Proving a Negative

In this post, I’m going to attempt to prove a negative:

1) P → Q

2) ¬Q

3) ∴ ¬P

Wow…that was easy. I wonder if I can take it one step further, and prove a universal negative? Hang on to your hats, cause this one is tricky:

1) ∀xPx → ∀xQx

2) ¬∀xQx

3) ∴ ¬∀xPx

Quod Erat Demonstratum.

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11 responses to “Proving a Negative”

  1. Tom B says :

    “¬∀xPx” is not a ‘universal negative’, in your sense. Px could still obtain, given ¬∀xPx. To make it universally negative, you’d need ∀x¬Px, surely?

    • Tom B says :

      Or, you’d need ¬∃xPx.

    • Robert says :

      Ah, I was wondering if someone was going to bring this up. You may be right. But the motivation behind this post was the increasingly common “You can’t prove a negative!” internet atheist/folk logic claim.

      Unfortunately for me, anyone making the claim in the first place probably won’t see the subtle difference between ¬∀xPx and ∀x¬Px, and even if they could, probably wouldn’t agree on which one is what they mean by “universal negative” (and this is all assuming that they will even recognize “∀”!).

      So ultimately, I just guessed at what they meant. I should note here in the interest of completeness (no, not that kind of completeness, the regular kind! :P) that similar proofs can be constructed for ∀x¬Px, and ¬∃xPx, both simply by using modus tollens or modus ponens.

      Now I guess I should sit back and wait for someone to claim that modus tollens is invalid.

      • Tom B says :

        I dunno: there’s a pretty substantial difference between ‘not for all x, Px’ and ‘for all x, not Px.’ The first is consistent with ‘there is some x such that Px’, the second is not.

        Presumably modus ponens will do the job for ∀x¬Px. But I don’t think this would be to prove a negative, since ∀x¬Px in its logical form is a positive claim, in that it is not preceded by ¬.

      • Tom B says :

        (in neither case has a negative been proven in predicate logic).

      • Robert says :

        But I don’t think this would be to prove a negative, since ∀x¬Px in its logical form is a positive claim, in that it is not preceded by ¬.

        This is why I’m usually against translating between logic and English – “for all x, not Px” seems like a universal negative if anything is (example: for all crows, this crow is not white; or loosely, there are no white crows). But anyway, consider this:

        1) P → {insert universal negative here}
        2) P
        2) ∴ {insert universal negative here} (1,2 MP)

        Now the question of what counts as a universal negative is irrelevant. 😛

  2. thebiblereader says :

    maybe you should actually try a real example

  3. thebiblereader says :

    fyi, there is an actual atheist blog called proving the negative

    http://atheismblog.blogspot.com/

  4. Christopher says :

    I’m quite certain this post just made my night. High five!

  5. Christopher says :

    Reblogged this on Prepared for the Worst and commented:
    May the phrase “You can’t prove a negative” never be uttered again.

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