278 PROCEEDINGS OP THE AMERICAN ACADEMY 



Short Reduction. Long Reduction. 



B A B A Br 



r B ' r B A B 



AT FA AT 



The verses of Shyreswood show how Celantes and Dabitis are to be 

 reduced in the short way, and Frisesomorum in the long way. Celan- 

 tes and its long reduction are as follows : — 



Any X is not Y, Any not-X is not Z, 



Any Z is X; Any Y is not-^; 



.'. Any Y is not Z. .'. Any Y is not Z. 



" Any X is not Y" becomes, by conversion, " Any Y is not X." 

 The terra "not-X" is then introduced, being defined as that which Y 

 is when it is not X. Then " Z is X" becomes " Any not- A" is not Z" ; 

 and, the premises being transposed, the reduction is effected. 



Dabitis and its long reduction are as follows : — 



Any Z is X, Any some-^ is Y, 



Some Y is Z; Some X is some-^; 



.*. Some X is Y. .*. Some X is Y. 



" Some Y is Z" becomes, by conversion, " Some Z is Y." Then 

 the terra "sorae--^" is introduced, being defined as that Z which is 

 Zif "some Z is Y" Then "Any Z is X" becomes •" Some X is 

 some-^," and, the premises being transposed, the reduction is effected. 



Frisesomorum is,' 



Some Y is Z, 



Any X is not Y; 



.'. Some Z is not X. 



Let some-Z be that Y which is Z when some Z^ is Z; and then 

 we have, 



