26 Mr Van Hum, An Aadoiii in Symholic Logic 



Theorem 1 



If }) is an elementary proposition, ~ p is an elementary pro- 

 position, 



Deni. 



Axiom 2 gives us ' p Ap' elementaiy when j) is elementary ; 

 'pAp' is ~ p, by Definition of Negation. Hence the theorem. 



This is a proof of Mr Russell's primitive proposition *1.7 given 

 above. 



Theorem 2 



Ifj} and q are elementary ])ropositions, ' pv (j' is an elementary 

 proposition. 



Dem. 



By Theorem 1 , if p and q are elementary so also are ~ p and 

 ~ q. Therefore, by Axiom 2, ' -^ p A ~ (/ ' is elementary ; but this, 

 by Definition of Disjunction, is ' pv q'. Hence the theorem. 



This is Mr Russell's primitive proposition *1.7l quoted above. 



Theorem 3 

 The propositions p and ~ p are of opposite truth-values. 



Dem. 



Two possibilities can occur : 



1°:^ true. By Axiom S, " p A p' is false; but this by 

 Definition of Negation is ^ p; hence in this case p and ~ p are 

 opposite in truth-value. 



2° : j9 false. By Axiom 3, 'pAp' is true; but this by 

 Definition of Negation is ~ jo ; hence in this case also ^j and f^ p 

 are opposite in truth-value. Hence the theorem. 



This theorem states in precise form the information usually 

 given in text-books on logic in more or less vague statements that 

 are called ' definitions ' of negation. 



Theorem 4 

 \-. pDp. 



Dem. 



[Th. 3] h. p and <^ p of opposite 



truth-values (1) 



[(1). Ax. 3] \-. pA r^ p (2) 



[(2). Def. of Implication] h. theorem. 



This is proposition *2.08f of the Principia. 



I 0]}. cit. Vol. I. p. 105. 



