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Math Help - Simple derivation in first order predicate logic

  1. #1
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    Simple derivation in first order predicate logic

    Ok, this must be a simple derivation, but it has eluded me. I'm trying to prove that the following argument is valid in first order predicate logic: 'Any creature with a kidney has a heart. Therefore, nothing with a kidney lacks a heart.'

    Symbolised, it's: (Ax) (Kx > Hx) therefore ~ (Ex) (Kx & ~Hx)

    Where (Ax) is a universal quantifier, (Ex) an existential quantifier, > is a material conditional.

    What's the derivation!?

    Huge thanks.
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  2. #2
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    Depends on the logical calculus you are using.

    In words, the derivation is the following. Suppose (Ex) (Kx & ~Hx) is true. Take that x0 that makes it true; then K x0 & ~H x0, which gives K x0 and ~H x0 separately.

    Then substitute x0 into (Ax) (Kx > Hx) to get K x0 -> H x0. By Modus Ponens and previously derived K x0 we get H x0. But we also have ~H x0, a contradiction. Therefore, (Ex) (Kx & ~Hx) is false, i.e., ~(Ex) (Kx & ~Hx).
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