256 PROCEEDINGS OF THE AMERICAN ACADEMY 



(33.) b & = l + b a _ a) (l-l) 



(34.) « b = i_L=l^ (i _ a) 



(35.) (9«)a = (9» (l))a- 



The application of the system to probabilities may best be ex- 

 hibited in a few simple examples, some of which I shall select from 

 Boole's work, in order that the solutions here given may be compared 

 with his. 



Example 1. Given the proportion of days upon which it hails, and 

 the proportion of days upon which it thunders. Recmired the propor- 

 tion of days upon which it does both. 



Let 1 == days, 



p ■==. days when it hails, 



q = days when it thunders, 



r = days when it hails and thunders. 



p,q = r 



Then by (29), r==p,q=pq p = qp q . 



Answer. The required proportion is an unknown fraction of the 

 least of the two proportions given. 



By p might have been denoted the probability of the major, and by 

 q that of the minor premise of a hypothetical syllogism of the following 

 form : — 



If a noise is heard, an explosion always tahes place ; 

 If a match is applied to a barrel of gunpowder, a noise is heard ; 

 . • . If a match is applied to a barrel of gunpowder, an explosion 

 always tahes place. 



In this case, the value given for r would have represented the proba- 

 bility of the conclusion. Now Boole (p. 284) solves this problem by 

 his unmodified method, and obtains the following answer : — 



r=pq-\-a(l — q) 



where a is an arbitrary constant. Here, if q = 1 and p = 0, r = 0. 

 That is, his answer implies that if the major premise be false and the 

 minor be true, the conclusion must be false. That this is not really so 



