OF ARTS AND SCIENCES : MARCH 12, 1867. 257 



is shown by the above example. Boole (p. 286) is forced to the 

 conclusion that "propositions which, when true, are equivalent, are 

 not necessarily equivalent when regarded only as probable." This is 

 absurd, because probability belongs to the events denoted, and not to 

 forms of expression. The probability of an event is not altered by 

 translation from one language to another. 



Boole, in fact, puts the problem into equations wrongly (an error 

 which it is the chief purpose of a calculus of logic to prevent), and 

 proceeds as if the problem were as follows : — 



It being known what would be the probability of T, if X were to 

 happen, and what would be the probability of Z, if Zwere to happen ; 

 what would be the probability of Z, if X were to happen ? 



But even this problem has been wrongly solved by him. For, 

 according to his solution, where 



p= T^ q = Zy r=z Z^, 



r must be at least as large as the product of p and q. But if X be 

 the event that a certain man is a negro, Zthe event that he is born 

 in Massachusetts, and Z the event that he is a white man, then neither 

 p nor q is zero, and yet r vanishes. 



This problem may be rightly solved as follows : — 



Let/= Y^ = X,T 



q' = Z^ = X,Z 



r'=Z,= X,Z 



Then, r' =^ p' , q' -, p' ■== p' , q ; q'- 



Developing these expressions by (18) we have 



=FP',q' + ^'p-.a' (?''?') + ^'p'.a' ip'^9')' 



The comparison of these two identities shows that 



r'=y,^' + r'p,,^, ip' ,q'). 



Ti.Tr I -^ > 2/ ) ^ 



Let F == r'piq, == , - 



VOL. VII. 33 . 



