# How to find a proof of a formula in propositional calculus?

$1.$ $\alpha$ $\rightarrow$$(\beta \rightarrow \alpha)$ --- (Ak)

$2.$ $(\alpha \rightarrow (\beta \rightarrow \gamma)) \rightarrow ((\alpha \rightarrow \beta ) \rightarrow (\alpha \rightarrow \gamma))$ --- (AS)

$3.$ $(\lnot \beta \rightarrow \lnot \alpha) \rightarrow ((\lnot \beta \rightarrow \alpha) \rightarrow \beta)$ --- (A$\lnot$)

I'm told that I can use any formula to prove something, but if I know that $\vdash \alpha \rightarrow \alpha$ why I can't use it to prove anything?

For example : To prove $\lnot (\alpha \rightarrow \beta)\vdash \lnot \beta$

$1.$ I can write $\lnot \beta \rightarrow \lnot \beta$, or,known $\lnot \alpha \rightarrow \alpha \vdash \alpha$,I can say $(\lnot \lnot \beta \rightarrow \lnot \beta)\rightarrow \lnot \beta$.

An other example : to prove $\lnot \alpha \rightarrow \alpha \vdash \alpha$

$1.$ I could say $\alpha \rightarrow \alpha$, all the solutions I provided are not correct, but that's what I understood, can you explain what I'm missing?

• In addition to axioms (like 1...3 above), you also need one or more rules of deduction. Feb 3 '18 at 12:27
• Maybe for the first example but not for the second one. My problem is that I don't know how to create formula to use axioms or other stuff Feb 3 '18 at 12:28
• Use Ax.1 : $\beta \to (\alpha \to \beta)$ and contrapose it. Feb 3 '18 at 12:28
• For the other one, see similar post. Feb 3 '18 at 12:30
• why the first derivation is not correct? Feb 3 '18 at 14:31

• Okay thank you. I have an other question : is $\alpha \rightarrow \lnot(\beta \rightarrow \alpha)$ still axiom Ak? Feb 3 '18 at 17:41