NUMERICALLY DEFINITE REASONING. 351 



minor limits of the probability are evidently q-^i —p 

 and q. 



Were the events A and B independent, vre should have 



Prob. of B = g+(i-i?) (prob. of B), 

 =_— 1_ =^. 



29. It may be truly remarked of what is given in this 

 paper, that all the results can be reached by the exertion of 

 common sense, or by ordinary mathematical calculation; and 

 I do not doubt that problems combining logical and mathe- 

 matical conditions of a more complicated character have 

 been solved, especially in the ^ Theory of Probabilities,^ by 

 those who were unconscious of using any peculiar logical 

 method ; but what I claim for my logical method and 

 notation is, that it is in no sense or way peculiar, but 

 represents truthfully and completely the natural course of 

 intelligent thought. The indirect method, first explained 

 in 1864 in my ^ Pure Logic,^ embodied in the mechanical 

 device called the Logical Abacus, explained to the Society 

 in April 1866, and further exemplified in the Logical 

 Machine lately brought before the Boyal Society, repre- 

 sents the exhaustive and necessary classification of objects 

 which the mind must make under any logical conditions. 

 Of previous systems, Boole^s mathematical method could 

 alone be said to do this ; and his method was deformed by 

 needless obscurity, and by at least one deep-seated error. 

 It has been my purpose in this paper to exemplify the way 

 in which a true and simple logical method lends its aid to 

 all such mathematical problems as involve logical con- 

 siderations. The number of such problems requiring solu- 

 tion is not great, unless, perhaps, in the theory of proba- 

 bilities ; but I believe that in the progress of science the 

 number will probably increase. And whether this be 



