proof intuitionist logic

Question

So here is the proposition I'd like to prove:

a ∨ ¬a ⊢((a→b)→a)→a

I have tried many way to find a proof, and always end up in a mess...

for example:

a ∨ ¬a ⊢((a→b)→a)→a

a ⊢((a→b)→a)→a (elimination of disjunction)

a ∧ (a→b)⊢ a→a

(a→b)∧ a ⊢ a→a

(a→b) ⊢ a→a→a

(a→b) ⊢ True?

Anybody has an idea how to get me out of my mess?

Thank you!

It seems you're doing it in the other direction. We should give a derivation arriving to the given formula, not starting from that. — Berci, May 02 '20 at 02:11
Ohhh I did start from the wrong direction smh... I started again with what I want to prove at the bottom and I think I found a solution . Thank you for pointing the obvious! — Phil, May 02 '20 at 18:22

score 2 · Accepted Answer · answered May 02 '20 at 05:19

Suppose $a$. Then $((a\rightarrow b)\rightarrow a)\rightarrow a$.

Now suppose $\lnot a$. Then $(a\rightarrow b)$. So if $(a\rightarrow b)\rightarrow a$, by modus ponens $a$. So $((a\rightarrow b)\rightarrow a)\rightarrow a$.

In either case, we have $((a\rightarrow b)\rightarrow a)\rightarrow a$, so $a\lor \lnot a\vdash ((a\rightarrow b)\rightarrow a)\rightarrow a$.

proof intuitionist logic

1 Answers1