206 THE PRINCIPLES OF SCIENCE. [CHAP. 



or incapable of happening together, like a clear sky and 

 rain, or a new inoon and a full moon, then the events are 

 not really A or B, but A not-B, or B not- A, or in symbols 

 Ab -I- aB. Now if we take /* = probability of Ab and 

 v = probability of aB, then we may add simply, and the 

 probability of Ab -\- aB is /* + v. 



Let the reader carefully observe that if the combi- 

 nation AB cannot exist, the probability of Ab is not the 

 product of the probabilities of A and b. When certain 

 combinations are logically impossible, it is no longer 

 allowable to substitute the probability of each term for 

 the term, because the multiplication of probabilities pre- 

 supposes the independence of the events. A large part of 

 Boole's Laws of Thought is devoted to an attempt to 

 overcome this difficulty and to produce a General Method 

 in Probabilities by which from certain logical conditions 

 and certain given probabilities it would be possible to 

 deduce the probability of any other combinations of 

 events under those conditions. Boole pursued his task 

 with wonderful ingenuity and power, but after spending 

 much study on his work, I am compelled to adopt the 

 conclusion that his method is fundamentally erroneous. 

 As pointed out by Mr. Wilbraham, 1 Boole obtained his 

 results by an arbitrary assumption, which is only the most 

 probable, and not the only possible assumption. The 

 answer obtained is therefore not the real probability, 

 which is usually indeterminate, but only, as it were, the 

 most probable probability. Certain problems solved by 

 Boole are free from logical conditions and therefore may 

 admit of valid answers. These, as I have shown, 2 may be 

 solved by the combinations of the Logical Alphabet, but 

 the rest of the problems do not admit of a determinate 

 answer, at least by Boole's method. 



Comparison of the Tlieory with Experience. 



The Laws of Probability rest upon the fundamental prin- 

 ciples of reasoning, and cannot be really negatived by any 



1 Philosophical Magazine, 4th Scries, vol. vii. p. 465 ; vol. viii. 

 p. 91. 



* Memoirs of the Manchester Literary and Philosophical Society, 

 3rd Series, vol. iv. p. 347. 



