30 



Mr. Murphy tm a 



These arc very simple, and are self-evident as soon as 

 understood, yet very unfamiliar; they are like no generally 

 recognised logical forms. They are, however, easily de- 

 ducible from the property of transitivcncss, by application 

 of the principles already stated. 



It will be observed that the two sets of converse pro- 

 positions are identical in their formal properties, differing 

 only in the indices being reversed. It will consequently be 

 necessary to give the demonstrations of those of the first 

 column only. 



Proposition i is proved by combining the definition of 

 a relative term with that of transitiveness. It belongs to 

 the definition of any possible relative, that it stands in the 

 specified relation to all its correlatives. Thus any ancestor 

 E is ancestor of all his own descendants ; which is expressed 

 in our notation by 



E' = Ex lE-^xE' ; 

 combining this with 



iE^E = E, 

 we get 



iEy.E' = Ex lE-'x E\ 



that is to say every ancestor of E' is ancestor of all the 

 descendants of E' ; or, more briefly, 

 iE = ExiE-\ 



which asserts that every ancestor (of any man) is an ancestor 

 of all descendants (of that man). 



Proposition 2 is directly derived from 

 iExE = E, 

 which may be written 



iExE=i-'E, 

 whence by transposition 



iE = E-'xi-'E. 



