Nexv System of Logical Notation. 31 



Propositions 3 and 4 are the complements of i and 2 

 respectively. Proposition 5 is obtained by the contraposition 

 of 



for, as we have seen above, the negative oi Ex E — ancestor 



oi any ancestor — is ^x \E — non-ancestor oi every ancestor ; 



so that the contra-position of the above equation gives 



\e = e y. \E. 



And Proposition 6 



\c = E y. i~^e 



is the complement of proposition 5, 



We have worked these out with De Morgan's examples, 



derived from the relation of ancestor and descendant. But 



they are true of any transitive relation whatever, such as 



before and after, and cause and effect (if we so define cause 



that a cause of the cause is a cause of the effect) ; and 



among others, of the relation of whole and part, which is 



the fundamental relation of the common logic when the 



terms are interpreted in extension ; so that if E is taken to 



mean the relation of a part to the whole, 



ExE^E, 



means that a part of a part is a part of the whole ; or, as I 



propose to express it, an enclosure of an enclosure is an 



enclosure ; and conversely 



E~^xE-' = E-\ 



or, an includent of an includent is an includent. Then 



e and e~'^ 



will mean respectively non-enclosure and non-includcnt ; 



and the expressions 



A = EB, B = E-A, 



A = ^B. B = ^-^A, 



will mean respectively 



A is (included in) B. B includes A. 



Some A is not (included in) B. B does not include all A. 



