By A. R. D. Mathias, H. Rogers

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E. d into c, a proof a p r o o f of [A ~ (B ~ C)] of A -~ (B ~ C). F r o m the a b o v e it f o l l o w s that w e h a v e (A A B -* C). A ~ - - A or A ~ ((A By 1. w e may consider construction c that converting Solution: define = d(a)~ , ~here ia a d is a con- some care. e(f) is d e c i d a b l e w e for all d or c = X x y . y ( x ] . 'falsumsymbol' that I has for a p r o o f f of I s u c h that the p a i r (a,d) Required a into f, into f (where f is a p r o o f of ±). c(a~d) to r e a s o n I ~ A requires ion' i n s t e a d A a (A ~ I) ~ I.

However, semantics, 68] . (for c l o s e d The p r o o f of the c o m p l e t e n e s s intuitionistic a closer analysis of K r i p k e m o d e l s are shows to (classical) In a n u m b e r of i n s t a n c e s , that r e s u l t s intuitionistically 31 rather obtained acceptable. J~'(-~P -~ p) ~ Proof: and some l(y,P) clearly the ~ A8 <~ e quantifiers). m (P VY Because with I(a,P) = I(a,Q) = I(B,Q) I(6,P) = = t. I(y,Q) (P ~ 3 x Qx) Let D(~) I(~,P) -~ 3 x ( P P and s i d e r o@ w i t h A : {0}, = f, 3x(P that 3x(P -* Qx).

Conversion. fact' (B ~ C) into a p r o o f of A a B -~ C? p r o o f of C. Now w e h a v e to i n d i c a t e verts of this and u n i v e r s a l of A into a p r o o f of B ~ C. That gives of the p r o o f - i n t e r - (A ^ B -~ C). How to t r a n s f o r m of A ~ examples a proof implication into account. 1 . e. (a,b) a into a = c. e. d into c, a proof a p r o o f of [A ~ (B ~ C)] of A -~ (B ~ C). F r o m the a b o v e it f o l l o w s that w e h a v e (A A B -* C). A ~ - - A or A ~ ((A By 1. w e may consider construction c that converting Solution: define = d(a)~ , ~here ia a d is a con- some care.