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I'm trying to prove the expression ¬¬A→A. I can use 3 axioms:

  • A1: a→(b→a)
  • A2: (a→(b→c))→((a→b)→(a→c))
  • A3: (¬b→¬a)→(a→b),

MP rule and deduction theorem.

What am I doing wrong?

⊢ ¬¬A → A

¬¬A ⊢A deduction th.

1.¬¬A

2.¬¬A→(¬A→¬¬A) [A1]

3.¬A→¬¬A [MP(1,2)]

4.(¬A→¬¬A)→(¬A→A) [A3]

5.¬A→A [MP(3,4)]

Evster
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  • Here's a MathJax tutorial :) – Shaun Dec 21 '19 at 19:15
  • I don’t think I can use the deduction theorem inside the proof like him – Evster Dec 21 '19 at 19:32
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    The Deduction Theorem is indeed typically not an inference rule. But, if you just take off that last line from the prloof that Matt Daly referenced, you can say that that is a proof of $\neg \neg a \vdash a$. And therefore, by the Deduction Theorem, it follows that $\vdash \neg \neg a \to a$. Also, here are a few more proofs: https://math.stackexchange.com/questions/2962525/derive-p-to-neg-neg-p-in-a-structure-with-not-and-implies – Bram28 Dec 21 '19 at 19:44
  • Thank you for the help:) – Evster Dec 21 '19 at 19:50

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