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  • 1 Post By emakarov

Math Help - First-order prefixes

  1. #1
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    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
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    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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