244 H. M^COLL OX THE GROAVTH AND 



small trouble^ even though I had previously more than 

 once wondered under Avhat circumstances the symbol 

 {x -.y)' would be required. For a long time I did not 

 recognize this {x :y)' as the equivalent (in classication) of 

 "Some X is not Y/' and {oe-.y)' as the equivalent of 

 " Some X is Y." In my second communication to the 

 Mathematical Society I used the symbol v. xy to denote 

 " Some X is Y;" and it was only when I had read the 

 very just objection made by one of the referees to my 

 introduction of the arbitary and possibly non-existent 

 class V that it suddenly flashed upon me that the true 

 symbolical expression for '' Some X is Y " should be 

 {x:y'y, the denial of the implication x:y', and that the 

 true symbolical expression for '' Some X is not Y " should 

 be {x : y)', the denial of x:y. 



The next new symbol which I introduced into my 

 symbolic system was the symbol x^, to express the chance 

 of X being true on the assumption that a is true. The cir- 

 cumstances which suggested this symbol to me are curious 

 and instructive. My first idea was to use the symbol x^ 

 to denote the chance of x being true, the suffix c being 

 merely suggestive of the word chance and not denoting a 

 statement. In fact^ this was the notation which originally 

 formed the basis of my fourth paper, " On the Calculus of 

 Equivalent Statements/^ when it was first communicated 

 to the London Mathematical Society. While this paper 

 was in the hands of the referees, I was occupied with a 

 problem proposed to me by Mr. C. J. Monro, and 

 involving among other things the consideration of a 

 chance {xz)^, which I at first considered as equal to 

 X [x : z)^, being under the idea that, since x : z expressed 

 the conditional statement " If x is true z is true,''^ {x : z)^ 

 would be the proper symbol to express the chance that if 

 X is true z is true. On reflexion I discovered that this, 



