Write down a proof for $\bot\Rightarrow q$ in proposition calculus

Question

I am given the hint in the question that I will need to use the axiom $(((s\Rightarrow \bot)\Rightarrow \bot)\Rightarrow s)$.

The axioms I am using are $$(s\Rightarrow (t \Rightarrow s)) \\((s\Rightarrow(t\Rightarrow u))\Rightarrow((s\Rightarrow t)\Rightarrow(s\Rightarrow u)) \\ (((s\Rightarrow \bot)\Rightarrow \bot)\Rightarrow s)$$

In a proof every step is either an axiom or deduced by modus ponens.

I don't know what system it is called. I have three axioms called K, S, and T and each line of a proof is either a hypothesis, an axiom, or something deduced by modus ponens. Can you see what system this is?@ChrisEagle — Spook, Oct 19 '12 at 14:36
I could see it better if you actually gave the axioms. Also, you should be editing this into the question, not leaving it in the comments. — Chris Eagle, Oct 19 '12 at 14:37
@copper.hat it is part of "left to the reader to prove" in my lecture notes. — Spook, Oct 19 '12 at 14:42

score 2 · Accepted Answer · answered Oct 19 '12 at 14:49

Using your first two axioms you can prove the deduction theorem. So to prove $\vdash \bot \Rightarrow q$, it's enough to prove $\bot \vdash q$. The hint suggests using the third axiom. With that, you can show that $((q \Rightarrow \bot)\Rightarrow \bot)\vdash q$. So you're done if you can prove $\bot \vdash ((q \Rightarrow \bot)\Rightarrow \bot)$. Can you see how to do this?

Write down a proof for $\bot\Rightarrow q$ in proposition calculus

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