94 OF INTERPRETATION. [CHAP. VI. 



which is interpretable into the following Proposition: Unclean 

 beasts are all which divide the hoof without chewing the cud, all 

 which chew the cud without dividing the hoof, and all which neither 

 divide the hoof nor chew the cud. 



Ex. 2. The same definition being given, required a descrip- 

 tion of beasts which do not divide the hoof. 



From the equation x = yz we have 



x 



y---, 



therefore, . z - x 



z 



and developing the second member, 



Here, according to the Rule, the term whose coefficients is , 

 must be separately equated to 0, whence we have 



whereof the first equation gives by interpretation the Proposition : 

 Beasts which do not divide the hoof consist of all unclean beasts which 

 chew the cud, and an indefinite remainder (some, none, or all) of un- 

 clean beasts which do not chew the cud. 



The second equation gives the Proposition : There are no clean 

 beasts which do not chew the cud. This is one of the independent 

 relations above referred to. We sought the direct relation of 

 " Beasts not dividing the hoof," to " Clean beasts and beasts 

 which chew the cud." It happens, however, that independently 

 of any relation to beasts not dividing the hoof, there exists, in 

 virtue of the premiss, a separate relation between clean beasts 

 and beasts which chew the cud. This relation is also necessarily 

 given by the process. 



Ex. 3. Let us take the following definition, viz. : " Respon- 

 sible beings are all rational beings who are either free to act, or 



