A new Symbolic Treatment of the Old Logic. 207 



group and of the N group, which are expressed by these 

 two equations respectively, make it unnecessary to use the 

 index — i in the present system. 



It is to be observed that the negative of a contrapositive 

 is identical with the contrapositive of the negative. 



Contraposition is an invertible operation. It is true of 

 any relation Ry that 



The formal properties of any two propositions contra- 

 positive to each other are the same, at least in all relations 

 treated of in this paper. 



We have now got eight relations, which are to be divided 

 into two groups of four each in four different ways. I 

 proceed to tabulate, thus arranged, in parallel columns, the 

 propositions asserting these relations, in language and in 

 my notation. 



They are divided as opposites, into the E group and the 

 N group. The relatives of the former group are uninvertiblei 

 those of the latter invertible. 



All A is B 

 A is enclosure of B 

 A = ^B 



Some A is not B 

 A is non-enclosure of B 

 A = ^B 



All not A is not B ; whence 

 All B is A 

 -A is includent of B. 

 A = ^'B 



Some not A is B ; whence 

 Some B is not A 

 A is non-includent of B 

 A = tf^B 



Nothing is both A and B 

 A and B are excludents 

 A = iV^B 



Some A is B 

 A and B are participants 

 A = ^B 



Everything is either A or B 

 A and B are alternatives 



Some things are neither A nor B 



A and B are non-alternatives 

 A = ;i'B 



