^fr Van Horti, An A.iiom, in Si/)iibolic Logic 29 



Theorem 11 

 H: pvp . D .p. 



Dem. 



[Ax. 3] ■ h. ~ jj and ' <^ p A f^ p' of 



opposite truth- values (1) 



[(1). Ax. 3] |-:~_p A ~^9. A . ~_p (2) 



[(2). Def. Disjunc. Implica.] I-. theorem. 



This is Mr Russell's primitive proposition *1.2 given above. 



Theorem 12 



h: q.'^.pwq. 



Dem. 



Two cases need only be treated : 



I'' : q true. Then (Th. 3) ~ q is false. Hence (Th. 6) 

 ' ~ jj A <^ q ' is true. Hence ~ (~ ^j A ~ q) is false, by Th. 3. 

 Hence 



[Ax. 3] h : (/ . A . <^ ( ~ p A ~ (/) ( 1) 



2° : q false. 



[Th. r, |-~(~pA ~g)j F. , . A . ~ ( ~ ,, A ~ ,/) (2) 



[(1). (2). Def. Disjunc. Implica.] h. theorem. 



This is Mr Russell's primitive proposition *l.o given above. 



Theorem 13 

 h: pv q .D . qy P' 



Dem. 



[Th. 7] h : ' ~ jt) A ~ g ' and ' ~ g A ~ p ' 



of the same truth- value (1) 



[(1). Th. 3. Ax. 3] h: ~ jj A ~ (/ . A . ~ ( ~ ry A ~ p) (2) 



[(2). Def Disjunc. Implica.] h: theorem. 



This is Mr Russell's primitive proposition *1.4 given above. 



Theorem 14 

 V : p y {q y r) ."^ . qy {p M r). 



Dem. 



[Th. 9] I-: '~p A ~(~(/ A ~?-)' a-nd '~(/ A ~(~|) A ~?-)' 

 of the same truth-value (1) 



[(1). Th. 3. Ax. 3] h: ~ ;j A ~ (~ (? A ~ /•) 



. A . ~ [~ q A '^ {r^ p A ~ r)] (2) 



[(2). Def. Disjunc. Implica.] h: theorem. 



This is Mr Russell's primitive proposition *1.5 given abt)ve. 



