OF ARTS AND SCIENCES : APRIL 9, 1867. 279 



Some T is some-J^ 



Any X is not Y; 



.'. Some some-Z^ is not X. 



Then let not-Xbe that which any l^is when some l^is not X, and 

 we have, 



Some some- 1^ is not-X, 



which yields by conversion, 



Some not-X is some- J[^; 



and we thus obtain the reduction, 



Any some-1^ is Z, 



Some not-X is some-I^; 



.*. Some not-X is Z. 



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

 rum is justified as follows : — 



Some not-X is Z, 



Any X is not not-X; 



.•. Some Z is not X. 



Another mode of effecting the short reduction of Frisesomorum 

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

 have 



Some 7" is Z, 



Any not-1^ is not Y; 



.'. Some Z is not not- 2^ 



Let some-^ be that Z which is not not- T' when some Z is not- 7", 

 and we have, 



Any some-^ is not not-1^ 

 and by conversion, 



Any not- 7" is not some-^. 



