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Constructive Proof of Material Implication

  1. Jun 3, 2010 #1
    Hi,

    I'm struggling to find a constructive proof (through natural deduction) of the material implication replacement rule (i.e., that (a => b) <=> (~a \/ b). I believe the only possible way would be through contradiction, but I can't seem to get to it. Is it even possible?

    Thx.
     
  2. jcsd
  3. Jun 4, 2010 #2
    IF by "constructive proof" in a natural deduction system you mean a proof using only intuitionistic rules, then it's not possible: the implication

    [tex]\left(p\rightarrow q\right)\rightarrow\left(\neg p \vee q\right)[/tex]

    is not intuitionistically valid, whereas the reverse one:

    [tex]\left(\neg p \vee q\right)\rightarrow\left(p\rightarrow q\right)[/tex]

    is, so this one may be proved only with intuitionistic rules.

    The first implication is only classically valid, so its proof must use a non-intuitionistic rule, like

    [tex]\Phi,\neg\alpha\vdash\bot \Rightarrow \Phi\vdash\alpha[/tex]
     
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