30 Mr Van Horn, An Aj'iunt in Sfjtnbolic Logic 



Theorem 15 



J 



\-:.qDr.D:pvq.D.2i'v r. 



Bern. 



There are three cases to be discussed : 

 1" : li p is true, or if r is true, or if both p and r are true, 

 q being any elementary proposition. 



[Th. 8] }-: r^l. A.^i^p A '^r) (1) 



[(1). ::ii. Th. 10] h: Z. A .~(-2J A ~?-) (2) 



[(2). ~i^^ ~5j h: ~p A ~(/. A . ~(~2) A ~ r)(3) 



[(3). Th. 3. Th. 6] 



h: q A ^ r . A . <^ [^ p A '^ q . A . <^ (r^ p A '^ r)] (4) 



Taken together with the Definitions of Implication and 

 Disjunction, (4) gives the theorem in this case. 



2° : If both p and r are false, but q true. In this case ~ ^j and 

 oo r are true by Th. 3. Hence (Ax. S) ' ^ p A ~ r ' is false. The 

 proof in this case proceeds as folloAvs : 



[Th. 3] 1-: ~(-2) A ~ r) (5) 



Since q is true, '^ q is false (Th. 3). 



[Th. 6] h. - 19 A ~ 5 (6) 



[(5). (6). Th. 3. Ax. 3] V: ^[^^p A ^q. A.'^i^p A ^r)]{1) 



By Ax. S, ' q A <^ ?' ' is in this case false. 



[(7). Ax. 3] 



h: q A ~r.A.'^[~/jA <^5.A.~(~jjA ~ ?•)] (8) 



As in the previous case this result gives the theorem. 



3° : All three false. Hence ~ p and ~ ?■ true as before. In 

 this case ' ^ p A <-^ q' is false by Ax. 3. The proof in this last 

 case proceeds thus : 



[Th. 3, as in 2°] h. ~(~pA~?') (9) 



[(9). Ax. 3] h: ~p A ~ g. A . ~(~p A ~ r) (10) 



In this case q and f--' r are of opposite truth- values. 



[Ax. 3] h: f^ A~r (11) 



[(10). Th. 3. (11). Ax. 3] 



h: ^A~?'.A.'^[~pA'^^.A.'^ (~i^ ^ ~ ''')] (12) 



As in the two preceding cases, this result, together with the 

 Definitions of Implication and Disjunction, gives the theorem. 



No other cases can arise. Hence the theorem. 



This is Mr Russell's primitive proposition *1.6 given above. 

 It asserts that an alternative may be added to both premise and 



