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  • 1 Post By emakarov

Thread: First-order prefixes

  1. #1
    Oct 2007

    First-order prefixes

    Explain the difference between the first-order prefixes:
    (a) \exists x\forall y and \forall x\exists y;
    (b) \exists x\forall y\exists z and \forall x\exists y\forall z;
    (c) \forall x\exists y\forall z\exists w and \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):
    \forall x\exists y\forall z\exists w(x+z=y+w) is true, but \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 \exists x\forall y\,A(x,y) implies \forall x\exists y\,A(x,y) and vice versa, and similarly for (b) and (c).
    Thanks from andrei
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