OF ARTS AND SCIENCES : APRIL 9, 1867. 271 



are reduced to 



No not-P is M, 



Any or some S is not-P ; 



.\ Any or some S is not M. 



The short reduction of Disamis and Bocardo is effected by intro- 

 ducing the term some-*S', defining it as that part of 8 which is or is 

 not P when some S is or is not P. We can therefore substitute for 

 the first premise, Some S is or is not P, All some-AS is or is not 

 P ; while, for the second premise, All S is M, can be contraposed into 

 " Some M is some-/S " : and thus the forms 



Some S is (or is not) P, 



Any S is M; 



.'. Some M is (or is not) P, 

 are reduced to the following : — 



Any some-AS is (or is not) P, 



Some M is some-^; 



.*. Some M is (or is not) P. 



To reduce Cesare, Festino, and Baroco in the long way, it is neces- 

 sary to introduce the terms not-P and some-*S'. Not-P is defined as 

 that class to which any M belongs which is not P. Hence for the 

 first premise of Cesare and Festino we can substitute " Any M is 

 not-P." Some-$ is defined as that class of S which is (or is not) 

 P, when some S is (or is not) P. Hence for the second premises of 

 Festino and Baroco we can first substitute "Any some-S is (or is 

 not) P"; and then, by contraposition or conversion, we obtain "Any P 

 (or not-P) is not some-S." Then, by the transposition of the prem- 

 ises, we obtain from Cesare, which is 



No M is P Any not-P is not S 



Any S is P Any M is not-P 



(.-. No S is M). .: Any M is not-5. 



