of the Primitive Propositions of Logic 37 



Now, Qi I 7r| TT fulfils (a) and {^). For (a) tt being the complex 

 expression t\t\t, i s a case of the form q \ r, and (/3) we have, by 

 (c) above, tt i tt/Qi \ Qi/tt \ tt, and by (a) tt | tt | Qj. 



To obtain the strictest development of the proof we have only 

 to write Qi/tt tt for P and ir ; tt/Qi for T all through the preceding 

 argument. 



Permutation, s | p | p I s 

 Gives sv p .1) . py s hy ^ , 



Dem. : Prop. - — — — - , Id., and Rule. 

 j3 q r 



Tautology, p/p \ p/p \ pjp 



i.e. py p . -p 



Dem.: Id.^, Perm., and Rule. 

 P 



Addition, s\p\sls 



Gives s ."D .py s by — . 

 Dem. : By Perm, (twice), p \ s/s\sjs \p (a) 



By Prop, ^-y qrs ' ^ (")' ^ W. +, p \ s/s \ s 



By Perm., result. 



Return froivi Generalised Implication P \ tt/Q to P Q/Q. 



Lemma, pjp \ s/j) 



Dem. : By Perm, (twice), s/p \ p/s (a) 



By Prop. -^ — , I- {a), 



-^ ^ p q r s 



u\p\ s/p I It 

 Write p/p for ii : by Id. and Perm, (twice), result. 



t \- (a) means the use of the Rule to pass from a to b iu a sjl). 



