ADDITION ETC. OF LOGICAL RELATIVES. 201 



XXIV. On the Addition and Multiplication of Logical 

 Relatives. By Joseph John Murphy, F.G.S. Com- 

 municated by the Rev. Robert Harley, F.R.S. 



Read January 25th, 1881. 



In every science, and most of all perhaps in Logic, it is 

 desirable to begin at the beginning. In order the more 

 effectually to do so, I shall avail myself of the method 

 introduced, I believe, by De Morgan, of working in an 

 arbitrarily limited universe. 



The simplest possible case is that of a universe contain- 

 ing only two individuals, exactly alike but with different 

 names, A and X. Each of these is the logical negative of 

 the other : whatever is not A is X, and conversely. Let 

 A have also the names B, C, &c., and let X have also the 

 names Y, Z, &c. We call the names of individuals and 

 classes absolute terms. Let the relation between identical 

 names be indicated by i, and that between contrary 

 names by — i . Both of these relatives are invertible ; 

 that is to say, each is its own reciprocal, and, if used as a 

 multiplier, may be transposed to the other side of an 

 equation without change. Thus, 



ifA=iB, then B=iA, 



and 



ifA=-iX, then X=-iA. 



But, as we shall see, in another equally important respect 

 their properties differ. 



These two relatives form the following four syllogistic 

 combinations : — 



