222 MR. JOSEPH JOHN MURPHY ON ADDITION 



that is to say, coincludent of participant is coparticipant, 

 and conversely. But the syllogisms 



Z-" X n° and n° x L~^ 



are inconclusive, because the factors, having different 

 letters, are not understood as referring to the same unex- 

 pressed middle term. 



Considered exclusively with respect to their logical form, 

 there are four classes o£ relatives. All of them have 

 representatives in this table. 



1 . Every relative of the first class is transitive, or equal 

 to its own second power — and invertible, or equal to its 

 own reciprocal. The only numerical coefficient which 

 unites these two properties is unity. To this class belong 

 all terms with zero index. 



2. Every relative of the second class is transitive, but 

 not invertible. The only numerical coefficients which unite 

 these two properties are zero and its reciprocal infinity. 

 To this class belong L and Z~^ 



3. Every relative of the third class is not transitive, but 

 is invertible. The only numerical coefficient which unites 

 these two properties is negative unity. To this class belong 

 N and ili, with their denials n and m. 



4. Every relative of the fourth class is neither transitive 

 nor invertible. These properties are united in all nume- 

 rical coefficients whatever except unity, zero, infinity, and 

 negative unity. To this class belong /and /~^ 



The denial of a relative of the first class belongs to the 

 third class (e. g. " equal •'^ is denied by " unequal ''•') . 



The denial of a relative of the second class belongs either 

 to the second class (e. g. " greater than " is denied by " no 

 greater than ") or to the fourth (e. g. " enclosure " or " A 

 is B " is denied by " indifferent '' or "■ some A is not B ") . 



