28 Jllr Van Horn, An Axiont in H[iinholic Logic 



Th. 8. Hence (Ax. 3) '~ </ A '^ ?• " is false, making ~ (~ (/ A ~ /•) 

 true (Th. 3). Hence (Ax. 3) in this case the proposition is false. 



Hence the theorem. 



Theorem 9 

 The propositions 



' <>•' p A f^ (<^ q A f^ r)', 'r^(/A~(~^:>A~ r) ', 



always have the same truth-valm. 



This follows at once from Th. 8. 



At this point I introduce Mr Russell's definition of Equivalence f 

 as it occurs in the Principia. 



Equivalence: p = q. = .pDq.qDp Df. 



Theorem 10 



h. p= <^ (^ p). 



Dem. 



We first prove h. jj D ~ ( ~ p). Two cases arise : 

 1°: p true. By Theorem 3, '^ ^ is false, ~ (~p) is true, and 

 f^ [f^ {^ py] is false. Hence 



[Ax. 3] h. I? A ~ [~ (~ jj)] (1) 



[(1). Def. Implica.] Kj9D~(~p) (2) 



2° : p false. By Th. 3, r^ p is true, ^^ {<^ p) is false, and 

 ,^ [r^ ('^i^)] is true. 



This is proposition *"4.13 of the Principia^. It is the Principle 

 of Double Negation, and asserts that any proposition is logically 

 equivalent to the denial of its negation. 



t Op. cit. Vol. I. p. 120, *4.0l. 

 % Op. cit. Vol. I. p. 122. 



