270 PROCEEDINGS OF THE AMERICAN ACADEMY 



These are the two ways of contraposing the Universal Affirmative. 



There are two ostensive reductions of each mood of the second and 

 third figures. I shall distinguish them as the short reduction and the 

 long reduction. The short reduction is effected by converting or con- 

 traposing that premise which is not the denial of the result. The long 

 reduction is effected by transposing the premises, contraposing or con- 

 verting the denial of the result, and contraposing or converting the 

 conclusion. The alteration thus produced in the order of the terms is 

 shown in the following figure : — 



Short Reduction. Long Reduction. 



N M M N M 3 



3 M 3 M N 31 



3 N 3 N N 3 



2 n 2 n 2 p 



2 P P 2 n 2 



p n p n n p 



The names bestowed by Shyreswood, or Petrus Hispanus, upon the 

 moods indicate the possibility of the short reduction in the case of 

 Cesare and Festino of the second figure, and of Datisi and Ferison 

 of the third figure ; also the possibility of the long reduction of 

 Camestres of the second figure and of Disamis of the third. 



The short reduction of Camestres and Baroco is effected by intro- 

 ducing the term not-P, and defining it as that which S is when it is 

 not P. Hence for the second premise (Any or some S is not P) we 

 substitute "Any or some S is not-P"; and as the first premise, 

 Any M is P, gives by contraposition Any not-P is not M, the 

 moods 



Any M is P, 



Any or some S is not P; 



.*. Any or some S is not M, 



