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Math Help - derive formulas in mathematical logic

  1. #1
    Junior Member
    Joined
    Jan 2008
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    Waipahu, HI
    Posts
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    derive formulas in mathematical logic

    Hey everyone,

    Here's the problem:

    Show that if \Gamma \vdash \phi and \Delta,\phi \vdash \psi, then \Gamma,\Delta \vdash \psi

    I soooort of know how to start, it's supposed to be like,
    We already have a derivation of \psi from \Gamma, so start with:
    .
    .
    .
    (k) \phi
    (k+1)
    (k+2)
    .
    .
    .

    I have a hard time getting this naturally, because to me it feels like I should be able to assume \Delta in line (k+1) and then have \psi... but then, that's not right because it doesn't use Modus Ponens or any of the three axioms.

    Any help or hints as to how I should be thinking?
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  2. #2
    Junior Member
    Joined
    Oct 2006
    Posts
    71
    Quote Originally Posted by sfitz View Post
    Hey everyone,

    Here's the problem:

    Show that if \Gamma \vdash \phi and \Delta,\phi \vdash \psi, then \Gamma,\Delta \vdash \psi

    I soooort of know how to start, it's supposed to be like,
    We already have a derivation of \psi from \Gamma, so start with:
    .
    .
    .
    (k) \phi
    (k+1)
    (k+2)
    .
    .
    .

    I have a hard time getting this naturally, because to me it feels like I should be able to assume \Delta in line (k+1) and then have \psi... but then, that's not right because it doesn't use Modus Ponens or any of the three axioms.

    Any help or hints as to how I should be thinking?
    From what you've described (three axioms schemes, and one rule that you've identified as MP), I'd say you might be thinking:

    First, an application of the Deduction Theorem, followed by an application of MP.
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