2IO 



Mr. J. J. Murphy on 



All A is B 

 All B is A 



A = JS'B 

 A = ^^B 



Therefore A and B are identical. 



All A is B 

 No A is B 



A = ^B 

 A = iV^B 



Therefore A does not exist. 



E.N^o, 



All B is A 

 No B is A 



A = ^^B 



A = A^B 



Therefore B does not exist. 



Nothing is both A and B, A = iVB 

 Everything is either A 



orB, A = iV^B 



Therefore A and B are 



exclusive alternatives 



All B is A 



A = J^'B 

 All not-B is A (or, 

 everything is either 

 BorA) A = iV'^B 



Therefore everything is 

 A. 



All A is B A = ^B 



All not-A is B (or, 

 Everything is either A or B) 



A = iV^B 

 Therefore everything is B. 



In the foregoing combinations, there is no middle term* 

 and no elimination. A syllogism is defined as a combination 

 of two propositions into one, with elimination of a middle 

 term, and it may be treated as a multiplication of the two 

 relations asserted by the two premises. Every two pairs 

 of factors R and 5 admit of two multiplications 



^ X ^ and Sy,R 



and when R and ^S represent any two of the eight relations 

 which we have considered, the two multiplications in no case 

 give the same result. This however is not true of all 



logical relations : — e.g. let i mean the whole and - a part, 



then 



o o 

 I X - =- X I 



o o 



