ABSTRACT
D u r in g th e d iscu ss io n o f m o d u s ponens c a lc u li in th e la s t C h a p te r , I e s ta b lis h e d v a r io u s d e d u c ib i li t ie s h u-+v fo r p o s it iv e a n d in tu i t io n is t ic lo g ic , a n d in v ie w o f th e t ra n s la t io n s b e tw e e n H i lb e r t ty p e a n d s e q u e n tia l c a lc u li, th e s e d e d u c ib i li t ie s g iv e r is e to d e d u c ib i li t ie s u = > v in th e c o rre s p o n d in g s e q u e n tia l c a lc u li (w h ic h a c tu a lly a re m uch s im p le r to d e r iv e ) . In th e l is t o f ta u to lo g ie s 9 . ( M 1 ) - ( M 4 ) fo r m in im a l lo g ic , th e re w a s n u m b e r o f im p lic a tio n s fo r w h ic h th e converse im p lic a t io n s w e re n o t d e d u c e d . T h is w a s no a c c id e n t :
LE M M A 1 T h e m is s in g im p lic a t io n s
cannot be d e r iv e d in in tu i t io n is t ic lo g ic ( b u t can e a s ily be d e r i v e d in th e c lass ica l c a lc u li w i t h m u lt ip le s e q u e n ts ) .
