the rule, c is the probability that if 

 the event denoted by C take place, the event x^ will take place. 

 Hence C being equal to 1, and, therefore, embracing all possible 

 contingencies, c must be interpreted as the absolute probability of 

 the occurrence of the event x t . 



It may be interesting to determine in a particular case the 

 actual form of the final logical equation. Suppose, then, that the 

 elements from which the data are derived are the records of 

 events distinct and mutually exclusive. For instance, let the 

 numerical data a l9 tf 3 , . . a m , be the respective probabilities of 

 distinct and definite majorities. Then the logical functions 

 X 19 X z , . . X m being mutually exclusive, must satisfy the con- 

 ditions 



X l X> = 0, . . X l X m = 0, X 2 X m = 0, &c. 

 Whence we have, 



ti t, = 0, t, t m = 0, &c. 

 2c 



