THE LOGIC OP RELATIVES. 325 



denotes all men, and, besides, all women. This signification for this sign is needed 

 for connecting the notation of logic with that of the theory of probabilities. But 

 if there is anything which is denoted by both the terms of the sum, the latter no 

 longer stands for any logical term on account of its implying that the objects de- 

 noted by one term are to be taken besides the objects denoted by the other. I or 



example, 



f+u 



means all Frenchmen besides all violinists, and, therefore, considered as a logical term, 

 implies that all French violinists are besides themselves. For this rea-on alone, in a 

 paper which is published in the Proceedings of the Academy for March 17, 3 S, '»7, 1 

 preferred to take as the regular addition of logic a non-invertible process, such that 



m-k-b 



stands for all men and black things, without any implication that the black things 

 are to be taken besides the men ; and the study of the logic of relatives has supplied 

 me with other weighty reasons for the same determination. Since the publication 

 of that paper, I have found that Mr. W. Stanley Jevons, in a tract called " Pure 

 Logic, or the Logic of Quality," had anticipated me in substituting the same opera- 

 tion for Boole's addition, although he rejects Boole's operation entirely and v ' 

 the new one with a + sign while withholding from it the name of addition* 

 plain that both the regular non-invertible addition and the invertible addition satisfy 

 the absolute conditions. But the notation has other recommendations. The con- 

 ception of taking together involved in these processes is strongly analogous to that 

 of summation, the sum of 2 and 5, for example, being the number of a collection 

 which consists of a collection of two and a collection of five. Any logical equation 

 or inequality in which no operation but addition is involved may be converted into 

 a numerical equation or inequality by substituting the numbers of the several ten. 

 for the terms themselves, - provided all the terms summed are mutually exclusive. 

 Addition beine taken in this sense, nothing is to be denoted by zero, for then 



It 



*4r<> 



X 



u * -a i a k r~ • ™d this is the definition of zero. This interpretation i 



whatever is denoted by x\ and tms is wie uc 



i. t> i a ' «n tW on account of the resemblance between the ord 

 given by Boole, and is very neat, on account. 



. n j a a + nf nothing and because we shall thus have 



conception of zero and that ot notmng, 



nary 



[0] = 0. 



book he uses the sign .|. instead of -f 



VOL. IX. 



45 



