DESCRIPTION OF A NOTATION FOR THE LOGIC OF RELAT1H 373 



means, everything which exists, exists only if there is not anything which does not 

 exist. So, * 



0x= (i 



means that there is nothing which exists it', and only if, $<nne t does not exist Tin 

 reason of this is that some x means some existing x. 



It "lightens" and " it thunders " might have been expressed by equations in the 

 forms 



Ad, Bsl, 



In that case, in order to express that if it lightens it thundejs, in the form 



(pA -<^ qt>B , 



it would only be necessary to find a function, yx, which should vanish unless x were 

 1 , and should not vanish if x were 1. Such a function is ] .r . We must therefore in- 

 terpret 1 as "that which exists if, and only if, there is — ," x x as "that which <\i> - if ' 

 and only if, there is nothing but #,'Vmd Ix as "that which exists if. aud only if, there 

 is some x." Then the equation 



1*= 1 



j 



means everything exists if, and only if, whatever x there is exists. 



Every hypothetical proposition may be put into four equivalent forms, as follow 



If X, then Y . 

 If not Y, then not X . 

 Either not X or Y . 

 Not both X and not Y . 



If the propositions X and Y are A = 1 and B = 1, these four forms are naturally 

 expressed by 



1(1 - A)-< 1(1.- B), 



4 



1(1-A) + B«1j 



ia in — B^ = 0. 



A,l(l 



For l x we may always substitute (X 1 — *>• 



Particular propositions are expressed by the consideration that they 



* * • i •*• „o Thn« sm h (1 — b) = means every horse is black 



tory of universal propositions. Inus, asn,^± u, * 



. r . -. • * KiooL-. ami as h.b = means that nt 



0* 



b) = means that some horse is not black ; and as h,b 



VOL. IX. 51 



