OF ARTS AND SCIENCES : APRIL 9, 1867. 279 



Some Y is some-T^ 



Any A is not Y; 



.'. Some some-J r is not A. 



Then let not- A' be that which any Y'\s when some Yis not X, and 

 we have, 



Some some- J" is not- A, 



which yields by conversion, 



Some not- A is some-T"; 



and we thus obtain the reduction, 



Any soine-T" is Z, 



Some not-A is some- J"; 



.•. Some not-A is Z. 



From the conclusion of this reduction, the conclusion of Frisesomo- 

 rum is justified as follows : — 



Some not-A is Z, 



Any A is not not-A; 



.*. Some Z is not A. 



Another mode of effecting the short reduction of Frisesomorum 

 is this : Let not-T" be that which any A" is when no A is Y, and we 

 have 



Some Y is Z, 



Any not-T' is not Y; 



.*. Some Z is not not- Y. 



Let home-Z be that Z which is not not- Y when some Z is not- Jj 

 and we have, 



Any some-Z is not not- 7", 

 and by conversion, 



Any not- Y is not some-Z 



