N'ezu System of Logical Notation. 



27 



Consequently, we may write the second and third of the 

 above three forms thus : — 



iA = i?x iB, 

 -^-xi-V;, 



iA = ^x i-iB, 

 = ry. lb. 

 The fourth form, obviously, ought to be related to the third 

 as the first to the second ; so that the completed system is 

 constituted by the following four propositions, whereof two 

 are singly and two doubly total. 



A = i?xB, 



I A 



That is to say^, 

 Every A is a teacher of 

 some Bs. 



Kvery A is a not-teacher of 

 some not-Bs. 



iA = ^=iB, 



= ;' X i~^b, 

 iA = Iix i-^B, 



= rx lb. 



Every A is a teacher of ever)^ 



B, and a not-teacher of 



none but not-Bs. 

 Every A is a teacher of none 



but Bs, and a not-teacher 



of all not-Bs. 



The two forms of proposition 



iA=Sx iB, 



ma}' be called the complements of each other, or comple- 

 mentary to each other. Their equivalence is self-evident ; 

 nevertheless it is worth while to show it symbolically. 



iA = J?xiB 

 becomes by conversion 



iB = i?-ix lA, 

 which becomes by contraposition and inversion 



1-1/' = ^"^ X I A, 

 and this again by conversion 



iA = ?- X i~^^. 



