276 PROCEEDINGS OP THE AMERICAN ACADEMY 



contrapositions are of essentially different kinds, and that the reduc- 

 tions per impossibile of the second and third figures respectively involve 

 the following formal inferences.* 



Figure 2. 



The Result follows from the Case ; 

 . • . The Negative of the Case follows from the Negative of the Result. 



Figure 3. 



The Result follows from the Rule ; 

 . • . The Rule changed in Quantity follows from the Result changed in 

 Quantity. 



But these inferences may also be expressed as follows : — 



Figure 2. 

 Whatever {S) is M is n^p-, 

 .: Whatever {S) is "°^^ is not M. 



Figure 3. 

 Any Bojf e s is whatever (P or not-P) M is ; 



.-. Some M is whatever {P or not-P) ^°7 ^ is. 



Now, the limitations in parentheses do not affect the essential nature 

 of the inferences ; and omitting them we have, 



Figure 2. 

 Any M is „ofi', 



.-. Any ""j.-^ is not M. 



Figure 3. 

 Any some 5 is 31; 



.-. Some 31 is =°'5f ^^ 

 * A formal inference is a substitution having the form of an inference. 



