26 Mr. Murphy on a 



and " every B learns from every A," which is thus expressed 

 in our notation 



iA = i?x iB, iB = i?-'x lA 



is manifestly only one proposition in two equivalent and 

 converse forms. Its doubly total character is visible to the 

 eye as printed above, but this is not so under all its transfor- 

 mations. It may be stated in the form 



«= ir X B 



i.e. " not-As are the only not-teachers of Bs ; " but this 

 again is shown to be doubly total by writing below it the 

 equivalent form 



i.e., " not = Bs are the only not-pupils of As." 



De Morgan, in the paper already quoted, states three 

 elementary forms of proposition containing a single relative 

 term. These are, when stated in our notation and with our 

 examples : — 



iA = i?xB, iA = ^xiB, 



iA = i2x i-^B, 

 that is to say : — 

 Every A teaches a B, Every A teaches every B, 



Every A teaches none but Bs. 

 But as Prof Peirce has shown,* the symmetry of the system 

 requires a fourth form, which in our notation is thus sup- 

 plied. 



A teacher of every B is necessarily a not-teacher of 

 none but not-Bs ; and the converse is also true. This is 

 expressed by the equation 



ii?x iB = irx i-V;; 



*"0n the Algebra of Logic," by C. S. Peirce, reprinted from the 

 Avc\enc7i.n /oiivjial of Mathematics, Vol. III. 



