Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
What proof system and what axioms are you using? – Chris Eagle Oct 19 '12 at 14:33
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 – Montez 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
I presume this is homework? – copper.hat Oct 19 '12 at 14:39
@copper.hat it is part of "left to the reader to prove" in my lecture notes. – Montez Oct 19 '12 at 14:42
up vote 1 down vote accepted

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?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.