TRANSFORMATIONS OP A LOGICAL PROPOSITION. 



135 



of any one implies the truth or falsehood of the rest, 

 x and y signifying respectively not-X and not-Y, as in 

 De Morgan's notation : — 



iX<RY. 

 in Y<x. 

 i?- I iX<Y. 



XnY<o. 

 That is to say : — 



Every X teaches a Y. 

 Whoever does not teach any Y is a not-X. 

 Pupils of every X are contained among Y's. 

 There is no X who does not teach some Y. 



A doubly-total proposition of the above form admits of 

 the following four pairs of forms : — 



iX<i?iY. 



i r~*X<y. 

 i X< i _I ry. 

 XrY<o. 



That is to say : — 



iY<R~ l iX. 

 i r Y<x. 

 i Y< i-V -I x. 

 Yr- x X<o. 



Every X teaches every Y. Every Y learns from every 



X. 



Whoever is a not-teacher 

 of any Y is not an X. 



Every Y is a not-pupil of 

 none but not-X's. 



There is no Y who does 

 not learn from every X. 



Whoever is a not-pupil of 

 any X is not a Y. 



Every X is a not-teacher 

 of none but not-Y's. 



There is no X who does 

 not teach every Y. 



