272 PROCEEDINGS OF THE AMERICAN ACADEMY 



And from the conclusion of this reduced form we obtain the conclu- 

 sion of Cesare by simple conversion. So Festino and its long reduc- 

 tion are 



Any 31 is not P, Any not-P is not some-^S!, 



Some *S' is P\ Any M is not-P; 



(.-, Some S is not 31.) .*. Any 31 is not some-*S ; 



and the conclusion of Festino is obtained from that of the reduced 

 form by a substitution which may be made syllogistically thus : — 



Any 31 is not some-/S', 



Some S is some-zS'; 



.*. Some S is not M. 



Baroco and its long reduction are 



Any 31 is P, Any P is not some-A^, 



Some S is not P ; Any J/ is P ; 



(.•. Some S is not 31.) .'. Any 3f is not some-zS; 



and the conclusion of Baroco is obtained from the conclusion of the 

 reduction in the same way as that of Festino. 



In order to reduce Datisi, Bocardo, and Ferison in the long way, 

 we must define Some-A^as that ;S' which is TJfwhen some iS' is 3f, and 

 Not-P as that which some (or any) S is when it is not P. Hence for 

 " Some /Sis M" we can substitute "Any some-^S is 3P'; and for "Some 

 (or any) S is not P," " Some (or any) S is not-P." " Some S is 

 not-P" may be converted simply; and "Any S is not-P" may be con- 

 traposed so as to become " Some not-P is some-A^." Then Datisi 

 and its long reduction are 



Any S is P, Any some-*S^ is 3f, 



Some S is 3f', Some P is some-*S'; 



(.-. Some if is P.) .-. Some P is 3f. 



And from the conclusion of the reduction, the conclusion of Datisi is 



