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Math Help - 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:

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

    The rule \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 \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
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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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