THE THEORY OF PROBABILITY. 235 



tide, or an earthquake and a storm, the probability of 

 A or B happening is not the sum of their separate proba- 

 bilities. For by the Laws of Thought we develop A I B 

 into AB | Kb I aB, and substituting a and /3, the proba- 

 bilities of A and B respectively we obtain a./3 + a.(i /3) + 

 ( i a) ./3 or a + /3 a./3. But if events are incompossible or 

 incapable of happening together, like a clear sky and rain, 

 or a new moon and a full rnoon, then the events are not 

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

 Kb | aB. Now if we take 



p. = probability of Kb 



v = probability of aB, 



then we may add simply, and probability of Kb \ aB = /x + v. 



Let the reader observe that since the combination AB 

 cannot exist, the probability of Kb is not the product of 

 the probabilities of A and b. 



But 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 presupposes the independence of the events. 

 A large part of the late Dr. Boole's Laws of Thought is 

 devoted to an attempt to overcome this difficulty and 

 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 &, Boole 

 obtains his results by an arbitrary assumption, which is 

 only the most probable, and not the only possible assump- 



? ' Philosophical Magazine,' 4th Series, vol. vii. p. 465 ; vol. viii. 

 p. 91. 



