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Math Help - Problem with Fitch

  1. #1
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    Problem with Fitch

    Hi to all, please can anyone help me to understand fitch program. I understand logic part, but not at all how to implement in this program.

    Ok, if I have, (p ⇒ q) and (q ∧ p ⇒ r) and p, and I have to use the Fitch System to prove r.
    p => ~q - Premise
    ~q & p => r - Premise
    p - premise
    Goal is r?

    And also if I have ((p ⇒ q) ⇒ r), I have to use the Fitch system to prove ((p ⇒ q) ⇒ (p ⇒ r)).

    I really appreciate your help.
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  2. #2
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    Re: Problem with Fitch

    Ok, I understand the first one...
    Get ~q, then ~q and p to and.Int. to get ~q&P, then ~q&p i ~q&p=>r done Imp.Elm. and get r

    Still have problem with other...
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  3. #3
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    Re: Problem with Fitch

    Quote Originally Posted by Alea View Post
    Hi to all, please can anyone help me to understand fitch program.
    By Fitch program, do you mean some software? I am not familiar with it, but I know Fitch calculus as it is studied in mathematics.

    Quote Originally Posted by Alea View Post
    p => ~q - Premise
    ~q & p => r - Premise
    p - premise
    Goal is r?
    From p and p => ~q derive ~q. Then from ~q and p derive ~q & p. From that and ~q & p => r derive r.

    Quote Originally Posted by Alea View Post
    And also if I have ((p ⇒ q) ⇒ r), I have to use the Fitch system to prove ((p ⇒ q) ⇒ (p ⇒ r)).
    To prove ((p ⇒ q) ⇒ (p ⇒ r)), assume p ⇒ q and p. From p ⇒ q and (p ⇒ q) ⇒ r derive r.
    Last edited by emakarov; April 17th 2013 at 06:03 AM. Reason: Removed unnecessary formatting
    Thanks from Alea
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  4. #4
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    Re: Problem with Fitch

    Tnx a loooooooooooooooooooooot!

    I have few more problems if you have time and will
    _____________________________________________

    Given (p ⇒ q) and (r ⇒ s), use the Fitch System to prove (p ∨ r ⇒ q ∨ s).
    p => q - premise
    r => s - premise

    Goal - p | r => q | s

    __________________________________________________ ______
    Given (pq) and (qr), use the Fitch System to prove ((pr) ⇒ p).
    ~p => q - premise
    q => r - premise

    Goal: (~p => ~r) => p
    Last edited by Alea; April 17th 2013 at 06:37 AM. Reason: error in writing
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  5. #5
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    Re: Problem with Fitch

    Quote Originally Posted by Alea View Post
    Given (p ⇒ q) and (r ⇒ s), use the Fitch System to prove (p ∨ r ⇒ q ∨ s).
    p => q - premise
    r => s - premise

    Goal - p | r => q | s
    Code:
     1. p => q		Assumption
     2. r => s		Assumption 
     3. p | r		Assumption
     4.   p			Assumption
     5.     q		1, 4: =>E
     6.     q | s		5: |I
     7.   r			Assumption
     8.     s		2, 7: =>E
     9.     q | s		8: |I
    10.   q | s		3, 4-6, 7-9: |E
    11. p | r => q | s	3-10: =>I
    Quote Originally Posted by Alea View Post
    Given (pq) and (qr), use the Fitch System to prove ((pr) ⇒ p).
    ~p => q - premise
    q => r - premise

    Goal: (~p => ~r) => p
    This one requires double-negation elimination rule. The most one can derive without it is ~~p from the contraposition of ~p => q.

    Assume ~p => ~r and ~p. Derive ~r, q and r by =>E. Then r and ~r give a contradiction, so ~~p. From there double-negation elimination gives p.

    Rules dealing with negation may vary from one version of Fitch calculus to another.
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  6. #6
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    Re: Problem with Fitch

    I can get this at all..

    10. q | s 3, 4-6, 7-9: |E
    11. p | r => q | s 3-10: =>I

    Problem with Fitch-1.jpg

    Have any clue what I m doing wrong?
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  7. #7
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    Re: Problem with Fitch

    You should apply Or Elimination to p | r. Or Elimination has three components: a disjunction (p | r in this case) and two suberivations. The first derives q | s from an open assumption p, and the second derives the same q | s from an open assumption r. Both subderivations should be at the same nesting level, i.e., same indent. So, after deriving q | s from p, the subderivation should close and a new assumption r should be made at the same indent as p.

    I don't understand, by the way, why indent is not increased after the assumption p. E.g., a subderivation that uses the assumption p | r (starting in step 4) is indented compared to p | r, but q in step 5 is not indented compared to p. But maybe I don't know the details of the program.
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  8. #8
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    Re: Problem with Fitch

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  9. #9
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    Re: Problem with Fitch

    Thanks from emakarov
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