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Thread: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

  1. #1
    CDs
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    In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B?

    I believe I need to assume A and do something Elim V, but I am really stumped as to what to do exactly. Any help would be appreciated.
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    Is a truth table not an acceptable proof for this? That's what I would use.
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    Quote Originally Posted by CDs View Post
    In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B?
    What do you mean by "corresponds to"?

    To prove (~A v B) -> (A -> B), assume ~A v B and A and do Elim V on ~A v B. If ~A, then you get get a contradiction with A, so you can derive B by ex falso. Otherwise, you already have B.

    To prove the converse implication is a little more difficult and requires double negation elimination.
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    Quote Originally Posted by CDs View Post
    In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B?

    I believe I need to assume A and do something Elim V, but I am really stumped as to what to do exactly. Any help would be appreciated.
    Fitch-Style Proof Builder
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  5. #5
    CDs
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    This is what I've done so far. It's how to get from step 5 to 6 that confuses me.

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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    Nevermind, I figured it out. And I even know how to do the reverse. Thanks a lot guys!
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    So, could you say how you derived B in step 6 for everybody's benefit?
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  8. #8
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    Fitch says that I've proved what I tried to prove, yet upon more reflection I'm worried it might due to a bug in the program (view steps 6-9). Could somebody confirm if I did this properly?

    In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B-proof2.jpg
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    Usually natural deduction is presented with the following two rules:

    $\displaystyle \frac{\bot}{A}\bot\ \mbox{Elim}$ and $\displaystyle \frac{\neg\neg A}{A}\mbox{DNE}$ (double-negation elimination, called $\displaystyle \neg$ Elim in your program) for all formulas $\displaystyle A$

    The rule $\displaystyle \bot$ Elim can be derived from DNE, which is what you did in your derivation:

    Code:
      ~B    Assumption
      ...
      _|_   _|_ Intro
    ~~B     ~ Intro
    B       DNE
    Here the assumption ~B can be vacuous, in which case you derive B just from $\displaystyle \bot$.

    The reason to keep these two rules separate is that the logic without DNE is quite interesting on its own.

    Which program do you use to create Fitch-style derivations?
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  10. #10
    CDs
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    Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

    The program is called Fitch by a CD called Language, Proof and Logic.
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