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I have been stuck on this proof for a while. Here's where I'm at:

Goal $(\neg B \to \neg A) \leftrightarrow (A \to B)$

l 1. $A \to B$

ll 2. $\neg B$

lll 3. $A$

lll 4. $B$ Elim 1,3

lll 5. $\neg B$ Reit

ll 6. $A \to \neg B$ Into 3-5

ll 7. $\neg A$ ????

l $\neg B \to \neg A$ Into 2-7

If someone can help it will be great. Thank you

choco_addicted
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user3727903
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  • This is the second question in propositional calculus which you tagged as first order logic. Please be aware of the difference. – Git Gud Jun 10 '14 at 23:33

2 Answers2

4

I don't get where you're going from step 6, but note the following.

In 4. you got $B$ and in the same subproof you got $\neg B$. You can infer a contradiction and proceed with $\neg$-$\text{Intro}$.

The other subproof should be similar.

Edit: See what I mean below.

enter image description here

Git Gud
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  • I am sorry but I am not quite sure what u mean. Can u please explain? Thank you – user3727903 Jun 11 '14 at 00:01
  • @user3727903 Don't apologize. See the edit. – Git Gud Jun 11 '14 at 00:19
  • lol that was so helpful. thank you but any suggestions on connecting both sentences or that's how I would end the proof? I am still new at this. thanks again – user3727903 Jun 11 '14 at 00:28
  • @user3727903 From where I left off you should start a new subproof with premise $\neg B\to \neg A$ and infer $A\to B$. When you have both subproofs, you can use $\leftrightarrow$-$\text{Intro}$ on them to conclude. – Git Gud Jun 11 '14 at 00:31
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    @user3727903 In case it wasn't clear, the new subproof I mention should be at the same level as the one I did, not inside it. – Git Gud Jun 11 '14 at 00:37
  • Thank You very much for your help. I was able to correctly answer the proof. Would it be possible to e-mail you about other proofs that I am having trouble with? – user3727903 Jun 11 '14 at 01:01
  • @user3727903 I don't want to give you my e-mail. Not only that, everybody wins if you post your questions here on MSE. If I see them, if I have the time and no helps by the time I can answer, I will help you. Between me, Mauro and Doug, it is unlikely that you'll not get an answer in this sort of thing. – Git Gud Jun 11 '14 at 01:02
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Premises:

  1. A ∨ B
  2. C → ¬B
  3. ¬C → ¬A
  4. ¬B

Logical consequence: (A ↔ C)??

david
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