28 Mr. Murphy on a 



It is to be observed that, somewhat as in the common 



logic a total proposition, such as " every A is B," contradicts 



and is contradicted by a corresponding partial proposition, 



such as " some A's are not B " ; so that one of the pair must 



be true and the other false, — so in the logic of relative terms 



the same relation of contradiction subsists between a doubly 



total proposition such as " every A is a teacher of every B " 



and a singly total proposition, such as " every A is a not- 



teacher of some B." 



The proposition 



iA = i?x iB 



admits of the following equivalent forms. It will be 

 observed that they arrange themselves in pairs of converses. 



iA = i2xiB iB = ;?-^xiA 



i~-'« = /'xB i~-'(^ = r~^A 



iA = /-x i"^^ iB = ?-"^xi"^« 



a= i^-x B h= ir~^ X A 



.iArxB = o iBr^'^ X A = o 



lAR X i-^b = o iBR~'^ X i-^a = 



All that has been yet stated is equally true, whether the 

 relation is transitive or not. A transitive relation is such a 

 one that 



if A = 7? X B and B = A^C, then A = i^C, 

 or more briefly 



RxR=R, ox R^ = R. 

 This is the algebraic expression of the common " syllogism 

 in Barbara." But it expresses nothing except the transi- 

 tiveness of the relation, and is not restricted to relations of 

 identity and co-existence. As De Morgan says in the 

 paper already quoted, " The law which governs every 

 possible case (of Syllogism) ... is this : — Any relation 

 of X to F, compounded with any relation of Y to Z, gives 

 a relation of X to Z." The following is a valid syllogism : — 

 " Abraham was the father of Isaac ; Isaac was the father of 

 Jacob ; therefore Abraham was the grandfather of Jacob." 



