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Thread: First-order prefixes

  1. #1
    Oct 2007

    First-order prefixes

    Explain the difference between the first-order prefixes:
    (a) $\displaystyle \exists x\forall y$ and $\displaystyle \forall x\exists y$;
    (b) $\displaystyle \exists x\forall y\exists z$ and $\displaystyle \forall x\exists y\forall z$;
    (c) $\displaystyle \forall x\exists y\forall z\exists w$ and $\displaystyle \exists x\forall y\exists z\forall w$.
    First, I notice that each prefix can be obtained by negation from the other one in the pair. Also, I can show a formula such that different prefixes produce different truth values. For example, consider (c):
    $\displaystyle \forall x\exists y\forall z\exists w(x+z=y+w)$ is true, but $\displaystyle \exists x\forall y\exists z\forall w(x+z=y+w)$ is false. I ask whether I missed something.
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  2. #2
    MHF Contributor
    Oct 2009

    Re: First-order prefixes

    Quote Originally Posted by andrei View Post
    I ask whether I missed something.
    Your answer is correct. Another thing you may add is whether $\displaystyle \exists x\forall y\,A(x,y)$ implies $\displaystyle \forall x\exists y\,A(x,y)$ and vice versa, and similarly for (b) and (c).
    Thanks from andrei
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