242 H. AI'COLL ON THE GROWTH AND 



note logical statements rather than mathematical chances, 

 I could not for some time turn to any account the inde- 

 pendence of mathematics which I had thus secured for the 

 method. The notion of the mutual exclusiveness of events 

 (or statements) connected by the sign + clung to the 

 method up to a very late period ; in fact^ I was in the very 

 act of writing my first article " On Symbolical Reasoning'' 

 for the ' Educational Times/ when the needlessness of this 

 restriction occurred to me. I had written down my defi- 

 nitions of the equations ABC=i and ABC = o in the fol- 

 lowing words : — 



The equation ABC = i asserts that all the three statements are true ; the 

 equation ABC = o asserts that all the three statements are not true, i. e. that 

 at least me of the three is false ; 



and I had to consider suitable definitions of the equations 

 A.-f B + C=i and A-|-B + C = o. It was quite evident 

 that the equation A + B + C = o_, whether the statements 

 A, B, C were mutually exclusive or not, must assert that 

 all the three statements are false, and the very words used 

 in the previous definitions of ABC=i and ABC=o 

 suggested that, as a symmetrical complement of this, the 

 equation A + B + C=i should assert that all the three 

 statements are not false, i. e. that at least one of the three 

 is true. The only question to decide was whether the rule 

 of multiplication, (A + B)(C+D)=AC+AD + BC + BD, 

 would still hold good. A very little consideration showed 

 that it would ; so, though the method was correct, so far 

 as it went, on either supposition, I judged it wiser to leave 

 room for possible future development by adopting the 

 wider rather than the narrower hypothesis for its basis. 



Finding myself thus, at the end of my investigation, on 

 logical instead of mathematical ground, I naturally began 

 to study the relation in which my method stood towards 



